1. Introduction
Vector differential and integral operators are important in various fields of mechanics and physics. In standard vector calculus, integrals and derivatives of integer order are used, and therefore this mathematical tool cannot be used to describe systems, media and fields with nonlocality in space. In this regard, it is important to generalize the vector calculus for applications to the description of non-local media and systems. Nonlocality in space means a dependence of the process, states and variable at the current point of space on the changes on other points of the space.
To describe nonlocality in space, we can use derivatives and integrals of non-integer orders. There operators form a calculus if these operators of non-integer orders satisfy nonlocal analogs of fundamental theorems of standard calculus. These theorems connect the integral and differential operators of non-integer orders. Such calculus is called fractional calculus (FC), and operators are called fractional derivatives (FD) and fractional integrals (FI) [
1,
2,
3,
4,
5,
6,
7]. The FD and FI have different nonstandard properties. For example, the standard product rule, the standard chain rule, and standard semigroup rule are violated for FD of non-integer order [
8,
9,
10]. Fractional derivatives and integrals are actively used to describe non-standard properties of systems, media and fields with nonlocality in space and time in varios subjects of mechanics and physics [
11,
12,
13,
14,
15], economics [
16,
17], and biology [
18].
Attempts to generalize some differential operators of vector calculus have been made since the beginning of the 21st century. The history of the development of fractional vector calculus can be conditionally divided into three stages.
- (1)
At the first stage, definitions of various fractional generalizations of differential vector operators (gradient, divergence, rotor, Laplace operator) were constructed. This stage began with the work of Riesz published in 1949 [
19,
20,
21] (see also [
22,
23]). The definition of fractional generalizations of the gradient was proposed in the works of Adda in 1998 [
24], which is actually based on the Sonin-Letnikov fractional derivative [
25], and the works by Tarasov in 2005 [
26,
27], which are based on the Caputo fractional derivative. The definition of a fractional curl operators was proposed by Engheta in 1998 [
28,
29] and Meerschaert, Mortensen and Wheatcraft in 2006 [
30]. The definition of a fractional divergence was proposed by Meerschaert, Mortensen and Wheatcraft in 2006 [
30].
At the first stage, the proposed definitions of fractional vector differential operators were usually not consistent with each other. Fractional generalizations of integral vector operators (fractional circulation, fractional flux and fractional volume integral) have not been proposed. Fractional generalizations of fundamental theorems of vector calculus (such as the Green’s, Stokes’ and Gauss’s theorems) have not been suggested.
- (2)
At the second stage, definitions of fractional generalizations of differential and integral vector operators, which are consistent with each other, were suggested. Fractional generalizations of fundamental theorems of vector calculus were proved. This phase began with the work of Tarasov published in 2008 [
31] and book [
13], pp. 241–264, where the power-law spatial non-locality is considered by using the Caputo fractional derivatives and Riemann-Liouville fractional integrals. The fractional generalizations of the Green’s, Stokes’ and Gauss’s theorems are formulated and proved in [
13,
31].
After 2008, other articles began to appear, in which special aspects of self-consistent formulations of the fractional vector calculus are discussed. Let us note these aspects: (a) the product rule for FVC is discussed by Bolster, Meerschaert and Sikorskii in 2012 [
32]; (b) applications of fractional gradient to the fractional advection are considered by D’Ovidio, and Garra in 2014 [
33]; (c) the discrete fractional vector calculus on lattices is proposed by Tarasov in 2014 [
34]; (d) fractional generalizations of the Helmholtz decomposition are proposed by Ortigueira, Rivero and Trujillo in 2015 [
35]; (e) the fractional vector operators are considered on convex domain by Agrawal and Xu in 2015 [
36]; (f) the fractional vector calculus that is based on the Grunwald-Letnikov derivatives is discussed by Tarasov in 2015 [
37], and then by Ortigueira and Machado in 2018 [
38]; (g) the fractional Green and Gauss formulas are considered by Cheng and Dai in 2018 [
39].
- (3)
The third stages of the development of fractional vector calculus actually began in 2021. At this stage, the fractional vector calculus as a self-consistent mathematical theory is generalized for general form of non-locality and general form of kernels of fractional vector differential and integral operators. Self-consistent mathematical theory involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. This stage began with the work of D’Elia, Gulian, Olson and Karniadakis [
40] published in 2021 and based on the generalization of the Meerschaert, Mortensen and Wheatcraft approach to FVC [
30]. We can also state that this stage began with our proposed work, which generalizes the approach to formulation of FVC that is proposed in basic paper [
31] (see also Chapter 11 in book [
13], pp. 241–264) and gave first self-consistent formulation of FVC. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we proposed to use the general fractional calculus (GFC) in the Luchko approach [
41,
42,
43]. In our paper, we proposed the following:
- (A)
Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed.
- (B)
Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed.
- (C)
The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves.
All these three parts allow us to state that in this paper, for the first time, a calculus is proposed, which is a general fractional vector calculus (General FVC).
Below we describe the approach used in this paper.
In fractional calculus, nonlocality is described by the kernel of the operators, which are fractional integrals (FI) and fractional derivatives (FD) of non-integer orders. To take into account various types of nonlocality in space, we can use operators with various types of kernels. It is important to have a general fractional calculus that allows us to describe nonlocality in the most general form. We proposed [
44,
45] to use general fractional calculus (GFC) to describe systems, media and fields with general form of nonlocality in space.
In this paper, we proposed to use the general fractional calculus (GFC) to formulate a general fractional vector calculus (General FVC). The term “general fractional calculus” (GFC) was proposed by Kochubei in work [
46] in 2011 (see also [
47,
48,
49]). In the papers [
46,
47], the general fractional derivatives (GFDs) and general fractional integral (GFIs) are defined, and the fundamental theorems of the GFC are proved. The GFC is based on the concept of kernel pairs, which was proposed by N. Y. Sonin (1849–1915) in 1884 article [
50] (see also [
51]). (“Sonin” is more correct name of the Russian sientist [
52] instead of “Sonine” that is use in French [
50]). The very important form of the GFC was proposed by Luchko in 2021 [
41,
42,
43]. In works [
41,
42], GFD and GFI of arbitrary order are suggested, and the general fundamental theorems for the GFI and GFDs are proved. Operational calculus for equations with general fractional derivatives is proposed in [
43]. The GFC is also developed and applied in physics in works [
44,
45,
53,
54,
55,
56,
57,
58,
59,
60,
61,
62,
63,
64,
65,
66,
67,
68].
In this paper, we use general fractional derivatives (GFDs), general fractional integrals (GFIs) and fundamental theorems of GFC as mathematical tools to formulate General FVC. The proposed General FVC is built on the basis of the results in the GFC obtained in Luchko’s works in 2021 [
41,
42,
43].
In this paper, the proofs are detailed. This is due to the fact that many standard rules are violated for fractional integrals and derivatives, including Leibniz’s rule, the chain rule and the semigroup rule. These rules are often used in standard vector analysis to prove theorems. For example, the Stokes’ theorem is usually proved by using the chain rule, which cannot be applied to fractional derivatives and general fractional derivatives. Moreover, vector differential operators in fractional calculus become nonlocal, which creates additional difficulties for the accurate formulation and proof of theorems. Nonlocality also leads to the possibility of defining different vector differential operators, instead of one operator in the standard vector calculus. For example, we can define Regional GF Gradient, the Surface GF Gradient and Line GF Gradient. The situation is similar for GF Divergence and GF curl operators. Moreover, fractional vector analogs of fundamental theorems are not fulfilled for all these general fractional vector operations. In the general case, the general fractional (GF) gradient theorem should be considered for the line GF Gradient, the FG Stockes theorem should be considered for the surface GF Curl operator, and the GF Green theorem should be considered for the regional GF Divergence. This is due to violation of the chain rule for general fractional derivatives. In addition, violation of the chain rule leads to the fact that operators defined in different coordinate systems (Cartesian, cylindrical and spherical) are not related to each other by coordinate transformations.
