System Of Fractional Integro-differential Equations Research Articles

Solving linear systems of fractional integro-differential equations by Haar and Legendre wavelets techniques

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

A tau-Gegenbauer spectral approach for systems of fractional integro-differential equations with the error analysis

<abstract><p>This research paper focused on the solution of systems of fractional integro-differential equations (FIDEs) of the Volterra type with variable coefficients. The proposed approach combined the tau method and shifted Gegenbauer polynomials in a matrix form. The investigation of the existence and uniqueness of solutions for these systems was carried out using Krasnoselskii's fixed point theorem. The equations employed Caputo-style derivative operators, and to minimize computational operations involving derivatives and multiplications, integral and product operational matrices were derived. By introducing suitable polynomial approximations and employing the tau spectral method, the original system of FIDE was transformed into an algebraic system. Solving this algebraic system provided approximate solutions to the main system. Error bounds were computed in the Gegenbauer-weighted Sobolev space. The proposed algorithm was implemented and tested on two systems of integro-fractional differential equations to demonstrate its efficiency and simplicity. By varying the parameter $ \sigma $ in the Gegenbauer polynomials, the impact of this variation on the approximate solutions can be observed. A comparison with another method utilizing the block-by-block approach was also presented.</p></abstract>

A novel study for solving systems of nonlinear fractional integral equations

In this study, we explore the solution of a nonlinear system of fractional integro-differential equations based on the operational matrix method. We have modified the operational matrix method to accommodate such systems and have streamlined the resulting algebraic system in a practical manner. Instead of dealing with a system of equations, we have simplified the problem to finding the solution for a set of nonlinear equations. Here, m represents the number of block pulse functions, which is typically a large positive integer. This approach significantly reduces computational costs, especially for large values of m. We provide proofs for both the existence and uniqueness of the solution for this system. Additionally, we investigate two types of stability: Ulam–Hyers stability and generalized Ulam–Hyers stability. We have evaluated the effectiveness of the proposed method by applying it to three different problems and comparing our results with existing literature. The outcomes of these tests highlight the efficiency and efficacy of our proposed method.

A Novel Vieta–Fibonacci Projection Method for Solving a System of Fractional Integrodifferential Equations

In this paper, a new approach for numerically solving the system of fractional integrodifferential equations is devised. To approximate the issue, we employ Vieta–Fibonacci polynomials as basis functions and derive the projection method for Caputo fractional order for the first time. An efficient transformation reduces the problem to a system of two independent equations. Solving two algebraic equations yields an approximate solution to the problem. The proposed method’s efficiency and accuracy are validated. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests reinforce the interpretations of the theory.

Existence and Uniqueness Results for Different Orders Coupled System of Fractional Integro-Differential Equations with Anti-Periodic Nonlocal Integral Boundary Conditions

This paper presents a new class of boundary value problems of integrodifferential fractional equations of different order equipped with coupled anti-periodic and nonlocal integral boundary conditions. We prove the existence and uniqueness criteria of the solutions by using the Leray-Schauder alternative and Banach contraction mapping principle. Examples are constructed for the illustration of our results.

Existence, uniqueness and stability results for coupled systems of fractional integro-differential equations with fixed and nonlocal anti-periodic boundary conditions

In this paper, we investigate the existence and uniqueness of solutions for a new class of coupled system of two Caputo fractional derivatives of different orders and a Riemann- Liouville type integral with fixed and nonlocal anti-periodic boundary conditions, by using the fixed point approach in generalized metric spaces. The Ulam’s type stability of the proposed coupled system is also studied. An example is provided to illustrate the obtained theory.

Controllability of coupled fractional integrodifferential equations

AbstractIn this article, we examine a coupled system of fractional integrodifferential equations of Liouville–Caputo form with instantaneous impulsive conditions in a Banach space. We obtain the existence and uniqueness results by applying the theory of fixed point theorems. In a similar manner, we discuss Hyers–Ulam stability and controllability. We also present an example to show the validity of the obtained results.

A New Method to Solve Dual Systems of Fractional Integro-Differential Equations by Legendre Wavelets

The method that will be presented, is numerical solution based on the Legendre wavelets for solving dual systems of fractional integro-differential equations (FIDEs). First of all we make the operational matrix of fractional order integration. The application of this matrix is transforming FIDEs to a system of algebric equations. By this changing, we are able to solve it by a simple solution. In this way, the Legendre wavelets and their operator matrix are the most important keys of our solution. After explaining the method we test on some illustrative examples which numerical solutions of these examples demonstrate the validity and applicability of suggested method.

Existence and Ulam–Hyers Stability of a Fractional-Order Coupled System in the Frame of Generalized Hilfer Derivatives

In this research paper, we consider a class of a coupled system of fractional integrodifferential equations in the frame of Hilfer fractional derivatives with respect to another function. The existence and uniqueness results are obtained in weighted spaces by applying Schauder’s and Banach’s fixed point theorems. The results reported here are more general than those found in the literature, and some special cases are presented. Furthermore, we discuss the Ulam–Hyers stability of the solution to the proposed system. Some examples are also constructed to illustrate and validate the main results.

Existence of Mild Solutions for a Class of Impulsive Hilfer Fractional Coupled Systems

The aim of this paper is to give existence results for a class of coupled systems of fractional integrodifferential equations with Hilfer fractional derivative in Banach spaces. We first give some definitions, namely the Hilfer fractional derivative and the Hausdorff’s measure of noncompactness and the Sadovskii’s fixed point theorem.

