Nonlinear Fredholm Integro-differential Equations Research Articles

A numerical technique for solving nonlinear singularly perturbed Fredholm integro-differential equations

This study deals with two numerical schemes for solving a class of singularly perturbed nonlinear Fredholm integro-differential equations. The nonlinear terms are linearized using the quasi-linearization technique. On the layer adapted Shishkin mesh, the numerical solution is initially calculated using the finite difference scheme for the differential part and quadrature rule for the integral part. The method proves to be first-order convergent in the discrete maximum norm. Then, using a post-processing technique we significantly enhance the accuracy from first order to second order. Further, a hybrid scheme on the nonuniform mesh is also constructed and analyzed whose solution converges uniformly, independent of the perturbation parameter and directly gives second order accuracy. Parameter uniform error estimates are demonstrated and the theoretical results are validated through some numerical tests.

Hermite wavelets collocation method for solving a fredholm integro-differential equation with fractional Caputo-Fabrizio derivative

In this paper, we investigate the numerical study of nonlinear Fredholm integro-differential equation with the fractional Caputo-Fabrizio derivative. We use the Hermite wavelets and collocation technique to approximate the exact solution by reducing the Fredholm integro-differential equation to a nonlinear algebraic system. Furthermore, we apply this numerical method on certain examples to check its accuracy and validity.

A second-order numerical approximation of a singularly perturbed nonlinear Fredholm integro-differential equation

We consider a singularly perturbed problem for a nonlinear first-order Fredholm integro-differential equation. Our aim is to build and analyze a numerical approach with uniform convergence in the ε-parameter. The numerical solution of problem is discretized utilizing interpolation quadrature formulas for the differential part and the composite rectangular rule for the integral part. Error estimations for the approximate solution are established. In support of the idea, numerical examples are provided.

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

In this work, we consider first-order singularly perturbed quasilinear Fredholm integro-differential equation with integral boundary condition. Building a numerical strategy with uniform ε-parameter convergence is our goal. With the use of exponential basis functions, quadrature interpolation rules and the method of integral identities, a fitted difference scheme is constructed and examined. The weight and remainder term are both expressed in integral form. It is shown that the method exhibits uniform first-order convergence of the perturbation parameter. Error estimates for the approximation solution are established and a numerical example is given to validate the theoretical findings.

A solution to a nonlinear Fredholm integro-differential equation

This paper deals with a Fredholm integro-differential equation with nonlinear integral term. By using the Dzhumabaev parametrization method, we reduce the equation to a special Cauchy problem and establish sufficient conditions for the existence of its unique solution. The solution to the special Cauchy problem is used to construct a new general solution to the Fredholm integro-differential equation. We investigate some properties of the new general solution and apply it to boundary value problems.

Linearization-Discretization process to solve systems of nonlinear Fredholm integral equations in an infinite-dimensional context

In this paper, we propose a different way for solving systems of nonlinear Fredholm integral equations of the second kind. We construct our new strategy in two steps, through beginning with the linearization phase of the system of Fredholm integral equations by applying Newton method, then we pass to the discretization phase for some involved integral operator using Nystr\"{o}m method. The convergence analysis of our new method is proved under some necessary conditions. At last, a numerical application to approach a nonlinear Fredholm integro-differential equation by using this new process is taken to confirm its advantage.

Two-Dimensional Fredholm Integro-Differential Equation with Singular Kernel and Its Numerical Solutions

In this paper, we introduce the nonlinear Fredholm integro-differential equation of the second kind with singular kernel in two-dimensional NT-DFIDE. Furthermore, we study this new equation numerically. The existence of a unique solution of the equation is proved. The numerical results of NT-DFIDE are obtained by the following methods: Toeplitz matrix method (TMM) and product Nystrom method (PNM). The given applications showed the efficiency of these methods.

A Procedure for Constructing the Solution of a Nonlinear Fredholm Integro-Differential Equation of Second Order

In this work, a large class of integro-differential equations, arising from the description of heat transfer problems, is considered, particularly the nonlinear equations. We propose a procedure for constructing their solution in a very simple and reliable way in which the only needed tool is the same one employed to solve a linear second-order ordinary differential equation, subject to Robin boundary conditions. Proofs of the convergence, existence, and uniqueness are presented. Some special cases are simulated to illustrate the proposed tools.

Projection methods for approximate solution of a class of nonlinear Fredholm integro-differential equations

The aim of this article is to find the approximate solution of the nonlinear Fredholm integro-differential equations of second kind with smooth kernels with less computational complexity and investigate the asymptotic behavior of convergence of the approximate solutions by using global polynomials based projection methods. We develop the theoretical framework for the nonlinear Fredholm integro-differential equations to obtain the superconvergence results by Legendre polynomial based projection methods and their iterated versions. Numerical examples are considered to demonstrate the theoretical results.

