Fractal and Fractional

Research

23 pages, 367 KiB

Open AccessArticle

Quantum Laplace Transforms for the Ulam–Hyers Stability of Certain q-Difference Equations of the Caputo-like Type

by Sina Etemad, Ivanka Stamova, Sotiris K. Ntouyas and Jessada Tariboon

Fractal Fract. 2024, 8(8), 443; https://doi.org/10.3390/fractalfract8080443 - 28 Jul 2024

Viewed by 696

Abstract

We aim to investigate the stability property for the certain linear and nonlinear fractional

q

-difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear

q

-difference equations of the

q

-Caputo-like type [...] Read more.

We aim to investigate the stability property for the certain linear and nonlinear fractional

q

-difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear

q

-difference equations of the

q

-Caputo-like type are Ulam–Hyers stable by using the quantum Laplace transform and quantum Mittag–Leffler function. Moreover, after proving the existence property for a nonlinear Cauchy

q

-difference initial value problem, we use the same quantum Laplace transform and the

q

-Gronwall inequality to show that it is generalized Ulam–Hyers–Rassias stable. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

22 pages, 651 KiB

Open AccessArticle

Analytic Solution of the Time-Fractional Partial Differential Equation Using a Multi-G-Laplace Transform Method

by Hassan Eltayeb

Fractal Fract. 2024, 8(8), 435; https://doi.org/10.3390/fractalfract8080435 - 23 Jul 2024

Viewed by 623

Abstract

In several recent studies, many researchers have shown the advantage of fractional calculus in the production of particular solutions of a huge number of linear and nonlinear partial differential equations. In this research work, different theorems related to the G-double Laplace transform (DGLT) [...] Read more.

In several recent studies, many researchers have shown the advantage of fractional calculus in the production of particular solutions of a huge number of linear and nonlinear partial differential equations. In this research work, different theorems related to the G-double Laplace transform (DGLT) are proved. The solution of the system of time-fractional partial differential equations is addressed using a new analytical method. This technique is a combination of the multi-G-Laplace transform and decomposition methods (MGLTDM). Moreover, we discuss the convergence of this method. Two examples are provided to check the applicability and efficiency of our technique. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

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22 pages, 1507 KiB

Open AccessArticle

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

by Mohammed H. Alharbi, Abdullah F. Abu Sunayh, Ahmed Gamal Atta and Waleed Mohamed Abd-Elhameed

Fractal Fract. 2024, 8(7), 427; https://doi.org/10.3390/fractalfract8070427 - 20 Jul 2024

Cited by 1 | Viewed by 681

Abstract

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas [...] Read more.

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

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15 pages, 297 KiB

Open AccessArticle

The Existence and Ulam Stability Analysis of a Multi-Term Implicit Fractional Differential Equation with Boundary Conditions

by Peiguang Wang, Bing Han and Junyan Bao

Fractal Fract. 2024, 8(6), 311; https://doi.org/10.3390/fractalfract8060311 - 24 May 2024

Viewed by 582

Abstract

In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the [...] Read more.

In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the given equation, and the solution is unique. Afterwards, we give the criteria of Ulam–Hyers stability and Ulam–Hyers–Rassias stability. Additionally, we present an example to illustrate the practical application and effectiveness of the results. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

29 pages, 464 KiB

Open AccessArticle

On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam

by Sina Etemad, Sotiris K. Ntouyas, Ivanka Stamova and Jessada Tariboon

Fractal Fract. 2024, 8(4), 236; https://doi.org/10.3390/fractalfract8040236 - 17 Apr 2024

Cited by 1 | Viewed by 1112

Abstract

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, [...] Read more.

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned

(p; q)

-difference Navier problems in two single-valued and set-valued versions. We use the main properties of the

(p; q)

-operators in this regard. Application of the fixed points of the

ρ

-

θ

-contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential

(p; q)

-difference Navier problems. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

23 pages, 393 KiB

Open AccessArticle

Systems of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Boundary Conditions

by Alexandru Tudorache and Rodica Luca

Fractal Fract. 2023, 7(11), 816; https://doi.org/10.3390/fractalfract7110816 - 11 Nov 2023

Cited by 2 | Viewed by 1158

Abstract

We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply [...] Read more.

We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply various fixed point theorems, notably including the Banach contraction mapping principle, the Krasnosel’skii fixed point theorem applied to the sum of two operators, the Schaefer fixed point theorem, and the Leray–Schauder nonlinear alternative. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

12 pages, 304 KiB

Open AccessArticle

Affine-Periodic Boundary Value Problem for a Fractional Differential Inclusion

by Shanshan Gao, Sen Zhao and Jing Lu

Fractal Fract. 2023, 7(9), 647; https://doi.org/10.3390/fractalfract7090647 - 25 Aug 2023

Viewed by 747

Abstract

In the article, affine-periodic boundary value problem involving fractional derivative is considered. Existence of solutions to a Caputo-type fractional differential inclusion is researched by some fixed-point theorems and set-valued analysis theory. Specifically, we consider two cases in which the multifunction has convex values [...] Read more.