In
Section 2, we consider and proposed general fractional integrals and derivatives. In
Section 3, we consider and proposed general fractional integral and derivative for
. In
Section 4, we consider and proposed line general fractional integral (Line GFI). In
Section 5, we give formulation and proof for general fractional gradient theorem. In
Section 6, we proof general fractional Green theorem. In
Section 7, we proposed double and surface general fractional integral, and flux. In
Section 8, we give formulation and proof general fractional Stokes theorem. In
Section 9, the general fractional Gauss theorem is formulated and proved. In
Section 10, the equalities for general fractional differential vector operators are proposed. Basic interpretations of general fractional differential vector operators are described. In
Section 11, we consider and proposed General FVC for orthogonal curvilinear coordinates (OCC), which includes general fractional vector operators for the spherical and cylindrical coordinates.
4. Line General Fractional Integral (Line GFI)
4.1. Simple Line in
Let us define concept of simple line in of the -plane.
Definition 6. Let a line be described by the functionwhich is a continuously differentiable function for all , i.e., . Then the line L will be called Y-simple line on the plane. Remark 5. It is possible to weaken the condition of continuous differentiability of the function at all points of the interval , i.e., . We can consider the Y-simple lines, for whichfor finite number of points with . For example, these conditions can be used for broken lines (polygonal chains). If L is Y-simple line, then every line, which is parallel to the y-axis, intersects the line L at most once for .
Similarly, line
is called
X-simple line on the
plane, if
L can be described by the function
which is a continuously differentiable function for
,
.
Definition 7. The line L in of the plane is called simple line in of the plane, if L is X-simple and Y-simple line.
Let is X-simple line on the plane that is described by the single valued function for . If the functions has inverse function, then L is simple line on the plane. It is known that if the function has the derivative for such that (or ) for all , then there exists an inverse function for , where and .
Therefore we can formulate the following theorem.
Theorem 7. Let is Y-simple line on the plane, where and (or ) for all .
Then there is an inverse function , the line L is X-simple line with , and L is simple line.
Remark 6. In general FVC, we can consider smooth lines on that consist of lines, which are simple lines with respect to one of the axes and lines parallel to one of the axes (X, Y, Z).
4.2. Simple Line in
Let us define concepts of simple line in .
Definition 8. Let be a line that is described by the functionswhich are continuously differentiable functions for , i.e., . Then this line will be called -simple line.
If L is -simple line, then every plane, which is parallel to the -plane, intersects the line L at most once for .
Definition 9. The line L is called simple line in , if L is -, - and -simple line.
If the derivatives
and
are nonzero and do not change sign on the interval
, then there exist the inverse functions
and
. The derivatives of the functions
and
are defined by the equations
It is obvious that these derivatives of the functions are nonzero and do not change sign also.
We can also state that there exist the functions
and the derivatives of these functions that are are nonzero and do not change sign.
As a result, we proved the following theorem.
Theorem 8. The -simple line , for which the derivatives of the functions and are nonzero and do not change sign on the interval , is -simple, -simple line, and therefore simple line in .
Let L be a simple line in and the lines , and are projections of L on the -, -, planes. Then , and are simple lines in of the -, -, planes.
Remark 7. The simple line , which connects the points and , can be defined by three equivalent formswhere , , , .
Remark 8. We can also consider the simple lines, for which the function that are not differentiable at a finite number of points. These conditions allows us to use for broken lines (polygonal chains).
4.3. Problems with Definition of Line GFI of Vector Field
Let us consider the
Y-simple line
in the
-plane that is described by the equation
and the vector field
where
. Then the standard line (curve) integral of second kind can be defined by the equation
If the
X-simple line
in the
-plane is described by the equation
then the line integral can be defined by the equation
If
and
(or
) for all
. Then there is an inverse function
, and the
Y-simple line
is
X-simple line. Therefore the line integral (
49) can be represented as (
50), and we have the equality
This equality is based on the property
Equations (
49) and (
50) cannot be used to define the line general fractional integral (line GFI) since property (
52) and (
53) does not hold for general fractional integrals. In general fractional calculus, we have the inequality
that has the form
To solve this problem of definition of line GFI, we can use the fact that the line integral of second kind over simple line can be defined by the equation
Therefore the line GFI in can be defined by the following definition.
Definition 10. Let L be a simple line in of the -plane. Let the functions and belong to the function space .
Then line GFI for the line L is defined by the equation This line GFI exists, if the kernels and belong to the Luchko set .
The proposed approach to define line GFI for lines in
can be used to define line GFI for lines in
. Let
L be a simple line in
, which is defined in form (
46)–(
48), and the vector field
where
. The standard line integral of second kind over a simple line
L in
can be defined by the equation
The line GFI over a simple line
L in
can be defined by the equation
This line GFI exists, if the kernels , and belong to the Luchko set , and the functions , and belong to the function space .
Remark 9. The line integral can also be defined for wide class of lines in that are not simple lines, if the lines L can be split into finite number of simple lines in . These lines will be called piecewise simple lines.
4.4. Definition of Line GFI for Vector Field in
Let us define some conditions on the vector field. We will assume that the vector field
on the simple line
is described by the following functions that belong to the space
:
If these conditions are satisfied, then we will write .
In the case
, the line general fractional integral for the vector field
and the line
with endpoints
and
with all
is defined by the equation
For the proposed definition of a line general fractional integral, the kernels of the integral operator remain dependent on the difference of variables
and the line GFI itself is expressed through the Laplace convolution as a product in the ring
. Using the Laplace convolution, the line GFI can be written as
where
, and
where
or
with
and
. Using the property of GFI, we have
.
We can consider the variables
instead of the numbers
. Then
where
.
Remark 10. In general FVC, we can consider lines that consist of lines, which are simple lines and lines parallel to one of the axes (X, Y, Z). As an example of this type of lines, we can consider polygonal chains (broken lines).
4.5. Line General Fractional Integral for Polygonal Chains
Let us define a line GFI for the polygonal chains. A polygonal chain (broken line) is a geometric figure consisting of line segments connected in series by their endpoints. A polygonal chain L is a sequence of points , , …, that forms the successively connected line segments , , …, . The points , , …, are called the vertices of polygonal chain.
Let us consider the polygonal chain
where the vertices
have coordinates
, such that
for all
.
For polygonal chain (
57), the line GFI is defined by the equations
where
with the signum function
and
If
,
,
, then
If the opposite inequality holds, then a minus sign is put in front of the integral, and in the interval the numbers are set in ascending order. If equality ( or or ) holds, then the integral is considered equal to zero.
Let vector field
be defined by the equation
where
for all
.
Then the line GFI for the polygonal chain (broken line) is defined by Equation (
58), where
where the functions
,
,
,
,
,
are defined by the equations
if the denominators are not zero. For example,
Here
where
,
where
,
where
.
