Nonlinear Fractional Integro-differential Equations Research Articles

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.

Solutions of Second-Order Nonlinear Implicit ψ-Conformable Fractional Integro-Differential Equations with Nonlocal Fractional Integral Boundary Conditions in Banach Algebra

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

In literature, it is usually very difficult to investigate the analytical and numerical solutions of fractional integro-differential equations (FIDEs). In the current work, the solutions to linear and non-linear FIDEs and their systems have been analyzed by using the Aboodh transform decomposition method (ATDM). The Aboodh transformation is first utilized to simplify the given problem, and then the decomposition method is implemented to obtain the required results. The Adomian polynomials and Daftardar–Jafari polynomials are used to control the non-linear term in the system. The obtained results are compared with both polynomials and with different fractional orders. We show that Daftardar–Jafari polynomials work more accurately for the proposed method than Adomian polynomials. The graphical and tabular representations are made to show the effectiveness of the proposed technique. The accuracy of ATDM is remarkable while using the proposed technique. It is observed that ATDM solutions very closed to the exact solutions of the problems. ATDM has a lower computation cost and higher rate of convergence and thus can be used to solve other non-linear fractional order integro-differential equations and their systems. Finally, we propose some future research directions by utilizing ATDM.

Euler wavelets method for optimal control problems of fractional integro-differential equations

This research introduces an efficient direct method for finding the numerical solution to optimal control problems involving linear and nonlinear fractional integro-differential equations using orthonormal Euler wavelets. Using orthonormal Euler wavelets, fractional integral, and product operational matrices have been obtained. By using these fractional operational matrices and collocation points, optimal control problems of fractional Volterra integro-differential equations have been reduced to a system of linear or non-linear equations. We establish the convergence analysis and error bound of the proposed method. Orthonormal Euler wavelets have been utilized to solve numerical examples and determine the approximate value of the cost function by approximating state and control functions. This has been done in order to verify the effectiveness of the proposed numerical technique.

Existence analysis on multi-derivative nonlinear fractional neutral impulsive integro-differential equations

This article utilizes the Atangana–Baleanu (AB) fractional derivative to examine the behavior of multi-derivative fractional neutral impulsive integro-differential equations under non-local conditions. To demonstrate the existence, uniqueness, and controllability of the solutions, we utilize fixed point theory as our primary analytical tool. Furthermore, we include a detailed example to illustrate and validate the theoretical results obtained from our study.

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

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

Euler-Maruyama approximation for stochastic fractional neutral integro-differential equations with weakly singular kernel

This manuscript examines the problem of nonlinear stochastic fractional neutral integro-differential equations with weakly singular kernels. Our focus is on obtaining precise estimates to cover all possible cases of Abel-type singular kernels. Initially, we establish the existence, uniqueness, and continuous dependence on the initial value of the true solution, assuming a local Lipschitz condition and linear growth condition. Additionally, we develop the Euler-Maruyama method for numerical solution of the equation and prove its strong convergence under the same conditions as the well-posedness. Moreover, we determine the accurate convergence rate of this method under global Lipschitz conditions and linear growth conditions.

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>

Ulam-Hyers and generalized Ulam-Hyers stability of fractional functional integro-differential equations

Unique solvability conditions for general fractional functional integro-differential equations with nonlocal initial conditions are established using fixed-point theory. Ulam-Hyers and generalized Ulam-Hyers stability conditions for a solution of these problems are formulated. An example of the nonlinear fractional functional integro-differential equation related to wheel shimmy models is considered as well.

Coupled Nonlocal Boundary Value Problems for Fractional Integro-differential Langevin System via Variable Coefficient

In this paper, we aim to study a new coupled system of nonlinear fractional integro-differential Langevin equations with coupled multipoint boundary conditions. The existence and uniqueness of solution are investigated by using the Banach’s and Krasnoselskii’s fixed point theorems. The Ulam-Hyers stability of the mentioned equation is provided by applying the classical technique of functional analysis. Two examples are presented to verify our analysis.

On Hadamard-Caputo Implicit Fractional Integro-Differential Equations With Boundary Fractional Conditions

The purpose of this paper is to investigate the existence and uniqueness of solutions for nonlinear fractional implicit integro-differential equations of Hadamard-Caputo type with fractional boundary conditions. The reasoning is inspired by diverse classical fixed point theory, such as the Schauder and Banach fixed point theorems. The theoretical findings are illustrated through an example.

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.