Research

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15 pages, 3018 KiB

Open AccessArticle

Using a Mix of Finite Difference Methods and Fractional Differential Transformations to Solve Modified Black–Scholes Fractional Equations

by Agus Sugandha, Endang Rusyaman, Sukono and Ema Carnia

Mathematics 2024, 12(7), 1077; https://0-doi-org.brum.beds.ac.uk/10.3390/math12071077 - 03 Apr 2024

Viewed by 507

Abstract

This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the [...] Read more.

This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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14 pages, 287 KiB

Open AccessArticle

A Study of Some New Hermite–Hadamard Inequalities via Specific Convex Functions with Applications

by Moin-ud-Din Junjua, Ather Qayyum, Arslan Munir, Hüseyin Budak, Muhammad Mohsen Saleem and Siti Suzlin Supadi

Mathematics 2024, 12(3), 478; https://0-doi-org.brum.beds.ac.uk/10.3390/math12030478 - 02 Feb 2024

Viewed by 720

Abstract

Convexity plays a crucial role in the development of fractional integral inequalities. Many fractional integral inequalities are derived based on convexity properties and techniques. These inequalities have several applications in different fields such as optimization, mathematical modeling and signal processing. The main goal [...] Read more.

Convexity plays a crucial role in the development of fractional integral inequalities. Many fractional integral inequalities are derived based on convexity properties and techniques. These inequalities have several applications in different fields such as optimization, mathematical modeling and signal processing. The main goal of this article is to establish a novel and generalized identity for the Caputo–Fabrizio fractional operator. With the help of this specific developed identity, we derive new fractional integral inequalities via exponential convex functions. Furthermore, we give an application to some special means. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

8 pages, 651 KiB

Open AccessArticle

On Λ-Fractional Wave Propagation in Solids

by Kostantinos A. Lazopoulos and Anastasios K. Lazopoulos

Mathematics 2023, 11(19), 4183; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194183 - 06 Oct 2023

Viewed by 611

Abstract

Wave propagation in solids is discussed, based upon inherently non-local Λ-fractional analysis. Following the governing equations of Λ-fractional continuum mechanics, the Λ-fractional wave equations are derived. Since the variational procedures are only global, in the present Λ-fractional analysis, various jumpings, either in the [...] Read more.

Wave propagation in solids is discussed, based upon inherently non-local Λ-fractional analysis. Following the governing equations of Λ-fractional continuum mechanics, the Λ-fractional wave equations are derived. Since the variational procedures are only global, in the present Λ-fractional analysis, various jumpings, either in the strain or the stress, may be shown. The proposed theory is applied to impact-induced transitions in two-phase elastic materials and viscoelastic materials. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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12 pages, 319 KiB

Open AccessArticle

A Numerical Approach for Dealing with Fractional Boundary Value Problems

by Abeer A. Al-Nana, Iqbal M. Batiha and Shaher Momani

Mathematics 2023, 11(19), 4082; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194082 - 26 Sep 2023

Cited by 4 | Viewed by 619

Abstract

This paper proposes a novel numerical approach for handling fractional boundary value problems. Such an approach is established on the basis of two numerical formulas; the fractional central formula for approximating the Caputo differentiator of order

α

and the fractional central formula for [...] Read more.

This paper proposes a novel numerical approach for handling fractional boundary value problems. Such an approach is established on the basis of two numerical formulas; the fractional central formula for approximating the Caputo differentiator of order

α

and the fractional central formula for approximating the Caputo differentiator of order

2 α

, where

0 < α \leq 1

. The first formula is recalled here, whereas the second one is derived based on the generalized Taylor theorem. The stability of the proposed approach is investigated in view of some formulated results. In addition, several numerical examples are included to illustrate the efficiency and applicability of our approach. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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12 pages, 294 KiB

Open AccessArticle

Integro-Differential Boundary Conditions to the Sequential ψ₁-Hilfer and ψ₂-Caputo Fractional Differential Equations

by Surang Sitho, Sotiris K. Ntouyas, Chayapat Sudprasert and Jessada Tariboon

Mathematics 2023, 11(4), 867; https://0-doi-org.brum.beds.ac.uk/10.3390/math11040867 - 08 Feb 2023

Cited by 4 | Viewed by 979

Abstract

In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type

ψ_{1}

-Hilfer and

ψ_{2}

-Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the [...] Read more.

In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type

ψ_{1}

-Hilfer and

ψ_{2}

-Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the Banach contraction principle, while the existence of results is established by using the Leray–Schauder nonlinear alternative. Numerical examples are constructed illustrating the obtained results. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

29 pages, 4859 KiB

Open AccessArticle

An Investigation on Existence, Uniqueness, and Approximate Solutions for Two-Dimensional Nonlinear Fractional Integro-Differential Equations

by Tahereh Eftekhari and Jalil Rashidinia

Mathematics 2023, 11(4), 824; https://0-doi-org.brum.beds.ac.uk/10.3390/math11040824 - 06 Feb 2023

Cited by 3 | Viewed by 922

Abstract

In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In addition, we use operational matrices of two-variable shifted Jacobi polynomials [...] Read more.

In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In addition, we use operational matrices of two-variable shifted Jacobi polynomials via the collocation method to reduce the equations into a system of equations. Error bounds of the presented method are obtained. Five test problems are solved. The obtained numerical results show the accuracy, efficiency, and applicability of the proposed approach. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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9 pages, 273 KiB

Open AccessArticle

Topological Structure and Existence of Solutions Set for q-Fractional Differential Inclusion in Banach Space

by Ali Rezaiguia and Taher S. Hassan

Mathematics 2023, 11(3), 683; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030683 - 29 Jan 2023

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Abstract

In this work, we concentrate on the existence of the solutions set of the following problem [...] Read more.

