Fractional Integro-differential Equations Research Articles

Least Squares Approximation Method for Estimation of Volterra Fractional Integro-differential equation using Hermite Polynomial as the basis functions

This research work utilizes the least squares approximation method to estimate approximate solutions for fractional-order integro-differential equations, with Hermite polynomials as basis functions. The process begins by assuming an approximate solution of degree N, which is then substituted into the fractional-order integro-differential equation under investigation. After evaluating the integral, the equation is rearranged to isolate one side, allowing the application of the least squares method. Three examples were solved using this approach. In Example 1, the numerical results for α=0.9 and α=0.8 were compared to the exact solution for α=1. In Examples 2 and 3, the results for α=1.9 and α=1.8 were compared to the exact solution for α=2. These comparisons showed favorable alignment with the exact solutions. The numerical results and graphical illustrations demonstrate the validity, competence, and accuracy of the proposed method.

Solutions for non-autonomous fractional integrodifferential equations with delayed force term

Solutions for non-autonomous fractional integrodifferential equations with delayed force term

RESULTS ON IMPULSIVE ψ-CAPUTO FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS

In this research article, we investigate the existence and uniqueness of solutions for a type of boundary value problems involving fractional integro-differential equations with the ψ-Caputo fractional derivative. Our method utilizes several classical fixed point theorems, such as the Banach contraction principle and Krasnoselskii's fixed point theorem. We establish our main results through theoretical analysis and present a specific case study to illustrate the practical significance of our findings.

Existence of mild solutions for non-instantaneous impulsive ξ-Caputo fractional integro-differential equations

The aim of this paper is to investigate the existence of mild solutions for a nonlocal ξ-Caputo fractional non-instantaneous impulses semilinear integro-differential equation in a Banach space. The proofs are based on some fixed point theorems for condensing maps. As an application, an example is given to illustrate our theoretical results.

Existence of extremal solutions for a class of fractional integro-differential equations

In the study the existence of solutions of a class of fractional integro-differential equations with boundary conditions was considered. The main tool, we employ, is the conventional monotone iterative technique, which is highly effective method to examine the quantitative and qualitative characteristics of various nonlinear problems. This technique produces monotone sequences whose iterations are unique solutions of the certain linear problems. These bounds converge uniformly to the maximal solutions of the given problems. Some types of coupled solutions are considered to obtain the claim of the main results under suitable conditions.

An RBF Method for Time Fractional Jump-Diffusion Option Pricing Model under Temporal Graded Meshes

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method.

Theoretical and Numerical Studies of Fractional Volterra-Fredholm Integro-Differential Equations in Banach Space

This paper examines the theoretical, analytical, and approximate solutions of the Caputo fractional Volterra-Fredholm integro-differential equations (FVFIDEs). Utilizing Schaefer's fixed-point theorem, the Banach contraction theorem and the Arzel\`{a}-Ascoli theorem, we establish some conditions that guarantee the existence and uniqueness of the solution. Furthermore, the stability of the solution is proved using the Hyers-Ulam stability and Gronwall-Bellman's inequality. Additionally, the Laplace Adomian decomposition method (LADM) is employed to obtain the approximate solutions for both linear and non-linear FVFIDEs. The method's efficiency is demonstrated through some numerical examples.

On Fuzzy Fractional Volterra-Fredholm Model Under the Uncertainty θ-Operator of the AD Technique: Theorems and Applications

This article investigates the proper existence conditions and uniqueness results for a class of fuzzy fractional Caputo Volterra-Fredholm integro-differential equations (FFCV-FIDE) with initial conditions. The findings are based on Banach's contraction principle and Schaefer's fixed point theorem. Furthermore, the solution to the posed problem is found using the Adomian decomposition technique (ADT). We support the concept with several examples. The relationship between the upper and lower reduced approximation of the fuzzy solutions was demonstrated numerically and graphically using MATLAB.

The existence of a unique solution and stability results with numerical solutions for the fractional hybrid integro-differential equations with Dirichlet boundary conditions

In this paper, we investigate the fractional hybrid integro-differential equations with Dirichlet boundary conditions. We first prove the existence of a unique solution for the equation using a fixed point technique. Our main goal is to obtain the best approximation using optimal controllers. After studying the stability, we present the reproducing kernel Hilbert space numerical method to obtain approximate solutions to the equation. We finally conclude with numerical results.