4.6. Line GFI for Piecewise Simple Lines
Similarly to the case of a broken line (the polygonal chains), we can define line GFI for line, which consists of simple lines and lines parallel to the axes.
Let us consider a line
, which can be divided into several lines
,
that are simple lines or lines parallel to one of the axes:
where the line
connects the points
, and
with
for all
. Lines of this kind will be called the piecewise simple lines.
For piecewise simple line (
61) in
, and the vector field
, the line GFI is defined by the equation
where
where the functions
,
,
,
,
,
define the lines
that are simple or parallel to one of the axes.
4.7. General Fractional Circulation for Rectangle
Let us define concepts of general fractional circulation by using the GFC. Note that these concepts for the kernels
were proposed in [
31] (see also [
13], pp. 241–264).
We will consider the piecewise simple line (
61), where
, i.e.,
L is closed line. Let
L be the piecewise simple line
where the lines
with points
(
,
,
) are simple that are described as
where the derivatives of the functions
,
are nonzero and do not change sign on the interval
foe all
.
The last requirement for derivatives of the functions
,
allows us to represent the line in the form
where
,
,
,
.
Then, we can define the general fractional circulation in the following form
If you include segments parallel to some axes, then for such segments the integrals are equal to zero.
As a result, we can formulate the following definition.
Definition 11. A general fractional circulation is a general fractional line integral of the vector field along a piecewise simple closed line L that is defined bywhere , , , , , .
Example 10. Let us consider the piecewise simple closed line in -plane, which consists of the simple lines , , , and , with the coordinates of the pointswhere The lines and are Y-simple lineswhere the functions and belong to the space , whose derivatives are nonzero and do not change sign on the interval .
The lines and are X-simple lineswhere the functions and belong to the space , whose derivatives do not change sign on the interval .
Let us assume that the functions and satisfy the condition Then general fractional circulation along the piecewise simple closed lineis defined by the equation Example 11. Let us consider the piecewise simple line (61), where , i.e., L is closed line. For example, we consider the line GFI for the rectangle on with vertices at the pointsThe sides , , , of the rectangle form the line L. For closed line , the line GFI operator is written aswhere we used that For the vector fieldthe line GFI has the form Example 12. The general fractional circulation for the line L that is rectangle with sides , , , , where the points have coordinates (73), is written as For kernels (62), the general fractional circulation (77) has the formwhich was proposed in [31] (see also [13], pp. 241–264), where the Riemann-Liuoville fractional integrals are used. For , we get the standard circulation.
5. General Fractional Gradient Theorem
5.1. General Fractional Gradient
Let us give definitions of a set of scalar fields and a general fractional gradient for .
Definition 12. Let be a scalar field that satisfies the conditions Then the set of such scalar fields will be denoted as .
Definition 13. Let be a scalar field that belongs to the set .
Then the general fractional gradient for the region is defined as This operator will be called the regional general fractional gradient (regional GF gradient).
Remark 11. The formula defining the operator can be written in compact form. If the scalar field belongs to the function space , then the general fractional gradient for the region is defined aswhere Remark 12. The general fractional gradients can be defined not only for , but also for regions , surfaces and line .
The gradient theorem is very important for the vector calculus and its generalizations, since this theorem is actually a fundamental theorem for standard vector calculus for line integral and gradient, and their generalizations.
In the following sections, we will analyze the differences between regional and line general fractional gradients to formulate general gradient theorems.
Note that the general fractional gradient for line allows us to prove the general fractional gradient theorem for a wider class of lines and .
5.2. Difficulties in Generalization of Gradient Theorem
Let
L be simple line in
that is described by the equations
and
for
. Since the line is simple, the derivatives of the functions
and
are not equal to zero and do not change sign on the interval
. By definition, the linear integral of a vector field
for the simple line
L can be given by the equation
The standard gradient theorem is proved by using this definition of the line integral and the standard chain rule.
For the vector field
, we have
Using the standard chain rule
we get
In the fractional calculus and GFC, the standard chain rule is violated.
The line GFI for the simple line of the vector field
is defined by the equation
For the vector field
, where
we have
In this case, we should be emphasized that the fundamental theorem of GFC cannot be used. This fact is based on the following inequalities
which are satisfied if the functions
,
depend on the coordinate
x. Similarly, the fundamental theorem does not hold for other variables.
As a result, for the regional GF gradient with , the general fractional theorem can be proved only if the line consists of sections parallel to the axes.
5.3. General Fractional Gradient Theorem for Regional GF Gradient
The gradient theorem can be considered as a fundamental theorem of standard calculus for line integrals and gradients.
Let us consider a line L that consists of line segments that are parallel to the axes. To prove the general gradient theorem for such a broken line (polygonal chain), it is convenient to use the concept of an elementary broken line. We can state that any continuous line L that consists only of lines parallel to the axes can be represented as a sequence of elementary lines .
An elementary broken line is a line consisting of three (no more than three) segments parallel to different axes. There are 48 such elementary lines, eight of which differ in the directions of the segments along the axis or against the axis and six different different orders.
where
is defined by (
60) and
is defined by Equation (
59).
Let us prove the following general gradient theorem for the regional GF gradient with .
Theorem 9. (General Gradient Theorem for Regional GF Gradient)
Let L be continuous line in that consists only of lines parallel to the axes can be represented as a sequence of elementary lines, for which is the initial point and is the final point.
Let be a scalar field that belongs to the set .
Then the line GFI for the vector field with satisfies the equation Proof. For simplicity, we will consider only an elementary line
, when moving along which from the initial point
to the final point
coordinate values do not decrease, and with order
. Let the line
be represented by the points
For the vector field
, for which
the line GFI for the elementary line
is defined by the equation
Let us condider the vector field
where
, and
Then line GFI for the elementary line
is given as
Using the fundamental theorem fo GFC, we obtain
Equations for other elementary lines are proved in a similar way.
Using that any continuous line L that consists only of lines parallel to the axes can be represented as a sequence of elementary lines , we have that general gradient theorem holds. □
Corollary 2. The line GFI of the vector field is independent on the path, which is described by the lines L that consist only of line segments parallel to the axes, if the vector field can be represented as the regional GF gradient of a function .
5.4. Line General Fractional Gradient in
The general fractional gradients can be defined not only for the region . Using the fact that GFD is integro-differential operator, we can define general fractional gradients for line and .
Let us define a general fractional gradient for a simple line on the
-plane. In this definition of line general fractional gradient (line GF Gradienr), we can use the fact that GFD can be represented as a sequential action of a first-order derivative and a general fractional integral:
The general fractional vector operators can also be defined as a sequential action of first-order derivatives and general fractional integrals.
Definition 14. Let L be a simple line in of the -plane, and which means Then the line general fractional gradient for the line L is defined by the equationwhere the kernels and belong to the Luchko set .
We can use the definitions of the line GFI and line GF Gradient to prove the following theorem.
Theorem 10. (General Fractional Gradient Theorem for line GF Gradient)
Let L be a simple line in of the -plane, which connects the points and , and the scalar field belongs to the set .
Proof. Using that the line GFI of a vector field
for the simple line
of the
-plane is defined by the equation
we can consider the line GFI of the vector field that is defined by the line general fractional gradient
Therefore, we can get the line GFI of the line FG Gradient
Then, we can use the fact that the kernels
and
belong to the Luchko set
. In this case, we have the property
Similarly, we obtain the equation
Therefore, we get
where we use the standard gradient theorem.
