1. Introduction
The real-valued orders of derivatives and integrals are used in fractional differential equations (FDEs). These real-valued orders of derivatives and integrals enable FDEs to model physical and applied scientific phenomena more precisely. However, it is exceedingly challenging to develop analytical solutions to many types of FDEs due to the extra complexity caused by arbitrary orders of derivation and integration. Therefore, finding precise and effective numerical solution techniques for the FDEs is essential. In recent years, certain numerical methods have been used for FDEs, such as the finite difference method [
1], B-spline collocation method [
2], differential transform method [
3] Adomian decomposition method [
4,
5], variational iteration method [
6,
7], block-by-block method [
8], orthogonal polynomials method [
9,
10,
11], Galerkin method [
12], Bessel collocation method [
13], spectral method [
14], reproducing kernel method [
15,
16], and operational matrix methods [
17,
18,
19]. Similar to this, several wavelet types are currently being researched for issues including increased computational cost. Wavelets are mathematical operations that separate data into various time–frequency components. These functions are created by dilating and shifting a wavelet function known as the mother wavelet function. The fundamental benefit of the wavelet basis is that it simplifies the solution of the FDE problem to a set of algebraic equations. Additionally, the method converges quickly and easily due to the wavelets’ many advantages, including their orthogonality, singularity detection skills, compact support, and simultaneous representation of data in several resolutions. Many wavelet basis functions have been used to solve a wide range of FDEs. Legendre, Haar, Bernoulli, Euler, CAS, Taylor, Laguree, Chebyshev wavelets of first and second kind are employed in recent studies elsewhere [
20,
21,
22,
23,
24,
25,
26,
27,
28].
In this paper we aim to solve nonlinear FDEs using Hermite wavelets. To the best of our knowledge, Hermite wavelets have not been exploited often. Furthermore, it is obvious that the orthogonal basis functions will provide sparser operational matrices used for the numerical approximations of the fractional differential terms. In this paper, we first obtain the operational matrices for fractional integration using Hermite wavelets and block-pulse functions (BPF) for function approximation. The operational matrices for the fractional integration obtained using BPF is not the same as the ones obtained in [
29]. Our approximation produces fewer calculations and an easier conversion from the nonlinear FDEs in question into the system of algebraic equations. The novelty of the method lies in the fact that the method combines the orthogonal Hermite wavelets with their corresponding operational matrices of integration to obtain sparser conversion matrices, which smoothly convert the FDE to a corresponding algebraic equation in vector-matrix form. Calculating the algebraic equation for a few colocation points creates a system of algebraic equations. By solving for the coefficients, the approximate solution is also obtained. The proposed method consists of simple and clear steps, therefore it is straightforward to code in any programming language of choice.
The paper is organized as follows: In
Section 2, the fundamental definitions of fractional calculus are given. In
Section 3, The Hermite wavelets are defined. The operational matrices for nonlinear FDEs are obtained using Hermite wavelets in
Section 4. Convergence analysis is presented in
Section 5. The proposed method is presented in
Section 6. Numerical solutions for several nonlinear FDEs are provided in
Section 7. The paper is concluded in
Section 8.
6. Operational Matrices of HWs
In this section we obtain the fractional operational matrices of Hermite wavelets. Fractional operational matrices of HWs require less computation with Block Pulse Functions (BPFs). Consequently, it is simpler to convert the FDE into an algebraic vector-matrix-form equation. All the numerical calculations are performed using Matlab R2021b in this study.
An
set of BPFs is defined as:
where
. The functions
are disjoint and orthogonal. For
,
Any squarely integrable function
defined in [0, 1) can be expanded into an
set of BPFs as:
where
,
and
are given as
.
Definition :
Let
and
. By means of BPFs, we have:
The HW matrix can also be expanded to an
set of BPFs as:
The Block Pulse operational matrix for fractional integration
is defined as [
31]:
where
with
.
The fractional integration of the Hermite wavelet vector
defined in (
13) can be approximated as:
where matrix
is called the Hermite wavelet operational matrix.
Using Equations (26)–(29), we obtain:
The resulting Hermite wavelet operational matrix
becomes:
As an example, the Hermite wavelet operational matrix for
k = 2,
M = 3, and
yields
7. Numeric Solution Examples
This section includes several nonlinear FDE examples to show the effectiveness and compactness of the suggested approach.
We first analyze the following FDE, which can be used to model a solid material in a Newtonian fluid [
30], defined as:
Applying the proposed method, we have:
Using the approximations of (33)–(35) in (32), we obtain:
Equation (36) is written for a few collocation points to construct an algebraic system of equations. The solution of that system provides the values of the coefficient vector C, which in turn provides the approximate solution.
