Fractal and Fractional

Research

22 pages, 875 KiB

Open AccessArticle

Analysis of Time-Fractional Delay Partial Differential Equations Using a Local Radial Basis Function Method

by Kamran, Kalsoom Athar, Zareen A. Khan, Salma Haque and Nabil Mlaiki

Fractal Fract. 2024, 8(12), 683; https://doi.org/10.3390/fractalfract8120683 - 21 Nov 2024

Viewed by 180

Delay partial differential equations have significant applications in numerous fields, such as population dynamics, control systems, neuroscience, and epidemiology, where they are required to efficiently model the effects of past states on current system behavior. This work presents an RBF-based localized meshless method [...] Read more.

Delay partial differential equations have significant applications in numerous fields, such as population dynamics, control systems, neuroscience, and epidemiology, where they are required to efficiently model the effects of past states on current system behavior. This work presents an RBF-based localized meshless method for the numerical solution of delay partial differential equations. In the suggested numerical scheme, the localized meshless method is combined with the Laplace transform. The main attractive features of the localized meshless method are its simplicity, adaptability, and ease of implementation for complex problems defined on complex shaped domains. In a localized meshless scheme, a linear system of equations is solved. The Laplace transform, which is one of the most powerful techniques for solving integer- and non-integer-order problems, is used to represent the desired solution as a contour integral in the complex plane, known as the Bromwich integral. However, the analytic inversion of contour integral becomes very laborious in many situations. Therefore, a contour integration method is utilized to numerically approximate the Bromwich integral. The aim of utilizing the Laplace transform is to handle the costly convolution integral associated with the Caputo derivative and to avoid the effects of time-stepping techniques on the stability and accuracy of the numerical solution. We also discuss the convergence and stability of the suggested scheme. Furthermore, the existence and uniqueness of the solution for the considered model are studied. The efficiency, efficacy, and accuracy of the proposed numerical scheme have been demonstrated through numerical experiments on various problems. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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16 pages, 360 KiB

Open AccessArticle

On Riemann–Liouville Integral via Strongly Modified (h,m)-Convex Functions

by Ali N. A. Koam, Ammara Nosheen, Khuram Ali Khan, Mudassir Hussain Bukhari, Ali Ahmad and Maryam Salem Alatawi

Fractal Fract. 2024, 8(12), 680; https://doi.org/10.3390/fractalfract8120680 - 21 Nov 2024

Viewed by 207

Abstract

The generalization of strongly convex and strongly m-convex functions is presented in this paper. We began by proving the properties of a strongly modified

(h, m)

-convex function. The Schur inequality and the Hermite–Hadamard (H-H) inequalities are proved for [...] Read more.

The generalization of strongly convex and strongly m-convex functions is presented in this paper. We began by proving the properties of a strongly modified

(h, m)

-convex function. The Schur inequality and the Hermite–Hadamard (H-H) inequalities are proved for the proposed class. Moreover, H-H inequalities are also proved in the context of Riemann–Liouville (R-L) integrals. Some examples and graphs are also presented in order to show the existence of this newly defined class. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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20 pages, 1432 KiB

Open AccessArticle

An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel

by Sameeha A. Raad and Mohammed A. Abdou

Fractal Fract. 2024, 8(11), 644; https://doi.org/10.3390/fractalfract8110644 - 30 Oct 2024

Viewed by 454

Abstract

This work studies an integro-fractional differential equation (I-FrDE) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (I-DE) into a mixed integral equation (MIE) with an Able kernel [...] Read more.