As a result, we proved the general fractional gradient theorem for the line GF Gradient with the simple line in . □
5.5. Line General Fractional Gradient in
The proposed approach to define line GF Gradien for lines in
can be used to define line GF Gradient for lines in
. Let
L be a simple line in
, which is defined in form (
46)–(
48), and the vector field
belongs to the sset
. Then, the line GFI over a simple line
L in
can be defined by Equation (
56). The standard line integral of second kind over a simple line
L in
can be defined by the Equation (
55).
Definition 15. Let L be a simple line in , which is defined in form (46)–(48), and a scalar field satisfies the conditions Then the set of such scalar fields will be denoted as .
Definition 16. Let L be a simple line in , which is defined in form (46)–(48), and a scalar field belongs to the set .
Then the line general fractional gradient for the line is defined by the equationwhere the pairs of the kernels , and belong to the Luchko set .
Let us prove the general fractional gradient theorem for line GF Gradient with simple lines.
Theorem 11. (General Fractional Gradient Theorem for Line GF Gradient)
Let L be a simple line in , which is defined in form (46)–(48), and connects the points and .
Let be a scalar filed that belongs to the set .
Proof. Using that the line GFI of a vector field
for the simple line
is defined by the equation
we can consider the line GFI of the vector field that is defined by the line general fractional gradient
Therefore, we can get the line GFI of the line FG Gradient
Then, we can use the fact that the kernels
,
and
belong to the Luchko set
. In this case, we have the property
Similarly, we obtain the equations
Therefore, we get
where we use the standard gradient theorem.
As a result, we proved the general fractional gradient theorem for the line GF Gradient with the simple line in . □
Corollary 3. The line GFI of the vector field is independent on the path, which is described by the lines L that consist of simple lines, if the vector field can be represented as the line GF Gradient of a function .
Remark 13. The general fractional gradient theorem for line GF Gradient can be proves for the -simple line, which can be considered as a union of -simple lines and -simple lines.
8. General Fractional Stokes Theorem
The Stokes theorem connects the surface integral with the line integral. The Stokes theorem generalizes Green theorem from to . If the surface is a flat region lying in the plane, then the Stokes equation gives the Green equation.
8.1. Simple Domain on -Plane
We recall the definitions of simple domains.
Definition 24. The closed domain on the -plane will be called Y-simple domain, if can be represented in the formwhere , are continuous functions for .
Similarly, the closed domain
on the
-plane is called
X-simple domain, if
can be represented as
where
,
are continuous functions for
.
Definition 25. If is X-simple and Y-simple domain, then is called the simple domain on the -plane.
The simple domain on the -plane can be represented in the formwhere and , and , , , .
The boundary of a
Y-simple closed domain
in
can be represented as a closed line
consisting of the lines
To simplify the proofs, we can sometimes use the case, when there are no straight lines , that is, when and .
Similarly, we obtain a representation of the boundary for a X-simple domian .
8.2. Simple Surface
Let us give a definition of Z-simple surface.
Definition 26. Let S be a smooth oriented surface in that the surface S is described by the equationwhere the function is continuous in the closed domain (), which is a projection of the surface S onto the -plane. We will assume that is bounded by an oriented closed smooth line . The boundary of the domain is a projection of the line onto the -plane.
Then the surface S will be called the Z-simple surface, if the domain is simple on the -plane.
Definition 27. The surface S will be called the simple surface in , if S is simple with respect to the X, Y and Z axes.
The simple surface
S can be described by the continuous functions
where the domains
,
,
are simple domains in
, which are projections of the surface
S onto the
,
,
planes.
Remark 19. We can consider the Z-simple surfaces S in that can be described as a union of X-simple surfaces, as well as a union of Y-simple regions.
8.3. Vector Field on Surface
Let us define the properties of the scalar and vector fields in simple domains in .
Definition 28. Let be a simple closed domain on the -plane that can be described in the formswhere and . Let be a scalar field, which is defined in the simple domain on the -plane such that the following conditions are satisfied Then this property of the field will be denoted as .
Let us consider the vector fieldwhich is defined in the simple domain . The fact that the field satisfies the conditionswill be denoted as .
Let us define the properties of the scalar and vector fields on simple surfaces in .
Definition 29. Let S be a simple surface in that is described by Equations (136)–(138), and , , are simple domains, which are projections of the surface S onto the , , planes. Let us consider the field that is defined on the simple surface . The fact that the field satisfies the conditionswill be denoted as . Definition 30. Let S be a simple surface in that is described by Equations (136)–(138), and , , are projections of the surface S onto the , , planes.
Let us consider the vector fieldthat is defined on the simple surface .
The fact that the field satisfies the conditionswill be denoted as .
8.4. General Fractional Vector Integrals over Surface
Let us consider a smooth oriented surface S in , which is bounded by an oriented closed smooth line . We will assume that S is the simple surface in , and , , are simple domains, which are projections of the surface S onto the , , planes.
Definition 31. The surface GFI opereator over the simple surface is defined by the equationwhere the vectors , , are the normal vector to surfaces , , that are related with orientation of the closed lines , , bounding its. We can use the vectors , , instead of , , , if all orientation of the closed lines , , is positive.
For the simple domain
, the vector operator
is defined by the equation
where
for
, and
for
. The minus sign in front of the last expression is due to the relationship between the normal vector of the surface
and the orientation of the closed contours
.
Similarly, we obtain expressions for the simple domains , .
Definition 32. The surface GFI of the vector field over the simple surface is defined by the equationif , , .
Definition 33. Let be a Z-simple surface that surface can be represented as the union of X-simple surfaces and Y-simple surfaces such thatand , are projections of and on the and planes.
The surface GFI of the vector field over this surface is defined by the equationif , , .
8.5. General Fractional Curl Operators
8.5.1. Regional General Fractional Curl Operator
Let us give the definition of general fractional curl operator for .
Definition 34. Let be a vector field that belongs to the set .
Then the regional general fractional curl (Regional GF Curl) for the region is defined aswherewith .
Remark 20. If the vector fieldbelongs to the function space , then the regional general fractional curl operator for the region can be written in the compact formwhereand is Levi-Civita symbol, i.e., it is 1, if is an even permutation of , 0 if any index is repeated, and if it is an odd permutation, and 0 if any index is repeated.
Remark 21. The general fractional curl operator can be defined not only for , but also for regions , surfaces and line . Note that the surface general fractional curl operator is used in the general fractional Stokes theorem to be given and proved in the following sections.
Definition 35. Let belong to the space .
Then the regional general fractional curl (the regional GF Curl) for the regionis defined aswherewith .
8.5.2. Surface General Fractional Curl Operator
We have given the definition of the regional GF Curl operator. Let us now define the surface GF operator. In this definition we will use the notations
Let us give a definition of the set of vector fields that are used to define the surface GF Curl operator (compare with Definition 21 of the set that is used in definition of the surface GFI).
Definition 36. Let S be a piecewise simple surface ().
Let the vector fieldon the surface S satisfy the conditionsfor all , , .
The set of such vector fields on piecewise simple surface S will be denoted by .
Then the surface general fractional curl is defined in the following form. The piecewise simple surface is defined in Definition 20.
Definition 37. Let S be a piecewise simple surface () and a vector fields on this surface S belongs to the set .
Then the surface general fractional curl (surface GF Curl) of the vector field for the piecewise simple surface S is defined aswherewith .
8.6. Difficulties in Generalization of Stokes Theorem
In the standard Stokes theorem, the following equality should be satisfied
Equation (
140) can be regarded as the Stokes formula for the vector field
.