In the article, affine-periodic boundary value problem involving fractional derivative is considered. Existence of solutions to a Caputo-type fractional differential inclusion is researched by some fixed-point theorems and set-valued analysis theory. Specifically, we consider two cases in which the multifunction has convex values and nonconvex values, respectively. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

15 pages, 345 KiB

Open AccessArticle

Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives

by Vladimir E. Fedorov, Marina V. Plekhanova and Daria V. Melekhina

Fractal Fract. 2023, 7(6), 464; https://doi.org/10.3390/fractalfract7060464 - 7 Jun 2023

Cited by 2 | Viewed by 925

Abstract

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order [...] Read more.

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

24 pages, 390 KiB

Open AccessArticle

On a System of Hadamard Fractional Differential Equations with Nonlocal Boundary Conditions on an Infinite Interval

by Rodica Luca and Alexandru Tudorache

Fractal Fract. 2023, 7(6), 458; https://doi.org/10.3390/fractalfract7060458 - 3 Jun 2023

Cited by 5 | Viewed by 998

Abstract

Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann–Stieltjes [...] Read more.

Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main theorems, we employ the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

18 pages, 1009 KiB

Open AccessArticle

An Implicit Numerical Method for the Riemann–Liouville Distributed-Order Space Fractional Diffusion Equation

by Mengchen Zhang, Ming Shen and Hui Chen

Fractal Fract. 2023, 7(5), 382; https://doi.org/10.3390/fractalfract7050382 - 2 May 2023

Cited by 4 | Viewed by 1463

Abstract

This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit [...] Read more.

This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit scheme and prove that the numerical method is unconditionally stable and convergent with an accuracy of

O (σ^{2} + ρ^{2} + τ + h_{x} + h_{y})

. After employing an extrapolated technique, the convergence order is improved to second order in time and space. Furthermore, a fast algorithm is constructed to reduce computational costs. Two numerical examples are presented to verify the effectiveness of the numerical methods. This study may provide more possibilities for simulating diffusion complexities by fractional calculus. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

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22 pages, 1303 KiB

Open AccessArticle

Numerical Identification of External Boundary Conditions for Time Fractional Parabolic Equations on Disjoint Domains

by Miglena N. Koleva and Lubin G. Vulkov

Fractal Fract. 2023, 7(4), 326; https://doi.org/10.3390/fractalfract7040326 - 13 Apr 2023

Cited by 8 | Viewed by 1464

Abstract

We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we [...] Read more.

We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we decouple the full inverse problem into two Dirichlet problems at each time level. On this base, we develop decomposition techniques to obtain exact formulas for the unknown boundary conditions at point measurements. A discrete version of the analytical approach is realized on time adaptive mesh for different fractional order of the equations in each of the disjoint domains. A variety of numerical examples are discussed. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

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20 pages, 373 KiB

Open AccessArticle

Stability of p(·)-Integrable Solutions for Fractional Boundary Value Problem via Piecewise Constant Functions

by Mohammed Said Souid, Ahmed Refice and Kanokwan Sitthithakerngkiet

Fractal Fract. 2023, 7(2), 198; https://doi.org/10.3390/fractalfract7020198 - 17 Feb 2023

Cited by 3 | Viewed by 1194

Abstract

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (

L^{p (\cdot)}

). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers [...] Read more.

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (

L^{p (\cdot)}

). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers are established. Our results are obtained using two fixed-point theorems, then illustrating the results with a comprehensive example. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

23 pages, 390 KiB

Open AccessArticle

On a System of Sequential Caputo Fractional Differential Equations with Nonlocal Boundary Conditions

by Alexandru Tudorache and Rodica Luca

Fractal Fract. 2023, 7(2), 181; https://doi.org/10.3390/fractalfract7020181 - 12 Feb 2023

Cited by 6 | Viewed by 1147

Abstract

We obtain existence and uniqueness results for the solutions of a system of Caputo fractional differential equations which contain sequential derivatives, integral terms, and two positive parameters, supplemented with general coupled Riemann–Stieltjes integral boundary conditions. The proofs of our results are based on [...] Read more.

We obtain existence and uniqueness results for the solutions of a system of Caputo fractional differential equations which contain sequential derivatives, integral terms, and two positive parameters, supplemented with general coupled Riemann–Stieltjes integral boundary conditions. The proofs of our results are based on the Banach fixed point theorem and the Leray–Schauder alternative. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

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