The absolute errors, calculated as absolute difference between the exact and approximate solutions, is presented for several
m’ parameters in
Table 1. As can be seen from
Table 1, the Hermite Wavelet Method (HWM) error decreases with the increased resolution. The absolute errors are approximately on the order of E-4, E-5 for
, on the order of E-5, E-6 for
, and on the order of E-6, E-7 for
. A comparison with the Orthogonal Function Method (OFM) [
32], Variational Iteration Method (VIM) [
4], Adomian Decomposition Method (ADM) [
4], and the Finite Difference Method (FDM) [
1] is presented in
Table 2. As can be seen from the table, the proposed method can be said to converge better than the other methods. The exact solution and HWM results for
are plotted in
Figure 1. As can be seen from the figure, the numerical solution follows the exact solution closely.
Consider the Riccati FDE [
33] given below for
and
:
The exact solution for is given as .
The application of HWM to the Riccati FDE requires the approximate expressions listed below:
Defining
, we have:
Combining all approximations in the original Riccati FDE gives:
The solution for (41) is obtained as with the previous example through the algebraic system of equations which is constructed using a few collocation points.
Table 3 presents the method’s absolute errors for a number of
m’ parameters. The error diminishes with an increase in
m’, as seen in
Table 3. The absolute errors are approximately on the order of E-5 for
, on the order of E-5, E-6 for
, and on the order of E-6, E-7 for
. The HWM results and exact solution are given for
in
Figure 2. The FDE solutions for a few fractional values of
are presented in
Figure 3.
Figure 3 demonstrates that, as the fractional
approaches 1, the solution approximates to the exact solution obtained for integer order Riccati differential equation.
Consider the FDE [
30] given again for
and
:
The exact solution for is given as .
The application of HWM to the FDE results in:
where
.
The solution of (43) is obtained as in the previous examples.
Table 4 includes absolute errors for several m’. As expected, larger m’ values provide lower error values. The absolute errors are approximately on the order of E-5, E-6 for
, on the order of E-6, E-7 for
, and on the order of E-6, E-7 for
. The exact solution and HWM solution for
and
is plotted in
Figure 4 to illustrate the accuracy of the method. The FDE solutions for a few fractional values of
are presented in
Figure 5. As can be seen from
Figure 5, as the fractional
approaches 1, the approximate solution obtained for
approximates to the exact solution.
Consider the FDE [
34] given below for
and
:
The exact solution for is given as .
The application of HW method to the FDE yields:
The solution for (45) is obtained as with the previous example through the algebraic system of equations, constructed using a few collocation points. This example was chosen to provide comparison with several other numerical methods presented elsewhere.
Table 5 gives the results for several
values of the proposed method, and also includes the results of Modified Homotopy Perturbation Method (MHPM) [
34] and Iterative Reproducing Kernel Hilbert Spaces Method (IRKHSM) [
35] for
. The results show that HWM produces smaller errors for the most t values even when
, for
HWM is a better approximation method.
Table 6 summarizes the comparative results for fractional
of the proposed method and also of the Reproducing Kernel Method (RKM) [
36], Bernstein Polynomial Method (BPM) [
37], Iterative Reproducing Kernel Hilbert Spaces Method (IRKHSM) [
38], Haar Wavelet Operational Matrix Method (HWOMM) [
39] and Modified Homotopy Perturbation Method (MHPM) [
34]. For the fractional
there is not an exact solution, all results are relatively close to one another. Therefore, the results of the proposed method can be interpreted as producing an acceptable outcome for the fractional
.
The exact and HWM results are plotted in
Figure 6 for
.
Figure 7 includes plots for several
. As with the other examples, the fractional approximate solutions approach the exact result obtained for
as
approaches 1.
8. Conclusions
The motivation for this study stems from the need to employ Hermite wavelets for the numeric solution of nonlinear FDEs. To the best of our knowledge, the HWs have not been explored much in this regard. The operational matrices required for each of the fractional terms in the FDE to convert it to an algebraic equation are sparser due to the orthogonality of the Hermite wavelets. The orthogonality property is essential for the lower computational load and fast convergence of the method. The HW method is very accurate, even for the small number of collocation points, as demonstrated in the numerical examples. The maximum errors are generally on the order of E-5–E-7 for the collocation points up to . For higher accuracy, the number of the collocation points must be increased. Additionally, the resulting algebraic equation for numeric approximation is a vector-matrix equation. The compactness obtained in vector-matrix form facilitates the coding process of the method. The convergence analysis is also provided for the proposed method.
As can be seen from
Figure 3,
Figure 5 and
Figure 7, the numerical solutions for the fractional values of
approach the solution obtained for
as
approaches 1, which verifies the solutions obtained for fractional
values.
We believe the work presented here can be employed in a wide variety of applications such as variable-order models, systems of FDEs, systems of integro-fractional differential equations, optimal control problems, and fractional partial differential equations.