This work studies an integro-fractional differential equation (I-FrDE) with a generalized symmetric singular kernel. The scientific approach in this study was to transform the integro-differential equation (I-DE) into a mixed integral equation (MIE) with an Able kernel in fractional time and a generalized symmetric singular kernel in position. Additionally, the authors first set conditions on the singular kernels, whether related to time or position, and then transform the integral equation into an integral operator. Secondly, the solution is unique, which is proven by means of fixed-point theorems. In combination with the solution rules, the convergence of the solution is studied, and the error equation resulting from the solution is a stable error-integral influencer equation. Next, to solve this MIE, the authors apply a special technique to separate the variables and produce an integral equation in position with coefficients, in the form of an integral operator in time. As the most effective technique for resolving singular integral equations, the Toeplitz matrix method (TMM) is utilized to convert the integral equation into an algebraic system for the purpose of solving the position problem. The existence of a solution to the linear algebraic system in Banach space is then demonstrated. Lastly, certain applications where the functions of the generalized symmetric kernel are cubic or exponential and it assumes the logarithmic, Cauchy, or Carleman form are discussed. In each case, Maple 18 is also used to compute the error estimate. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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14 pages, 367 KiB

Open AccessArticle

Subclasses of Bi-Univalent Functions Connected with Caputo-Type Fractional Derivatives Based upon Lucas Polynomial

by Kholood M. Alsager, Gangadharan Murugusundaramoorthy, Daniel Breaz and Sheza M. El-Deeb

Fractal Fract. 2024, 8(8), 452; https://doi.org/10.3390/fractalfract8080452 - 31 Jul 2024

Viewed by 771

Abstract

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for functions in these subclasses. [...] Read more.

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for functions in these subclasses. Using the values of

|a_{2}|

and

|a_{3}|

, we determined Fekete–Szegő inequality for functions in these subclasses. Moreover, we pointed out some more subclasses by fixing the parameters involved in Lucas polynomial and stated the estimates and Fekete–Szegő inequality results without proof. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

17 pages, 499 KiB

Open AccessArticle

Analysis of Non-Local Integro-Differential Equations with Hadamard Fractional Derivatives: Existence, Uniqueness, and Stability in the Context of RLC Models

by Manigandan Murugesan, Saravanan Shanmugam, Mohamed Rhaima and Ragul Ravi

Fractal Fract. 2024, 8(7), 409; https://doi.org/10.3390/fractalfract8070409 - 12 Jul 2024

Cited by 2 | Viewed by 944

Abstract

In this study, we focus on the stability analysis of the RLC model by employing differential equations with Hadamard fractional derivatives. We prove the existence and uniqueness of solutions using Banach’s contraction principle and Schaefer’s fixed point theorem. To facilitate our key conclusions, [...] Read more.

In this study, we focus on the stability analysis of the RLC model by employing differential equations with Hadamard fractional derivatives. We prove the existence and uniqueness of solutions using Banach’s contraction principle and Schaefer’s fixed point theorem. To facilitate our key conclusions, we convert the problem into an equivalent integro-differential equation. Additionally, we explore several versions of Ulam’s stability findings. Two numerical examples are provided to illustrate the applications of our main results. We also observe that modifications to the Hadamard fractional derivative lead to asymmetric outcomes. The study concludes with an applied example demonstrating the existence results derived from Schaefer’s fixed point theorem. These findings represent novel contributions to the literature on this topic, significantly advancing our understanding. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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15 pages, 3655 KiB

Open AccessArticle

Dynamical Behavior of the Fractional BBMB Equation on Unbounded Domain

by Wei Zhang, Haijing Wang, Haolu Zhang, Zhiyuan Li and Xiaoyu Li

Fractal Fract. 2024, 8(7), 383; https://doi.org/10.3390/fractalfract8070383 - 28 Jun 2024

Viewed by 579

Abstract

The fractional-order Benjamin-Bona-Mahony-Burgers (BBMB) equation is a generalization of the classical BBMB equation. It’s dynamic behaviors is much more complex than that of the corresponding integer-order BBMB equation. The main purpose of this paper is to explore the dynamic behaviors of the fractional-order [...] Read more.

