Nonlinear Fractional Integro-differential Equations Research Articles

Numerical solution of nonlinear fractional delay integro-differential equations with convergence analysis

Numerical solution of nonlinear fractional delay integro-differential equations with convergence analysis

Correction: Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

Correction: Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types

Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems.

Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

Existence and Ulam stability of mild solutions for nonlinear fractional integro-differential equations in a Banach space

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

An efficient numerical algorithm for solving nonlinear fractional Volterra integro-differential equation

Legendre spectral collocation method for solving nonlinear fractional Fredholm integro-differential equations with convergence analysis

<abstract><p>The main purpose of this work was to develop a spectrally accurate collocation method for solving nonlinear fractional Fredholm integro-differential equations (non-FFIDEs). A proposed spectral collocation method is based on the Legendre-Gauss-Lobatto collocation (L-G-LC) method in which the main idea is to use Caputo derivatives and Legendre-Gauss interpolation for nonlinear FFIDEs. A rigorous convergence analysis is provided and confirmed by numerical tests. In addition, we provide some numerical test cases to demonstrate that the approach can preserve the non-smooth solution of the underlying problem.</p></abstract>

Investigation on integro-differential equations with fractional boundary conditions by Atangana-Baleanu-Caputo derivative.

We establish, the existence and uniqueness of solutions to a class of Atangana-Baleanu (AB) derivative-based nonlinear fractional integro-differential equations with fractional boundary conditions by using special type of operators over general Banach and Hilbert spaces with bounded approximation numbers. The Leray-Schauder alternative theorem guarantees the existence solution and the Banach contraction principle is used to derive uniqueness solutions. Furthermore, we present an implicit numerical scheme based on the trapezoidal method for obtaining the numerical approximation to the solution. To illustrate our analytical and numerical findings, an example is provided and concluded in the final section.

Existence of solutions for fractional integro-differential equations with integral boundary conditions

"The aim of this study is to prove the existence of solutions for Caputo boundary value problems of nonlinear fractional integro-differential equations with integral boundary conditions, by using the measure of non compactness combined with Mönch's fixed point theorem. Two examples are offered to demonstrate our outcomes."

A block-by-block approach for nonlinear fractional integro-differential equations

In this paper, a block-by-block scheme is proposed for a class of nonlinear fractional integro-differential equations. This method is based on the Gauss–Lobatto numerical integration method, which shows the high accuracy at all time intervals. Also, the method convergence for this type of equations is proved and it is shown that the order of convergence is at least eight. Finally, the high accuracy, fast calculations and good performance of the method are investigated by solving some numerical examples.

A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space

This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed.

A new wavelet method for fractional integro-differential equations with [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative is successfully used to define a new class of nonlinear fractional integro-differential equations. The extended Chebyshev cardinal wavelets (CCWs), as a proper set of wavelet functions, are employed to establish a computational procedure for such equations. To do this, a new operational matrix for the ψ-Riemann–Liouville fractional integral of the extended CCWs is extracted with the help of the block-pulse functions. The established approach obtains the problem solution by solving an algebraic system of nonlinear equations that is created from expanding the solution by the extended CCWs (with some unknown coefficients) and utilizing the expressed fractional operational matrix. In fact, by solving this system, the unknown coefficients and, subsequently the solution of the original problem are obtained. The convergence analysis of the proposed method is investigated. The accuracy and correctness of the scheme are illustrated by solving three examples.

Variational Iteration Approach for Solving Fractional Integro-Differential Equations with Conformable Differintegrals

Fractional integro-differential equations involving conformable differintegrals have promising applications but lack solution methods. This work develops a variational iteration approach for such equations. Conformable fractional derivatives and integrals that generalize integer-order equivalents are defined. Existence and uniqueness results are established for a class of nonlinear fractional integro-differential equations. These exist under less restrictive smoothness assumptions than integer-order cases. The variational iteration method (VIM) is then applied, providing an analytical approximate technique for integro-differential problems of fractional order. Convergence analysis demonstrates the efficiency and accuracy of the VIM solutions. Several test cases validate the VIM, matching analytical solutions available for simpler fractional differential sub-cases. The proposed technique advances available methods for this emerging class of fractional integro-differential equations. Significantly, it enables application of such models by allowing accurate solution of associated mathematical system representations. Extensions to include more singular equation classes, comparisons with other methods, and real-world applications are suggested as future work.
 The abstract summarizes the key points: significance of problem, fractional models used, existence/uniqueness proofs, VIM approach and analysis, test case validations, implications for applications

A hybrid technique for approximating the solution of fractional order integro differential equations

In this article, we present an effective approach for solving nonlinear fractional order integro-differential equations. The fractional order derivative will be in the Caputo sense. For this, we propose an approach combined the least squares method together with the Laplace transform and the shifted Legendre polynomials. Using the proposed approach we will converts the nonlinear fractional order integro-differential equation into a se of (N+1) algebraic equations, where the solution to the resultant equation provides us with the unknown coefficients of the infinite series that have been used to approximate the solution to the considered fractional order integro-differential equations. From the results of the given examples, one can conclude that the recommended approach in this article is simple to implement and accurate.

Nonexistence Results for a System of Nonlinear Fractional Integrodifferential Equations

Nonexistence Results for a System of Nonlinear Fractional Integrodifferential Equations

An efficient spline technique for solving time-fractional integro-differential equations

Spline curves are very prominent in the mathematics due to their simple construction, accuracy of assessment and ability to approximate complicated structures into interactive curved designs. A spline is a smooth piece-wise polynomial function. The primary goal of this study is to use extended cubic B-spline (ExCuBS) functions with a new second order derivative approximation to obtain the numerical solution of the weakly singular kernel (SK) non-linear fractional partial integro-differential equation (FPIDE). The spatial and temporal fractional derivatives are discretized by ExCuBS and the Caputo finite difference scheme, respectively. The present study found that it is stable and convergent. The validity of the current approach is examined on a few test problems, and the obtained outcomes are compared with those that have previously been reported in the literature.

Qualitative Analysis for Solving a Fractional Integro-Differential Equation of Hyperbolic Type with Numerical Treatment Using the Lerch Matrix Collocation Method

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space CEE, J. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters λ and time t as well as how the absolute error in each example is calculated.

A second-order finite difference scheme for nonlinear tempered fractional integrodifferential equations in three dimensions

A second-order finite difference scheme for nonlinear tempered fractional integrodifferential equations in three dimensions

On stability analysis of a fractional volterra integro delay differential equation in the context of Mittag–Leffler kernel

This paper deals with a nonlinear multi-derivative fractional Volterra integro-differential equation with finite delay involving a non-singular Mittag–Leffler kernel. Krasnoselskii's fixed-point theorem is used to obtain the current result. Also, we establish the generalization of Pachapate's inequality and provide its applications for the existence, uniqueness, stability, and data dependence of the suggested problem. All the results are justified by an example.

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