If
is
Z-simple and
Y-simple surface that is described by the equation
for
, and the equation
for
, then
where
and
are the projections of the surface
S on the
and
-planes.
The values of the function
on the line
are equal to the values of the function
on the line
, which is the projection of the line
L onto the
-plane
where
and
.
Therefore, equality (
140) means that
Equality (
140) is based on the standard chain rule
For the fractional derivatives and GFD, the standard chain is violated [
3], pp. 97–98.
For the general fractional Stokes theorem, we should have the equality
which is a fractional analog of Equation (
141), where
S is the
Z-simple and
Y-simple surface.
Identity (
142) does not hold in the general case, due to the inequalities
and
We can state that we have the equality only if
Z-simple surface
S is described by equation
for all
. Then the equalities have the form
and
A similar situation for Y-simple surface that is described by the equation with .
As a result, for the regional GF curl with , the general fractional theorem can be proved only if the surface consists of faces parallel to the , , planes.
8.7. General Fractional Stokes Theorem for Box without Bottom
Let us consider a box-shaped surface
S without a bottom, i.e., a parallelepiped surface without a bottom face (base). The parallelepiped can be described as
The vertices of the parallelepiped have the coordinates
The boundary of this parallelepiped is
where the surface
is a bottom face of the parallelepiped region, the surface
S consists of the following faces
where
without the face
.
The closed line
, which is the boundary of the surface
S in the form of rectangle
, consists of the segments
The surface GFI operator is desribed as
To calculate the expression
, we should consider the following surface GFIs
Theorem 15. (General fractional Stokes theorem for parallelepiped surface without bottom) Let S is a smooth oriented surface (144) in , which is bounded by an oriented closed smooth line that is given by (148), and , , are domains (145)–(147), which are projections of the surface S onto the , , planes. Then, for the vector field , we have the equationwhich is the general fractional Stokes equation.
For the surface S in the form of the parallelepiped without a bottom face, the general fractional Stokes equation has the form Proof. (YZ) Let us consider the GFI operator
. The surface
consists of two
X-simple surfaces
and
. Therefore, we get
where we took into account the direction of the normals to the outer surface with respect to the direction of the basis vector
.
(YZ1) For the first
X-simple surface
.
(YZ2) For the second
X-simple surface
.
(XZ) Let us consider the GFI operator
. The surface
consists of two
Y-simple surfaces
and
. Therefore, we get
Here we took into account the direction of the normals to the outer surface with respect to the direction of the basis vector . The minus in front of the operator is due to the fact that the normal to the surface is directed in the opposite direction with respect to the vector .
(XZ1) For the first
Y-simple surface
.
(XZ2) For the second
Y-simple surface
.
(XY) Let us consider the GFI operator
. The surface
consists of one
Z-simple surface
.
(XYZ) As a result, we get
Bringing down similar terms, we obtain
where
is the closed line in the
-plane, which is the boundary
of the surface
S. □
8.8. General Fractional Stokes Theorem for Surface GF Curl
Let us prove the general fractional Stokes theorem for surface general fractional gradient, where surface consist of simple surfaces or surfaces parallel to the coordinate planes.
Theorem 16. (General Fractional Stokes Theorem for Surface GF Curl)
Let be a simple surface (or a piecewise simple surface ), and the vector field belongs to the set .
Then, the equationholds, where is the surface GF Curl. Proof. Let us prove the general Stokes theorem for the vector field . The general Stokes equations for the vector fields and are proved similarly.
Let be a Z-simple and Y-simple surface that is described by the equation for , and the equation for .
The values of the function
on the line
are equal to the values of the function
on the line
, which is a projection of the line
L onto the
-plane,
where
and
.
Let us assume that
consists of two
Y-simple lines
and
that are described by equations
and
, where
. Then
Using the fundamental theorem of general fractional calculus, expression (
149) can be written as
Then using the definition of the general fractional derivative and the standard chain rule for the first-order derivative, we get
Let us use the property
for the expression of term (
152) in the form
Then we can use the equation, which is used in the standard Stokes theorem,
, and the fundamental theorem of GFC in the form
Using (
154), expression (
152) can be written in the form
Therefore, for two terms (
151) and (
152), we obtain the equations
for the vector field
.
As a result, we proved the general Stokes theorem for the surface FG Curl and the vector field
.
Equations for the remaining components of the vector field and are proved similarly. □
9. General Fractional Gauss Theorem
9.1. Definition of Triple GFI by Iterated GFI
Let us define concept of the Z-simple region W in .
Definition 38. Let W be region in that is bounded above and below by smooth surfaces , and a lateral surface , whose generatrices are parallel to the Z-axis. Let surfaces , be described by the equationswhere the functions are continuous in the closed domain that is a projection of the region W onto the -plane, and for all .
Then, the region W will be called the Z-simple region (simple area along the Z-axis). The region W is called simple, if W is simple along three axes (X, Y, Z). If W can be divided into a finite number of such regions with respect to all three axes, then W will be called piecewise simple region in .
Definition 39. Let be Z-simple domain that is that is bounded above and below by smooth surfaces , described by Equation (
158).
Let scalar field be satisfy the condition Then the triple general fractional integral (triple GFI) is defined by the equationwhere is double GFI.
Definition 40. Let be Z-simple domain that is that is bounded above and below by smooth surfaces , described by Equation (158). Let be projection of W on the -plane such that is Y-simple region in -plane that is bounded by the lines and , where and are continuous functions on the interval , and for all .
Let scalar fiels satisfy condition (159) and Then the triple general fractional integral (triple GFI) is defined in the form A volume general fractional integral (volume GFI) of a scalar field is a triple general fractional integral for the Z-simple region .
Example 18. Using the parallelepipedthe volume general fractional integral can be written as For the kernels (62), the general fractional flux (161) has the form For , Equation (162) is given aswhich is the standard volume integral for the function .
9.2. General Fractional Divergence
In this section, we give the definition of general fractional divergence for .
Let us define sets of vector fields that will be used in the definition of the regional general fractional divergence.
Definition 41. Let be a vector field that satisfies the conditions Then the set of such vector fields will be denoted as .
We can also consider vector fields that belong to the function space .
Definition 42. Let be a vector field that satisfies the conditions Then the set of such vector fields will be denoted as .
In other words, the condition means that all general fractional derivatives of all components of the vector field with respect to all coordinated belong to space .
Let us define the general fractional divergence.
Definition 43. Let be a vector field that belongs to the set or .
Then the general fractional divergence for the region is defined as Remark 22. The formula defining the operator can be written in compact form. If the vector fieldbelongs to the function space , then the general fractional divergence for the region is defined aswhere Remark 23. The general fractional divergence can be defined not only for , but also for regions , surfaces and line . Note that the general fractional divergence for regions is used in the general fractional Gausss theorem to be given and proved in the following sections.
Definition 44. Let be a vector field that belongs to the set , and the region W be defined in the form Then the general fractional divergence for the region W is defined as 9.3. General Fractional Gauss Theorem for Z-Simple Region
The standard Gauss theorem (the Gauss-Ostrogradsky theorem) relates the flux of a vector field through a closed surface to the divergence of the field in the region enclosed. The Gauss theorem states that the surface integral of a vector field over a closed surface, which is the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.
Let us define a set of vector field, for which general fractional Gauss theorem will be formulated.
Definition 45. Let W in be Z-simple region such that W is a piecewise Y-simple and X-simple regionwhere are the X-simple regions that is described by , for , and are the Y-simple regions that are described by the functions , for .