In this work, we concentrate on the existence of the solutions set of the following problem

^{c} D_{q}^{α} σ (t) \in F (t, σ (t),^{c} D_{q}^{α} σ (t)), t \in I = [0, T] σ (0) = σ_{0} \in E

, as well as its topological structure in Banach space E. By transforming the problem posed into a fixed point problem, we provide the necessary conditions for the existence and compactness of solutions set. Finally, we present an example as an illustration of main results. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

24 pages, 491 KiB

Open AccessArticle

New Hermite–Hadamard and Ostrowski-Type Inequalities for Newly Introduced Co-Ordinated Convexity with Respect to a Pair of Functions

by Muhammad Aamir Ali, Fongchan Wannalookkhee, Hüseyin Budak, Sina Etemad and Shahram Rezapour

Mathematics 2022, 10(19), 3469; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193469 - 23 Sep 2022

Cited by 1 | Viewed by 866

Abstract

In both pure and applied mathematics, convex functions are used in many different problems. They are crucial to investigate both linear and non-linear programming issues. Since a convex function is one whose epigraph is a convex set, the theory of convex functions falls [...] Read more.

In both pure and applied mathematics, convex functions are used in many different problems. They are crucial to investigate both linear and non-linear programming issues. Since a convex function is one whose epigraph is a convex set, the theory of convex functions falls under the umbrella of convexity. However, it is a significant theory that affects practically all areas of mathematics. In this paper, we introduce the notions of

(g, h)

-convexity or convexity with respect to a pair of functions on co-ordinates and discuss its fundamental properties. Moreover, we establish some novel Hermite–Hadamard- and Ostrowski-type inequalities for newly introduced co-ordinated convexity. Additionally, it is presented that the newly introduced notion of the convexity and given inequalities are generalizations of existing studies in the literature. Lastly, we look at various mathematical examples and graphs to confirm the validity of the newly found inequalities. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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27 pages, 471 KiB

Open AccessArticle

Sequential Fractional Hybrid Inclusions: A Theoretical Study via Dhage’s Technique and Special Contractions

by Sina Etemad, Sotiris K. Ntouyas, Bashir Ahmad, Shahram Rezapour and Jessada Tariboon

Mathematics 2022, 10(12), 2090; https://0-doi-org.brum.beds.ac.uk/10.3390/math10122090 - 16 Jun 2022

Cited by 2 | Viewed by 927

Abstract

The most important objective of the present research is to establish some theoretical existence results on a novel combined configuration of a Caputo sequential inclusion problem and the hybrid integro-differential one in which the boundary conditions are also formulated as the hybrid multi-order [...] Read more.

The most important objective of the present research is to establish some theoretical existence results on a novel combined configuration of a Caputo sequential inclusion problem and the hybrid integro-differential one in which the boundary conditions are also formulated as the hybrid multi-order integro-differential conditions. In this respect, firstly, some inequalities are proven in relation to the corresponding integral equation. Then, we employ some newly defined theoretical techniques with the help of the product operators on a Banach algebra and also with the aid of some special functions including

α

-

ψ

-contractions and

α

-admissible mappings to extract the existence criteria corresponding to the given mixed sequential hybrid BVPs. Some important useful properties such as the approximate endpoint property,

(C_{α})

-property, and the compactness play a key role in this regard. The final part of the manuscript is devoted to formulating and computing two applicable examples to guarantee the correctness of the obtained results. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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15 pages, 818 KiB

Open AccessArticle

Existence and Uniqueness Results for Fractional (p, q)-Difference Equations with Separated Boundary Conditions

by Pheak Neang, Kamsing Nonlaopon, Jessada Tariboon, Sotiris K. Ntouyas and Bashir Ahmad

Mathematics 2022, 10(5), 767; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050767 - 28 Feb 2022

Cited by 5 | Viewed by 1775

Abstract

In this paper, we study the existence of solutions to a fractional

(p, q)

-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence [...] Read more.

In this paper, we study the existence of solutions to a fractional

(p, q)

-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence results of solutions are obtained by applying Krasnoselskii’s fixed-point theorem and the Leary–Schauder alternative. Some examples illustrating the main results are also presented. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

Review

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28 pages, 370 KiB

Open AccessReview

A Survey on the Oscillation of Solutions for Fractional Difference Equations

by Jehad Alzabut, Ravi P. Agarwal, Said R. Grace, Jagan M. Jonnalagadda, A. George Maria Selvam and Chao Wang

Mathematics 2022, 10(6), 894; https://0-doi-org.brum.beds.ac.uk/10.3390/math10060894 - 11 Mar 2022

Cited by 5 | Viewed by 1762

Abstract

In this paper, we present a systematic study concerning the developments of the oscillation results for the fractional difference equations. Essential preliminaries on discrete fractional calculus are stated prior to giving the main results. Oscillation results are presented in a subsequent order and [...] Read more.

In this paper, we present a systematic study concerning the developments of the oscillation results for the fractional difference equations. Essential preliminaries on discrete fractional calculus are stated prior to giving the main results. Oscillation results are presented in a subsequent order and for different types of equations. The investigation was carried out within the delta and nabla operators. Full article

(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

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