Existence and uniqueness of solutions for coupled systems of Liouville-Caputo type fractional integrodifferential equations with Erdélyi-Kober integral conditions

Abstract In this paper, we examine a coupled system of fractional integrodifferential equations of Liouville-Caputo form with nonlinearities depending on the unknown functions, as well as their lower-order fractional derivatives and integrals supplemented with coupled nonlocal and Erdélyi-Kober fractional integral boundary conditions. We explain that the topic discussed in this context is new and gives more analysis into the research of coupled boundary value problems. We have two results: the first is the existence result of the given problem by using the Leray-Schauder alternative, whereas the second referring to the uniqueness result is derived by Banach’s fixed-point theorem. Sufficient examples were also supplemented to substantiate the proof, and some variations of the problem were discussed.

An Approximate Approach for Systems of Fractional IntegroDifferential Equations Based on Taylor Expansion

An Approximate Approach for Systems of Fractional IntegroDifferential Equations Based on Taylor Expansion

Existence Solution for Nonlinear System of Fractional Integrodifferential Equations of Volterra Type with Fractional Boundary Conditions

This article investigates existence, uniqueness and stability solutions of new fractional Volterra integro-differential equations system with fractional boundary conditions by using the existence and uniqueness theorem. Theorems on existence and uniqueness of solution are established under some necessary and sufficient conditions on compact space. A simple example of application of the main results of this article is presented.

Existence and stability of impulsive coupled system of fractional integrodifferential equations

Abstract In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

Numerical studies for solving fractional integro-differential equations

In this paper, we give a new numerical method for solving a linear system of fractional integro-differential equations. The fractional derivative is considered in the Caputo sense. The proposed method is least squares method aid of Hermite polynomials. The suggested method reduces this type of systems to the solution of systems of linear algebraic equations. To demonstrate the accuracy and applicability of the presented method some test examples are provided. Numerical results show that this approach is easy to implement and accurate when applied to integro-differential equations. We show that the solutions approach to classical solutions as the order of the fractional derivatives approach.

Generalized Spline Approach For Solving System of Linear Fractional Volterra Integro-Differential Equations

In this paper generalized spline method is used for solving linear system of fractional integro-differential equation approximately. The suggested method reduces the system to system of linear algebraic equations. Different orders of fractional derivative for test example is given in this paper to show the accuracy and applicability of the presented method.

Hybrid Bernstein Block–Pulse functions for solving system of fractional integro-differential equations

ABSTRACTThis paper deals with the numerical solution of system of fractional integro-differential equations. In this work, we approximate the unknown functions based on the hybrid Bernstein Block–Pulse functions, in conjunction with the collocation method. We introduce the Riemann–Liouville fractional integral operator for the hybrid Bernstein Block–Pulse functions. This operator will be approximated by the Gauss quadrature formula with respect to the Legendre weight function and then it is utilized to reduce the solution of the fractional integro-differential equations to a system of algebraic equations. This system can be easily solved by any usual numerical methods. The existence and uniqueness of the solution have been discussed. Moreover, the convergence analysis of this algorithm will be shown by preparing some theorems. Numerical experiments are presented to show the superiority and efficiency of proposed method in comparison with some other well-known methods.

Existence results for a coupled system of fractional integro-differential equations with time-dependent delay

Our purpose in this manuscript is concerned with the existence of solutions for a coupled system of fractional integro-differential equations with time-dependent delay. We use the Krasnoselskii’s fixed-point theorem along with fractional calculus. This work completes recent results on the subject and an example of partial integro-differential equation is presented.

New Solutions for System of Fractional Integro-Differential Equations and Abel’s Integral Equations by Chebyshev Spectral Method

Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations.

Recent Theory and Applications on Numerical Algorithms and Special Functions

In recent years, there has been extensive research activity on both the theory and application of efficient numerical methods for approximating the solutions of differential equations. Numerical algorithms and special functions have been a major thrust of this research and have numerous applications in science and engineering. Special functions play a tremendous role in mathematical analysis and in the development of newnumerical techniques. In particular, they allow a broad refinement of spectral methods for solving several types of differential equations in both finite and infinite domains. This is useful, in that there is a strong need for efficient and easy-to-use scalable techniques for the numerical solution of large scale problems. This special issue was focused on the analysis of numerical methods and special functions, while also addressing the practical nature of the two fields. In addition to the strong focus on computational mathematics, scientific computing, and theoretical physics, a special emphasis was given to orthogonal polynomials, the theory of numericalmethods for solution of differential equations, and applications of these fields. A modified generalized Laguerre-Gauss collocation scheme was reported for a class of fractional neutral differential equations on the half-line. Some efficient spectral algorithms were proposed for numerical solution third-order boundary value problems using novel operational matrices of generalized Jacobi polynomials. A mesh-free method for solving nonhomogeneous time-dependent problems was presented. The homotopy perturbation technique was developed for solving the Hamilton-Jacobi-Bellman equation. A 2-point block method for nonstiff higher-order ordinary differential equations was investigated. New bilateral type generating function associated with I-function, certain class of generating functions for the incomplete hypergeometric functions, complex transmission eigenvalues in one dimension, and certain matrices of Bernoulli numbers were other topics of our special issue. Fast and stable implementation of Clenshaw-Curtis and Fejer-type quadrature rules was investigated. An efficient computation of effective ground range using an oblate earth model and the boundary value problem for bihypermonogenic functions in Clifford analysis were presented. The bivariate spectral homotopy analysismethod for nonlinear evolution equations and the fractional Nagumo equationwith nonlinear diffusion and convection were distinct topics of our special issue. A coupled system of fractional integrodifferential equations with m-point boundary conditions on the half-line and global regularity for the ∂b-equation on CR manifolds of arbitrary codimension were analyzed.