Fractional Volterra-Fredholm Integro-differential Equations Research Articles

A Numerical Method for Investigating Fractionali Volterra-Fredholm Integro-Differential Model

In this article, we investigate the fractional Volterra-Fredholm integro-differential equations. These equations appear in several applications such as control theory, biology, and particle dynamics in physics. We derive a numerical method based on the operational matrix method to solve this class of integro-differential equations. We prove the existence and uniqueness of the ex-act solution. Additionally, we demonstrate the uniform convergence of the numerical solutions to the exact solution. We present several numerical examples to show the numerical efficiency of the proposed method. In the first example, we choose a linear problem and find that the approximate solution converges to the exact solution when the number of block pulse functions is very large. In the next two examples, we consider the nonlinear case and compute the L2-local truncation error since exact solutions are not available. The error was of order 10−12. Furthermore, we sketch the graph of the approximate solutions for different values of the fractional derivative to observe the influence of the fractional derivative on the profile of the solutions. Theoretical and numerical results show that the proposed method is accurate and can be applied to other nonlinear problems in science.

Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations

Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations

Solving Linear and Nonlinear Fuzzy Fractional Volterra-Fredholm Integro Differential Equations Using Shehu Adomian Decomposition Method

Objectives: In applied sciences and engineering, fuzzy fractional differential equations (FFDEs) and fuzzy fractional integral equations (FFIEs) are a crucial topic. The main objective of this work is to discover an analytical approximate solution for the fuzzy fractional Volterra-Fredholm integro differential equations (FFVFIDE). In the Caputo concept, fractional derivatives are regarded. Methods: The Shehu transform is challenging to exist for nonlinear problems. So, the Shehu transform is combined with the Adomian decomposition method is called the Shehu Adomian decomposition method (SHADM) and has been proposed to solve both linear and nonlinear FFVFIDEs. Findings: Both linear and nonlinear FFVIFIDEs can be solved using this technique. For nonlinear terms, Adomian polynomials have been used. The main benefit of this approach is that it converges quickly to the exact solution. Figures and numerical examples demonstrate the expertise of the suggested approach. Novelty: The comparison between the exact solution and numerical solution is shown in figures for various values of fractional order . The numerical evolution demonstrates the efficiency and reliability of the proposed SHADM. The proposed approach is rapid, exact, and simple to apply and produce excellent outcomes. Keywords: Fractional calculus, fuzzy number, Mittag ­ Leffler function, Shehu Adomian decomposition method, fuzzy fractional Volterra­-Fredholm integro differential equation

Existence, Uniqueness and Stability Results for Nonlinear Neutral Fractional Volterra-Fredholm Integro-Differential Equations

Existence, Uniqueness and Stability Results for Nonlinear Neutral Fractional Volterra-Fredholm Integro-Differential Equations

Symmetrical Solutions for Non-Local Fractional Integro-Differential Equations via Caputo–Katugampola Derivatives

Fractional calculus, which deals with the concept of fractional derivatives and integrals, has become an important area of research, due to its ability to capture memory effects and non-local behavior in the modeling of real-world phenomena. In this work, we study a new class of fractional Volterra–Fredholm integro-differential equations, involving the Caputo–Katugampola fractional derivative. By applying the Krasnoselskii and Banach fixed-point theorems, we prove the existence and uniqueness of solutions to this problem. The modified Adomian decomposition method is used, to solve the resulting fractional differential equations. This technique rapidly provides convergent successive approximations of the exact solution to the given problem; therefore, we investigate the convergence of approximate solutions, using the modified Adomian decomposition method. Finally, we provide an example, to demonstrate our results. Our findings contribute to the current understanding of fractional integro-differential equations and their solutions, and have the potential to inform future research in this area.

A New Approach for Solving a New Class of Nonlinear Optimal Control Problems Generated by Atangana-Baleanu- Caputo Variable Order Fractional Derivative and Fractional Volterra-Fredholm Integro-Differential Equations

In the sequel, the numerical solution of a new class of nonlinear optimal control problems (OCPs) generated by Atangana-Baleanu-Caputo (ABC) variable order (V-O) fractional derivative (FD) and fractional Volterra-Fredholm integro-differential equations (FVFIDEs) is found by Bezier curve method (BCM). The main idea behind this work is the use of the BCM. In this technique, the solution is found in the form of a rapid convergent series. Using this method, it is possible to obtain BCM solution of the general form of multipoint boundary value problems. To shown the efficiency of the developed method, numerical results are stated as the main results in this study.