Double Laplace Transform Method for Solving Fractional Fourth-Order Partial Integro-Differential Equations with Weakly Singular Kernel

Objectives: To investigates the solutions of fourth order partial integro-differential equations with high-order non-integer derivatives and weakly singular kernels. Methods: Weakly singular kernels present challenges in both analytical and numerical treatments due to their intricate behaviour near singular points. In this article, we introduce a novel approach utilizing the double Laplace transform method to effectively address these challenges. Findings: By solving a series of precise and understandable examples, the double Laplace transform clearly transforms the fractional partial integro-differential equation into an algebraic equation that can be solved easily. Novelty: The double Laplace transform is an effective instrument for solving fractional partial integro-differential equations, which are commonly used in fields such as physics, fluid mechanics, gas dynamics, and signal processing. Mathematics Subject Classifications: 35R09, 35R11, 44A10, 44A30. Keywords: Fractional PIDE, Weakly singular kernel, Double Laplace transform, Inverse double Laplace transform, Exact solution

The moving least square method for solving the time fractional partial integro-differential equation of Volterra type and its convergence analysis

The moving least square method for solving the time fractional partial integro-differential equation of Volterra type and its convergence analysis

Computational Approach for Two-Dimensional Fractional Integro-Differential Equations

Computational Approach for Two-Dimensional Fractional Integro-Differential Equations

A novel approach for solving weakly singular fractional integro-differential equations

In this essay, we introduce a novel idea to tackle the challenges of fractional integro-differential equations (FIDEs) featuring weakly singular kernels (WSKs). New idea leverages B-splines for solving such equations, offering a robust numerical solution technique. We delve into the operational matrices integral to this method, providing comprehensive insights into their functionality. Furthermore, we establish the convergence of our approach and substantiate its effectiveness through various numerical examples.

Editorial for the Special Issue of “Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis”

Editorial for the Special Issue of “Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis”

A numerical method for Ψ-fractional integro-differential equations by Bell polynomials

In this work, we focus on a class of Ψ− fractional integro-differential equations (Ψ-FIDEs) involving Ψ-Caputo derivative. The objective of this paper is to derive the numerical solution of Ψ-FIDEs in the truncated Bell series. Firstly, Ψ-FIDEs by using the definition of Ψ− Caputo derivative is converted into a singular integral equation. Then, a computational procedure based on the Bell polynomials, Gauss-Legendre quadrature rule, and collocation method is developed to effectively solve the singular integral equation. The convergence of the approximation obtained in the presented strategy is investigated. Finally, the effectiveness and superiority of our method are revealed by numerical samples. The results of the suggested approach are compared with the results obtained by extended Chebyshev cardinal wavelets method (EChCWM).

On One Point Singular Nonlinear Initial Boundary Value Problem for a Fractional Integro-Differential Equation via Fixed Point Theory

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order θ∈[0,1]. The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented.

A Numerical Scheme of a Fractional Coupled System of Volterra Integro-Differential Equations with the Caputo Fabrizio Fractional Derivative

In this paper, we have developed a new computational scheme for solving coupled systems of fractional order Volterra-type integro-differential equation (FVIDE). We construct new operational matrices, which serve as building blocks for converting the FVIDE into a Sylvester-type algebraic structure that is more easily solvable. The fractional derivatives and their inverses (integrals) are considered in the Caputo-Fabrizio sense. This computational scheme is based on Legendre polynomials. The assessment of the proposed method’s convergence involves utilizing appropriate error norms to measure the level of convergence. The algorithm is designed in such a way that it can be simulated using any computational package; we have used Matlab for simulating the proposed scheme. Graphical visualization and tabulation of data are performed to confirm convergence and to analyze errors.

New attitude on sequential Ψ-Caputo differential equations via concept of measures of noncompactness

In this paper, we have explored the existence and uniqueness of solutions for a pair of nonlinear fractional integro-differential equations comprising of the Ψ-Caputo fractional derivative and the Ψ-Riemann–Liouville fractional integral. These equations are subject to nonlocal boundary conditions and a variable coefficient. Our findings are drawn upon the Mittage–Leffler function, Babenko’s attitude, and topological degree theory for condensing maps and the Banach contraction principle. To further elucidate our principal outcomes, we have presented two illustrative examples.

The fast Euler-Maruyama method for solving multiterm Caputo fractional stochastic delay integro-differential equations

The fast Euler-Maruyama method for solving multiterm Caputo fractional stochastic delay integro-differential equations

Stochastic fractional integrodifferential equations with jumps: application to an averaging principle

In this work we deal with a stochastic fractional integrodifferential equation associated to a Poisson random measure. We first prove existence and uniqueness of solution in the case of Lipschitz coefficients but also in the non Lipschitz case. In the second part, we establish an averaging principle in the sense of Khasminskii approach for a class of this equation with non lipschitz coefficients and weak averaging conditions.