Let vector field satisfy the conditions Then the set of such vector fields will be denoted as . Then the set of such vector fields will be denoted as .
Theorem 17. (General fractional Gauss theorem for Z-simple region)
Let W in be Z-simple region such that W is a piecewise Y-simple and X-simple region. Let W is bounded above and below by smooth surfaces , , which is described by Equation (158), and a lateral surface , whose generatrices are parallel to the Z-axis. Let the vector field belongs to the sets and .
Then the general fractional Gauss equation has the fromthat can be written as Proof. Let us consider the triple GFI
for the function
where
for each point
.
Using the second fundamental theorem of GFC, we get
If we assume that
then we get
Therefore, the triple GFI can be represented through the surface GFIs in the form
where the surface GFIs are represented by the definition in the form
Then, taking into account that the surface GFI oves the surface
is equati to zero
we obtain
where
is closed surface that contains the region
W inside the surface
.
If
W is piecewise
Y-simple and
X-simple region, such that
where
are the
X-simple regions, and
are the
Y-simple regions. Then, we have
where
and
are projections of the region
W into
and
planes,
are projections of the
X-simple region
, and
are projections of the
Y-simple region
.
Then the following equations are proved similarly
Using that general fractional divercence is defined as
we can get the general fractional Gauss equation.
As a result, we derive the equation
that can be written as
□
Remark 24. The general fractional Gauss theorem is proved similarly for region that can be represented as unions of the Z-simple regions , which are piecewise Y-simple and X-simple regions in .
9.4. General Fractional Gauss Theorem for Parallelepiped
The standard Gauss theorem (the Gauss-Ostrogradsky theorem) states the folllwing. Let
W be a region in
with boundary
. Then the volume integral of the divergence of vector field
over
W and the surface integral of
over the boundary
are related by
For the parallelepiped region, a general fractional Gauss theorem can be formulated in the following form.
Theorem 18. (Fractional Gauss’s Theorem for a Parallelepiped)
Let , , belong to the function space , and the region has the form of the parallelepiped If the boundary of W be a closed surface , then Proof. For Cartesian coordinates, we have the vector field
, and the GFI operators
Then
and
where
,
,
are projections of
into
,
,
planes.
If
W is parallelepiped (
165), then GFI operators (
167) are
and
Using the fundanental theorem of GFC, we can realize the following transformations
This ends the proof of the fractional Gauss’s formula for parallelepiped region. □
11. General FVC for Orthogonal Curvilinear Coordinates
11.1. Orthogonal Curvilinear Coordinates (OCC)
Curvilinear system of coordinates, or curvilinear coordinates, is a coordinate system in the Euclidean space. In standard vector calculus. the curvilinear coordinates are usually used on a plane () and in space (). For such systems, the coordinate lines may be curved.
In Euclidean space, the use of orthogonal curvilinear coordinates (OCC) is of particular importance, since the formulas related look simpler in orthogonal coordinates than in the general case. The orthogonality can simplify the calculations. The well-known examples of such curvilinear coordinate systems in three-dimensional Euclidean space are cylindrical and spherical coordinates.
The curvilinear coordinates may be derived from a set of Cartesian coordinates by using nonlinear coordinate transformations. It should be emphasized that the violation of the standard chain rule leads to the fact that general fractional vector operators defined in different coordinate systems (Cartesian, cylindrical and spherical) are not related to each other by coordinate transformations. Due to this, it is impossible to obtain general fractional vector integral and differential operators in spherical and cylindrical coordinates by using the coordinate transformation. Therefore, the definitions of general fractional vector operators should be formulated separately.
The specific form of standard vector differential operators may differ, but these forms will be equivalent due to the Leibniz (product) rule. For fractional and generalized fractional calculus, the standard Leibniz rule does not hold. Due to this, such forms of notation cannot be equivalent. In this case, the specific form of the cylindrical and spherical general fractional operators must be such that the theorems of Green, Stock and Gauss hold.
It is known that the expressions for the gradient, divergence, curl and line, surface and volume integrals can be directly expressed. For orthogonal curvilinear coordinates (OCC), these integral and differential operators of the vector calculus can be expressed through the functions:
The positive values , which depend on a point in , are called the Lame coefficients or scale factors.
For the cylindrical coordinates
,
,
, the Lame coefficients are
where
,
,
.
For the spherical coordinates
,
,
, the Lame coefficients are
where
,
,
.
11.2. General Fractional Vector Differential Operators in OCC
In this section, we give only equations that will be used in definitions of the general fractional gradient, divergence and curl operators in OCC, and their examples for spherical and cylindrical coordinates. Complete definitions of these operators with function sets for which these operators are defined will be given in the following sections. In this section, formulas will be given only for regional GF vector differential operators. Definitions of line and surface general fractional gradient, divergence and curl operators in OCC will be given in the following sections.
11.2.1. General Fractional Grad, Div, Curl in OCC
The regional general fractional gradient for orthogonal curvilinear coordinates (the GF Gradient in OCC) is expressed in the form
where
where
, and
,
,
, and the function
belongs to the set
.
The regional general fractional divergence in orthogonal curvilinear coordinates (the GF Divergence in OCC) is expressed in the form
where
if
belong to the space
.
The regional general fractional curl in orthogonal curvilinear coordinates (the GF Curl in OCC) is expressed by the eqations
where
11.2.2. General Fractional Grad, Div, Curl in Spherical Coorditates
The Cartesian coordinates
and spherical coordinates
are connected by the equations
where
is the length of the radial vector connecting the origin to the point
,
is the polar angle,
is the azimuthal angle. The basic vectors of these coordinate systems
Let us emphasize that the violation of the standard chain rule leads to the fact that general fractional vector operators defined in Cartesian, and spherical coordinate are not related to each other by coordinate transformations. Because of this, it is impossible to obtain the fractional vector integral and differential operators in spherical coordinated by using coordinate transformation (
191). Therefore, the definitions of general fractional vector operators should be formulated separately.
Let us define the regions
We will consider the vector field
that belongs to
.
The general fractional gradient, divergence and curl operators in spherical coordinates are defined in the following form. For simplicity, we only present the definitions of regional operators. Linear and surface general fractional operators in spherical coordinates are defined similarly.
Definition 46. If , then the regional general fractional gradient in spherical coordinates for the region is defined aswhere Definition 47. If , then the regional general fractional divergence in spherical coordinates for the region is defined aswhere Let us note the violations of the standard product (Leibniz) rule. For example, we have the inequality
Definition 48. If , then the regional general fractional curl operator in spherical coordinates for the region is defined aswhere 11.2.3. General Fractional Grad, Div, Curl in Cylindrical Coorditates
The Cartesian and cylindrical coordinates
where
,
, and
. The basic vectors of these coordinate systems
Note that it is impossible to obtain the fractional vector integral and differential operators in cylindrical coordinated by using coordinate transformation (
192). Therefore, the definitions of general fractional vector operators should be formulated separately.
Let us define the regions
Let us consider the vector field
that belongs to
The general fractional gradient, divergence and curl operators in cylindrical coordinates are defined in the following form. For simplicity, we give definitions of regional operators. Linear and surface general fractional operators in cylindrical coordinates are defined similarly.
Definition 49. If , then the regional general fractional gradient in cylindrical coordinates for the region is defined aswhere Definition 50. If , then the regional general fractional divergence in cylindrical coordinates for the region is defined aswhere We should note that
since the standard Leibniz rule does not hold for fractional derivatives of non-integer order and for general fractional derivatives.