Solving Fractional Volterra–Fredholm Integro-Differential Equations via A** Iteration Method

In this article, we develop a faster iteration method, called the A∗∗ iteration method, for approximating the fixed points of almost contraction mappings and generalized α-nonexpansive mappings. We establish some weak and strong convergence results of the A∗∗ iteration method for fixed points of generalized α-nonexpansive mappings in uniformly convex Banach spaces. We provide a numerical example to illustrate the efficiency of our new iteration method. The weak w2-stability result of the new iteration method is also studied. As an application of our main results, we approximate the solution of a fractional Volterra–Fredholm integro-differential equation. Our results improve and generalize several well-known results in the current literature.

Spectral Collocation Algorithm for Solving Fractional Volterra-Fredholm Integro-Differential Equations via Generalized Fibonacci Polynomials

In this research article, we build and implement an efficient spectral algorithm for handling linear/nonlinear mixed Volterra-Fredholm integro-differential equations. First, we expand the exact solution as a truncated series of the generalized Fibonacci polynomials, and then we discretize the equation via Simpson's quadrature formula. Finally, we collocate the resulted residual at the roots of the shifted first-kind Chebyshev polynomials. Also, the rate of convergence is studied and the truncation error estimate is reported. Some numerical examples are exhibited to prove the applicability and accuracy of the algorithm.

OPERATIONAL MATRIX OF GENERALIZED BLOCK PULSE FUNCTIONS FOR SOLVING FRACTIONAL VOLTERRA–FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

This article aims to introduce two effective numerical approaches for solving the mixed linear fractional Volterra-Fredholm integro-differential equation (FVIDE) and multi-higher order with initial conditions. Even though these approaches transfer the integro-differential equations into a system of algebraic equations through operational matrices of generalized block pulse functions, these problems can be transferred to a system of algebraic equations by expanding the solution's multi-highest order derivative through the block pulse functions (BPFs). The block pulse operational matrices for fractional-order integration are derived by expanding the Riemann-Liouville fractional integral for repeated fractional integration in block pulse functions (BPFs). Also, the fractional derivative is done by using the generalized operational matrices of BPFs with the fractional calculus properties in the first scheme. The methods are attractive in the calculation, and examples are supplied to explain their use. The estimated and exact outcomes are compared in a table. In addition, they decrease the error terms in the specified domain using the least squares-error technique. For this, most general programs are written in Python V.3.8.8 software (2021).

Some New Uniqueness Results of Solutions for Fractional Volterra-Fredholm Integro-Differential Equations

Some New Uniqueness Results of Solutions for Fractional Volterra-Fredholm Integro-Differential Equations

A wavelet-based novel technique for linear and nonlinear fractional Volterra–Fredholm integro-differential equations

A wavelet-based novel technique for linear and nonlinear fractional Volterra–Fredholm integro-differential equations

Qualitative analysis of a fuzzy Volterra-Fredholm integrodifferential equation with an Atangana-Baleanu fractional derivative

<abstract><p>The point of this work was to analyze and investigate the sufficient conditions of the existence and uniqueness of solutions for the nonlinear fuzzy fractional Volterra Fredholm integro-differential equation in the frame of the Atangana-Baleanu-Caputo fractional derivative methodology. To begin with, we give the parametric interval form of the Atangana-Baleanu-Caputo fractional derivative on fuzzy set-valued functions. Then, by employing Schauder's and Banach's fixed point procedures, we examine the existence and uniqueness of solutions for fuzzy fractional Volterra Fredholm integro-differential equation with the Atangana-Baleanu-Caputo fractional operator. It turns out that the last interval model is a combined arrangement of nonlinear equations. In addition, we consider results by applying the Adams Bashforth fractional technique and present two examples that have been numerically solved using graphs.</p></abstract>

On Fractional Integro-Differential Equation with Nonlinear Time Varying Delay

In this manuscript, we analyze the solution for class of linear and nonlinear Caputo fractional Volterra Fredholm integro-differential equations with nonlinear time varying delay. Also, we demonstrate the stability analysis for these equations. Our paper provides a convergence of semi-analytical approximate method for these equations. It would be desirable to point out approximate results.

