Fractional Integro-differential Equations Research Articles

On Implicit Atangana–Baleanu–Caputo Fractional Integro-Differential Equations with Delay and Impulses

On Implicit Atangana–Baleanu–Caputo Fractional Integro-Differential Equations with Delay and Impulses

Investigating integrodifferential equations associated with fractal–fractional differential operators

This study focuses on integrodifferential equations involving fractal–fractional differential operators characterized by exponential decay, power law, and generalized Mittag–Leffler kernels. Utilizing linear growth and Lipschitz conditions, we investigate the existence and uniqueness of solutions, as well as the Hyers–Ulam stability of the proposed equations. For every instance, a numerical method is utilized to derive a numerical solution for the specified equation. The paper includes illustrations of fractal–fractional integrodifferential equations, with their precise solutions determined and subsequently compared with the numerical outcomes. This methodology can be applied to demonstrate convergence, and graphical presentations are included in relevant examples to illustrate our proposed approach.

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-Hyers stability results for a class of fractional integro-differential equations involving nonlocal fractional integro-differential boundary conditions

In this paper, we investigate the existence and uniqueness of solutions for a class of fractional integro- differential boundary value problems involving both Riemann–Liouville and Caputo fractional derivatives, and supplemented with multi-point and nonlocal Riemann-Liouville fractional integral and Caputo fractional deriv- ative boundary conditions. Our results are based on some known tools of fixed point theory. We also study the Ulam–Hyers stability for the proposed fractional problems. Finally, some illustrative examples are included to verify the validity of our results.

Uniqueness and Hyers–Ulam’s stability for a fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition

Abstract Introducing a pair-parameter matrix Mittag–Leffler function, we study the uniqueness and Hyers–Ulam stability to a new fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition using Banach’s contractive principle as well as Babenko’s approach in a Banach space. These investigations have serious applications since uniqueness and stability analysis are essential topics in various research fields. The techniques used also work for different types of differential equations with initial or boundary conditions, as well as integral equations. Moreover, we present a Python code to compute approximate values of our newly established pair-parameter matrix Mittag–Leffler functions, which extend the multivariate Mittag–Leffler function. A few examples are given to show applications of the key results obtained.

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

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

On Some Impulsive Fractional Integro-Differential Equation with Anti-Periodic Conditions

We investigate a class of boundary value problems (BVPs) involving an impulsive fractional integro-differential equation (IF-IDE) with the Caputo–Hadamard fractional derivative (C-HFD). We employ some fixed-point theorems (FPTs) to study the existence of this fractional BVP and its unique solution. The boundary conditions (BCs) established in this study are of a more general type and can be reduced to numerous specific examples by defining the parameters involved in the conditions. In this way, we extend some recent nice results. At the end, we use an example to verify our results.

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 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 progess 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.

Existence of solutions of infinite system of hybrid fractional integro-differential equation

Existence of solutions of infinite system of hybrid fractional integro-differential equation

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

This manuscript proposes a class of fractional stochastic integro-differential equations (FSIDEs) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of approximate solutions, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators, and fixed point approach. Furthermore, we examine the convergence of the Faedo–Galerkin (F–G) approximate solutions to the mild solution of our given problem. Finally, a concrete example involving a partial differential equation is provided to validate the main abstract results.

Laguerre Collocation Approach of Caputo Fractional Fredholm-Volterra Integro-Differential Equations

This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples.

Analysis of Some Semi-analytical Methods for the Solutions of a Class of Time Fractional Partial Integro-differential Equations

Analysis of Some Semi-analytical Methods for the Solutions of a Class of Time Fractional Partial Integro-differential Equations

Solution of fractional integro-differential equations using least squares and shifted Legendre methods

Solution of fractional integro-differential equations using least squares and shifted Legendre methods

LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations

Abstract Generally, fractional partial integro-differential equations (FPIDEs) play a vital role in modeling various complex phenomena. Because of the several applications of FPIDEs in applied sciences, mathematicians have taken a keen interest in developing and utilizing the various techniques for its solutions. In this context, the exact and analytical solutions are not very easy to investigate the solution of FPIDEs. In this article, a novel analytical approach that is known as the Laplace adomian decomposition method is implemented to calculate the solutions of FPIDEs. We obtain the approximate solution of the nonlinear FPIDEs. The results are discussed using graphs and tables. The graphs and tables have shown the greater accuracy of the suggested method compared to the extended cubic-B splice method. The accuracy of the suggested method is higher at all fractional orders of the derivatives. A sufficient degree of accuracy is achieved with fewer calculations with a simple procedure. The presented method requires no parametrization or discretization and, therefore, can be extended for the solutions of other nonlinear FPIDEs and their systems.

Adaptive loss weighting auxiliary output fPINNs for solving fractional partial integro-differential equations

We propose an adaptive loss weighting auxiliary output fractional physics-informed neural networks (AWAO-fPINNs) based on the fractional physics-informed neural networks (fPINNs) for solving fractional partial integro-differential equations. In this framework, the automatic differentiation technique and numerical differentiation algorithm are effectively combined to construct a universal numerical scheme for Caputo fractional derivatives of different orders. Secondly, using the multi-output form of neural networks, the main output represents the required numerical solution and the auxiliary outputs represent the integrals in the equation. The relationships and new constraints between all outputs are obtained through automatic differentiation, successfully avoiding the discretization of the integrals. In addition, an adaptive loss weighting strategy is introduced into the model, which is based on the maximum likelihood estimation to continuously update the adaptive weights to realize the automatic allocation of loss weights, thus improving the accuracy of the model. Finally, we verify the effectiveness and accuracy of the AWAO-fPINNs model via several numerical experiments.

Determination of a Coefficient and Kernel in a Two-dimensional Fractional Integrodifferential Equation

Determination of a Coefficient and Kernel in a Two-dimensional Fractional Integrodifferential Equation

A Spectral Collocation Method for Solving Caputo-Liouville Fractional Order Fredholm Integro-differential Equations

In this paper, a numerical method for solving the fractional order Fredholm integro-differential equations via the Caputo-Liouville derivative is presented. The method uses the well-known shifted Chebyshev expansion and a truncated series to represent the unknown function. It also incorporates numerical integration techniques like the Trapezoidal, Simpson’s 1/3, and Simpson’s 8/3 methods. The paper also provides an approximation for the derivative of an integer. The procedure converts the provided problem into a system of algebraic equations using shifted Chebyshev coefficients and collocation points. The coefficients are found by solving this system using well-known techniques like Newton’s method. Numerical results are presented graphycally to illustrate the applicability, efficacy, and accuracy of the approach presented in this work. All calculations in this study were performed using the MATHEMATICA software program.

Application of a matrix Mittag-Leffler function to the fractional partial integro-differential equation in R n

Application of a matrix Mittag-Leffler function to the fractional partial integro-differential equation in R n