Definition 51. If , then the regional general fractional curl operator in cylindrical coordinates for the region is defined aswhere 11.3. General Fractional Integral Operators in OCC
11.3.1. GFI in OCC
The following GGI operators can be used to define the line, surface and volume general fractional integrals in orthogonal curvilinear coordinates (the GF integrals in OCC).
Definition 52. Let be function in that satisfies the conditionsfor all .
Then the set of such functions is denoted by .
Let function
belong to the set
. For the orthogonal curvilinear coordinate
(
), the GFI operators are defined as
and
where notations (
193) means the following
11.3.2. GFI in Spherical Coordinates
To define the line spherical GFI, we can use the following GFI operators.
- (1)
- (2)
- (3)
The azimuthal GFI operator
We can define the operators on the positive intervals
for
. For
,
, we have
Integrals with respect to other variables are defined in a similar way.
and
For intervals
,
, and
, we define
11.3.3. GFI in Cylindrical Coordinates
Let us define the cylindrical GFI, we can use the following GFI operators.
- (1)
If
, then
and
where
, and
.
- (2)
If
, then
and
if
and
- (3)
If
, then
and
if
and
11.4. General Fractional Operators in Curvilinear Coordinates
11.4.1. Definition of Line GFI for Vector Field in OCC
Let us define the line GFI in orthogonal curvilinear coordinates (OCC) in of the -plane.
Definition 53. Let L be a simple line in of the -plane. Let the functionsbelong to the set .
Then line GFI for the line L is defined by the equation Line GFI (194) exists, if the kernel pairs and belong to the Luchko set .
Let us define the line GFI in orthogonal curvilinear coordinates (OCC) in
of the
-space. To give this definition, we will describe conditions on the vector field
. We will assume that the vector field
on the simple line
is described by the following functions that belong to the space
:
If these conditions are satisfied, then we will write .
In the case
, the line general fractional integral for the vector field
and the line
with endpoints
and
with all
is defined by the equation
Using the Laplace convolution, the line GFI can be written as
where
, and
where
or
with
and
.
11.4.2. Line GFI for Piecewise Simple Lines in OCC
Let us consider a line
, which can be divided into several lines
,
that are simple lines or lines parallel to one of the axes:
where the line
connects the points
, and
with
for all
. Lines of this kind will be called the piecewise simple lines.
For piecewise simple line (
196) in OCC of
, and the vector field
, the line GFI is defined by the equation
where
where the functions
,
,
,
,
,
define the lines
that are simple or parallel to one of the axes of OCC.
11.4.3. Regional GF Gradient in OCC
Let us give definitions of a set of scalar fields and a general fractional gradient in OCC for .
Definition 54. Let be a scalar field that satisfies the conditions Then the set of such scalar fields will be denoted as .
Definition 55. Let be a scalar field that belongs to the set .
Then the general fractional gradient in OCC for the region is defined as This operator will be called the regional general fractional gradient (regional GF gradient) in OCC.
11.4.4. Line GF Gradient in OCC for
The general fractional gradients (Line GF Gradient) in OCC can be defined not only for the region . Using the fact that GFD is integro-differential operator, we can define GF Gradients for line and . The general fractional vector operators can be defined as a sequential action of first-order derivatives and general fractional integrals.
Definition 56. Let L be a simple line in of the -plane, and that means Then, the line general fractional gradient in OCC (Line GF Gradient in OCC) for the line L is defined by the equationwhere the kernels and belong to the Luchko set .
We can use the definitions of the line GFI and line GF Gradient in OCC to prove the following theorem.
Theorem 19. (General Fractional Gradient Theorem for line GF Gradient for )
Let L be a simple line in of the -plane, which connects the points and , and the scalar field belongs to the set .
Then, the equalityholds, where and .
11.4.5. Theorem for Line GF Gradient in OCC of
In the theorem we will use the following definitions.
Definition 57. Let be a line in OCC that is described by the functionswhich are continuously differentiable functions for , i.e., . Then this line will be called -simple line. The line L is called simple line in OCC of , if L is -, - and -simple line.
Let us consider a simple line
in OCC, which connects the points
and
, and can be described by following equivalent forms
where
,
,
,
.
Definition 58. Let L be a simple line in , which is defined in form (198)–(200), and a scalar field satisfies the conditions Then the set of such scalar fields will be denoted as .
Definition 59. Let L be a simple line in , which is defined in form (198)–(200), and a scalar field belongs to the set .
Then the line general fractional gradient for the line is defined by the equationwhere the pairs of the kernels , belong to the Luchko set .
For OCC, the general fractional gradient theorem for the line GF Gradient is fomulated for simple lines in the form.
Theorem 20. (General Fractional Gradient Theorem for OCC and Line GF Gradient)
Let L be a simple line in OCC for , which is described in form (198)–(200), and connects the points and .
Let be a scalar filed that belongs to the set .
Proof. Using that the line GFI of a vector field
for the simple line
is defined by the equation
we can consider the line GFI of the vector field that is defined by the line general fractional gradient
Therefore, we can get the line GFI of the line FG Gradient
where we use
Further, transformations are made similar to the transformations performed in the proof of Theorem 20. Using the fact that the kernels
,
, belong to the Luchko set
, we obtain
Similarly, we can obtain the equations for and .
Therefore, we get
where the standard gradient theorem is used.
As a result, we proved the general fractional gradient theorem for OCC with the line GF Gradient and the simple line in . □
11.5. Regional and Surface GF Curl in OCC
In this section, we proposed definitions of two type GF Curl operators in orthogonal curvilinear coordinates (OCC) for .
11.5.1. Regional GF Curl in OCC
Let us give the definition of Regional GF Curl operator in OCC for .
Definition 60. Let be a vector field which satisfies conditionsfor all in OCC.
Then the regional general fractional curl in OCC for the region is defined aswherewith .
11.5.2. Surface GF Integral in OCC
Using Definition 20 piecewise simple surface () in OCC, we propose definition of surface GF integral in OCC.
Let us define a set . of vector fields that is used in definition of surface GF integral in OCC.
Definition 61. Let S be a piecewise simple surface (). Let the vector fieldon the surface S satisfy the conditionsfor all , , and , where we use the notations The set of such vector fields on piecewise simple surface S will be denoted by .
We can use the representation of the tilde
in the form
Let us give a definition of surface GF integral in OCC.
Definition 62. Let S be a piecewise simple surface () and a vector fields on this surface S belongs to the set .
Then, the surface general fractional vector integral in OCC (Surface GFI in OCC) of the second kindfor the vector fiels is defined by the equation Here the areasare the projections of the surface S onto the , , planes in OCC, where , and are simple areas in these planes (or simple along some axes).
11.5.3. Surface GF Curl in OCC
Let us now give a definition of the Surface GF Curl in OCC for for piecewise simple surface. The piecewise simple surface is given in Definition 20.
Let us give a definition of the set of vector fields that are used to define the surface GF Curl operator in OCC (compare with Definition 21 of the set that is used in definition of the surface GFI).
Definition 63. Let S be a piecewise simple surface ().
Let the vector fieldon the surface S satisfy the conditionsfor all , , and .
The set of such vector fields on piecewise simple surface S will be denoted by .
Then the surface general fractional curl is defined in the following form.
Definition 64. Let S be a piecewise simple surface () and a vector fields on this surface S belongs to the set .