The fractional-order Benjamin-Bona-Mahony-Burgers (BBMB) equation is a generalization of the classical BBMB equation. It’s dynamic behaviors is much more complex than that of the corresponding integer-order BBMB equation. The main purpose of this paper is to explore the dynamic behaviors of the fractional-order BBMB equations by using the Fourier spectral method. Firstly, the numerical solution is compared with the exact solution. It is proved that the proposed method is effective and high precision for solving the spatial fractional order BBMB equation. Then, some dynamical behaviors of fractional order BBMB equations are obtained by using the present method, and some novel fractal waves of the the fractional-order BBMB equation on unbounded domain are shown. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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18 pages, 355 KiB

Open AccessArticle

Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems

by Hongli Yang, Xindong Si and Ivan G. Ivanov

Fractal Fract. 2024, 8(5), 255; https://doi.org/10.3390/fractalfract8050255 - 25 Apr 2024

Cited by 1 | Viewed by 858

Abstract

This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order

0 < α < 1

. The objective is to establish the existence of conditions for a linear feedback control law within state constraints [...] Read more.

This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order

0 < α < 1

. The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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20 pages, 414 KiB

Open AccessArticle

Contributions to the Numerical Solutions of a Caputo Fractional Differential and Integro-Differential System

by Abdelkader Moumen, Abdelaziz Mennouni and Mohamed Bouye

Fractal Fract. 2024, 8(4), 201; https://doi.org/10.3390/fractalfract8040201 - 29 Mar 2024

Viewed by 925

Abstract

The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, [...] Read more.

The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, electrostatics, control theories, and the natural sciences. An effective approach solves the problem by reducing it to a pair of algebraically separated equations via a successful transformation. The proposed strategy uses first-order shifted Chebyshev polynomials and a projection method. Using the provided technique, the primary system is converted into a set of algebraic equations that can be solved effectively. Some theorems are proved and used to obtain the upper error bound for this method. Furthermore, various examples are provided to demonstrate the efficiency of the proposed algorithm when compared to existing approaches in the literature. Finally, the key conclusions are given. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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14 pages, 285 KiB

Open AccessArticle

Bang-Bang Property and Time-Optimal Control for Caputo Fractional Differential Systems

by Shimaa H. Abel-Gaid, Ahlam Hasan Qamlo and Bahaa Gaber Mohamed

Fractal Fract. 2024, 8(2), 84; https://doi.org/10.3390/fractalfract8020084 - 26 Jan 2024

Viewed by 1156

Abstract

In this paper, by using the controllability method, a bang-bang property and a time optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. We then state and prove the bang-bang theorem. [...] Read more.

In this paper, by using the controllability method, a bang-bang property and a time optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. We then state and prove the bang-bang theorem. Finally, we state the optimality conditions that characterize the optimal control. Some application examples are given to illustrate our results. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

15 pages, 371 KiB

Open AccessArticle

An Efficient Numerical Method Based on Bell Wavelets for Solving the Fractional Integro-Differential Equations with Weakly Singular Kernels

by Yanxin Wang and Xiaofang Zhou

Fractal Fract. 2024, 8(2), 74; https://doi.org/10.3390/fractalfract8020074 - 23 Jan 2024

Cited by 1 | Viewed by 1461

Abstract

A novel numerical scheme based on the Bell wavelets is proposed to obtain numerical solutions of the fractional integro-differential equations with weakly singular kernels. Bell wavelets are first proposed and their properties are studied, and the fractional integration operational matrix is constructed. The [...] Read more.

A novel numerical scheme based on the Bell wavelets is proposed to obtain numerical solutions of the fractional integro-differential equations with weakly singular kernels. Bell wavelets are first proposed and their properties are studied, and the fractional integration operational matrix is constructed. The convergence analysis of Bell wavelets approximation is discussed. The fractional integro-differential equations can be simplified to a system of algebraic equations by using a truncated Bell wavelets series and the fractional operational matrix. The proposed method’s efficacy is supported via various examples. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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