Existence, Uniqueness and Stability Results of Fractional Volterra-Fredholm Integro Differential Equations of ψ -Hilfer Type

In this paper, we establish some new conditions for the existence and uniqueness of solutions for a class of nonlinear $\psi$-Hilfer fractional Volterra-Fredholm integro differential equations with boundary conditions. In addition, the Ulam-Hyers stability for solutions of the given problem are also discussed. The desired results are proved by using generalized Gronwall inequality, aid of fixed point theorems due to Banach and Schauder in weighted spaces.

Fixed Point Results via Least Upper Bound Property and Its Applications to Fuzzy Caputo Fractional Volterra–Fredholm Integro-Differential Equations

In recent years, complex-valued fuzzy metric spaces (in short CVFMS) were introduced by Shukla et al. (Fixed Point Theory 32 (2018)). This setting is a valuable extension of fuzzy metric spaces with the complex grade of membership function. They also established fixed-point results under contractive condition in the aforementioned spaces and generalized some essential existence results in fixed-point theory. The purpose of this manuscript is to derive some fixed-point results for multivalued mappings enjoying the least upper bound property in CVFMS. Furthermore, we studied the existence theorem for a unique solution to the Fuzzy fractional Volterra–Fredholm integro-differential equations (FCFVFIDEs) as an application to our derived result involving the Caputo derivative.

Uniqueness and Stability Results for Caputo Fractional Volterra-Fredholm Integro-Differential Equations

In this paper, we established some new results concerning the uniqueness and Ulam’s stability results of the solutions of iterative nonlinear Volterra-Fredholm integro-differential equations subject to the boundary conditions. The fractional derivatives are considered in the Caputo sense. These new results are obtained by applying the Gronwall–Bellman’s inequality and the Banach contraction fixed point theorem. An illustrative example is included to demonstrate the efficiency and reliability of our results

On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative

<abstract><p>This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional ($ \mathcal{GPF} $) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-$ \mathcal{GPF} $ operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations $ (\mathcal{FVFIE}s) $ via generalized fuzzy Hilfer-$ \mathcal{GPF} $ Hukuhara differentiability ($ \mathcal{HD} $) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy $ \mathcal{FVFIE}s $ which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.</p></abstract>

New results on nonlocal fractional Volterra-Fredholm integro-differential equations

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence and uniqueness of solutions. Results of this paper are based on nonstandard assumptions and hypotheses and provide a supplementary result concerning the regularity of solutions.

Darbo-Type $ \mathcal{Z}_{\rm{m}} $ and $ \mathcal{L}_{\rm{m}} $ contractions and its applications to Caputo fractional integro-differential equations

Darbo type $ \mathcal{Z}_{\rm{m}} $-contraction and Darbo type $ \mathcal{L}_{\rm{m}} $-contraction are introduced and some fixed point results are established for such contraction mappings. As an application, we prove the existence of solution of a Caputo fractional Volterra-Fredholm integro-differential equation via integral type boundary conditions and verify the validity of our application by an appropriate example.

Existence and Uniqueness of Solutions for Fractional Neutral Volterra-Fredholm Integro Differential Equations

The topic fractional calculus can be measured as an old as well as a new subject. In the fractional calculus the various integral inequalities plays an important role in the study of qualitative and quantitative properties of solution of differential and integral equations. In this paper, we study the existence and uniqueness of solutions for the neutral Caputo fractional Volterra-Fredholm integro differential equations with fractional integral boundary conditions by means of the Arzela-Ascoli's theorem, Leray-Schauder nonlinear alternative and the Krasnoselskii fixed point theorem. New conditions on the nonlinear terms are given to pledge the equivalence. An example is provided to illustrate the results.