Then the surface general fractional curl in OCC (Surface GF Curl in OCC) of the vector field for the piecewise simple surface S is defined aswherewith .
Let us give the expression of the surface GFI of the vector field in the form of the Surface GF Curl in OCC.
Remark 25. The surface GFI of the surface GF Curl in OCC has the form For the case,
, we have
and
This case will be considered in the proof of the general fractional Stokes theorem for Surface GF Curl with simple surface, which is given in the next section.
11.6. General Fractional Stokes Theorem for Surface GF Curl in OCC
Let us prove the general fractional Stokes theorem for surface general fractional curl operator, where surface consists of simple surfaces or surfaces parallel to the coordinate planes.
Theorem 21. (General Fractional Stokes Theorem for Surface GF Curl in OCC)
Let be a simple surface (or a piecewise simple surface ), and the vector field belongs to the set .
Then, the equationholds, where is the surface GF Curl that is defined by Definition 64.
Proof. Let us prove the theorem for the vector field . The general Stokes equations for the vector fields and are proved similarly. The proof for the vector field is realized by the sum of the vector fields .
Let be a -simple and -simple surface that is described by the equation for , and the equation for .
The values of the function
on the line
are equal to the values of the function
on the line
, which is a projection of the line
L onto the
-plane,
where
and
.
Let us assume that
consists of two
-simple lines
and
that are described by equations
and
, where
. Then
where
Using the fundamental theorem of general fractional calculus, expression (
210) can be written as
Then using the definition of the general fractional derivative and the standard chain rule for the first-order derivative, we get
Let us use the property
for the expression of term (
213) in the form
Then we can use the equation, which is used in the standard Stokes theorem,
, and the fundamental theorem of GFC in the form
Using (
215), expression (
213) can be written in the form
Therefore, for two terms (
212) and (
213), we obtain the equations
and
for the vector field
.
As a result, we proved the general Stokes theorem for the surface FG Curl and the vector field
.
Equations for the remaining components of the vector field and are proved similarly. □
11.7. General Fractional Gauss Theorem in OCC
11.7.1. Definition of Triple GFI by Iterated GFI in OCC
Let us use the concept Definition 38 of Z-simple region for OCC in .
Definition 65. Let be -simple domain in OCC that is bounded above and below by smooth surfaces , described by the equationswhere the functions are continuous in the closed domain that is a projection of the region W onto the -plane, and for all .
Let scalar field be satisfy the conditionwhere Then the triple general fractional integral in OCC is defined by the equationwhere is double GFI.
Definition 66. Let be -simple domain that is that is bounded above and below by smooth surfaces , described by Equation (219). Let be projection of W on the -plane such that is -simple region in -plane that is bounded by the lines and , where and are continuous functions on the interval , and for all .
Let scalar field satisfy condition (220) and Then the triple general fractional integral (triple GFI) is defined in the form A volume general fractional integral (volume GFI) of a scalar field is a triple general fractional integral for the Z-simple region .
Example 19. Using the parallelepiped regionthe volume general fractional integral can be written as For the power-law kernels, the general fractional flux (223) has the form For , Equation (224) is given aswhich is the standard volume integral for the function .
11.7.2. General Fractional Divergence in OCC
In this section, we give the definition of general fractional divergence in OCC for .
Let us define sets of vector fields that will be used in the definition of the regional general fractional divergence.
Definition 67. Let be a vector field that satisfies the conditionswhere Then the set of such vector fields will be denoted as .
We can also consider vector fields that satisfy the condition .
Definition 68. Let be a vector field that satisfies the conditionsfor all .
Then the set of such vector fields will be denoted as .
Let us define the general fractional divergence in OCC.
Definition 69. Let be a vector field that belongs to the set or .
Then the regional general fractional divergence in OCC for the region is defined aswhere .
Divergence (
226) is the regional general fractional differential operator in OCC. We can also define the line and surface general fractional divergence in OCC.
Remark 26. The equation, which defines the regional general fractional divergence in OCC, can be written in compact form. If the vector fieldbelongs to the function space , then the regional general fractional divergence in OCC for the region can be defined aswhere Remark 27. The general fractional divergence in OCC can be defined not only for , but also for regions , surfaces and line . Note that the general fractional divergence in OCC for regions is used in the general fractional Gausss theorem for OCC to be given and proved in the following sections.
11.8. General Fractional Gauss Theorem for OCC
Let us define a set of vector field, for which general fractional Gauss theorem will be formulated.
Definition 70. Let W in be -simple region such that W is a piecewise -simple and -simple regionwhere are the -simple regions that is described by , for , and are the -simple regions that are described by the functions , for .
Let vector field satisfy the conditionswhere Then the set of such vector fields will be denoted as .
Definition 71. Let vector field satisfy the conditions Then the set of such vector fields will be denoted as .
Theorem 22. (General fractional Gauss theorem for OCC)
Let W in be -simple region such that W is a piecewise -simple and -simple region. Let W is bounded above and below by smooth surfaces , , which is described by Equation (219), and a lateral surface , whose generatrices are parallel to the -axis. Let the vector field belongs to the sets and .
Then the general fractional Gauss equation has the fromthat can be written as Proof. Let us consider the triple GFI
for the scalar field
where
for each point
.
Using Equations (
221), (
227) and (
228), we get
since
.
Using the second fundamental theorem of GFC, we get
Substituting expression (
230) into Equation (
229) and assuming that the condition
is satisfied, we obtain the equation
Therefore, the triple GFI can be represented through the surface GFIs in the form
where the surface GFIs are represented by the definition in the form
Then, taking into account that the surface GFI over the surface
is equal to zero
we obtain
where
is closed surface that contains the region
W inside the surface
.
Further, similarly with the proof of general fractional Gauss theorem for the Cartesian coordinate system and and by analogy with the transformation described above, we obtain the equation of the theorem to be proved. □
Example 20. As an example, let us give the general fractional Gauss theorem in cylindrical coordinates.
Let us consider the simple domain in the form The vertices of the domain are the following points Let , then For simple domain (231) the cylindrical general fractional Gauss equation has the form 12. Conclusions
The general fractional vector calculus (General FVC) is proposed as a generalization of the fractional vector calculus suggested in [
13,
31]. The formulation of the General FVC is based on the results in the calculus of general fractional integrals and derivatives that is proposed in Luchko’s work [
41].
The formulation of General FVC is self-consistent form, i.e., definitions of fractional generalizations of differential and integral vector operators are consistent with each other, and generalizations of fundamental theorems of vector calculus were proved. In this paper, the definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. Definitions of general fractional differential vector operators, including the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence, are suggested. Fundamental theorems of General FVC, which are general analogs of the standard gradient, Green’s, Stokes’ and Gauss’s theorems, are proved for simple and complex regions. Let us emphasize that the fractional vector analogs of fundamental theorems (such as the gradient, Stock’s and Gauss theorems) are not fulfilled for all type (regional, surface and line) of the general fractional vector operators (the gradient, curl and divergence). In the general case, the general fractional (GF) gradient theorem should be considered for the line GF Gradient, the FG Stock’s theorem should be considered for the surface GF Curl operator, and the GF Green theorem should be considered for the regional GF Divergence. This is due to violation of the chain rule for general fractional derivatives. The General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is described.
The proposed General FVC can be used as a mathematical tool in general fractional dynamics (GFDynamics) [
44,
45], in which non-locality in space is taken into account in general form.