A numerical approach for solving nonlinear Fredholm integro-differential equation with boundary layer

A numerical approach for solving nonlinear Fredholm integro-differential equation with boundary layer

A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations

Physics informed neural networks (PINNs) are a novel deep learning paradigm primed for solving forward and inverse problems of nonlinear partial differential equations (PDEs). By embedding physical information delineated by PDEs in feedforward neural networks, PINNs are trained as surrogate models for approximate solution to the PDEs without need of label data. Due to the excellent capability of neural networks in describing complex relationships, a variety of PINN-based methods have been developed to solve different kinds of problems such as integer-order PDEs, fractional PDEs, stochastic PDEs and integro-differential equations (IDEs). However, for the state-of-the-art PINN methods in application to IDEs, integral discretization is a key prerequisite in order that IDEs can be transformed into ordinary differential equations (ODEs). However, integral discretization inevitably introduces discretization error and truncation error to the solution. In this study, we propose an auxiliary physics informed neural network (A-PINN) framework for solving forward and inverse problems of nonlinear IDEs. By defining auxiliary output variable(s) to represent the integral(s) in the governing equation and employing automatic differentiation of the auxiliary output to replace integral operator, the proposed A-PINN bypasses the limitation of integral discretization. Distinct from the neural network in the original PINN which only approximates the variables in the governing equation, in the proposed A-PINN framework, a multi-output neural network is constructed to simultaneously calculate the primary outputs and auxiliary outputs which respectively approximate the variables and integrals in the governing equation. Subsequently, the relationship between the primary outputs and auxiliary outputs is constrained by new output conditions in compliance with physical laws. By pursuing the first-order nonlinear Volterra IDE benchmark problem, we validate that the proposed A-PINN can obtain more accurate solution than the conventional PINN. We further demonstrate the good performance of A-PINN in solving the forward problems involving nonlinear Volterra IDEs system, nonlinear 2-dimensional Volterra IDE, nonlinear 10-dimensional Volterra IDE, and nonlinear Fredholm IDE. Finally, the A-PINN framework is implemented to solve the inverse problem of nonlinear IDEs and the results show that the unknown parameters can be satisfactorily discovered even with heavily noisy data.

On solvability of boundary value problem for a nonlinear Fredholm integro-differential equation

The paper proposes a constructive method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation. Using D.S. Dzhumabaev parametrization method, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integrodifferential equations with parameters on the subintervals. When applying the parametrization method to a nonlinear Fredholm integro-differential equation, the intermediate problem is a special Cauchy problem for a system of nonlinear integro-differential equations with parameters. By substitution the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem. The iterative methods are used to solve both the constructed system of algebraic equations in parameters and the special Cauchy problem. An algorithm for solving boundary value problem under consideration is provided.

Solutions of Linear and Nonlinear Fractional Fredholm Integro-Differential Equations

The present paper analyzes a class of first-order fractional Fredholm integro differential equations in terms of Caputo fractional derivative. In the literature, such kind of fractional integro-differential equations have been solved using several numerical methods, while the exact solutions were not obtained. However, the exact solutions are obtained in this paper for various linear and nonlinear examples. It is shown that the exact solution of the linear problems is unique, while multiple exact solutions exist for the nonlinear ones. Moreover, the obtained results reduce to the classical ones in the relevant literature as the fractional order becomes unity. The obtained exact solutions can be further invested by other researchers to validate their numerical/approximation methods.

Numerical Solutions of Nonlinear Fredholm Integro Differential Equations using Fifth Order Runge-kutta Method

This paper is considered with the numerical solution of a type of second-order nonlinear Fredholm integro-differential Equations (FIDs). The proposed scheme is based on Runge - kutta methods of order fifth. Moreover, two numerical examples are considered to show the capability and the accuracy of the proposed methods compared with other Runge-kutta methods of less order. Finally, we apply the least square errors (LSE) formula to make numerical comparisons between the numerical and the exact solutions. The obtained results show that the proposed method is superior and more accurate than Runge - kutta methods of orders two, three, and four.

Alpert wavelet system for solving fractional nonlinear Fredholm integro-differential equations

In this paper, we first construct Alpert wavelet system and propose a computational method for solving a fractional nonlinear Fredholm integro-differential equation. Then we create an operational matrix of fractional integration and use it to simplify the equation to a system of algebraic equations. By using Newtons iterative method, this system is solved and the solution of the fractional nonlinear Fredholm integro-differential equations is achieved. Thresholding parameter is used to increase the sparsity of matrix coefficients and the speed of computations. Finally, the method is demonstrated by examples, and then compared results with CAS wavelet method show that our proposed method is more effective and accurate.

Solving nonlinear Fredholm integro-differential equations via modifications of some numerical methods

The paper presents the modifications of the variational iteration method (MVIM), Laplace Adomian decomposition method (MLADM), and the homotopy perturbation method (MHPM) for solving the nonlinear Fredholm integro-differential equation of the second kind. In these methods a series is created, the summation of which gives the solution of the discussed equation. Conditions ensuring convergence of this series are presented in the paper. An example illustrating the usage of the investigated methods is presented as well and the results reveal that the proposed methods are very effective, able, and simple. comparison between our proposed methods with the exact solution and some traditional methods is presented during a numerical example. The results reveal that (MHPM) and (MLADM) lead to an exact solution and (MVIM) leads to a limited solution. The uniqueness of the solutions and the convergence of the proposed methods are also proved.

Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative

In the present article, our aim is to approximate the solution of Fredholm-type integrodifferential equation with Atangana–Baleanu fractional derivative in Caputo sense. For this, we propose a method based on Laplace transform and inverse LT. In our numerical scheme, the given equation is transformed to an algebraic equation by employing the Laplace transform. The reduced equation will be solved in complex plane. Finally, the solution of the given problem is obtained via inverse Laplace transform by representing it as a contour integral. Then, the trapezoidal rule is used to approximate the integral to high accuracy. We have considered linear and nonlinear fractional Fredholm integrodifferential equations to validate our method.

Nonlinear Fredholm integro-differential equation in two-dimensional and its numerical solutions

<abstract> <p>This paper proposes a new definition of the nonlinear Fredholm integro-differential equation of the second kind with continuous kernel in two-dimensional (NT-DFIDE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions of NT-DFIDE are obtained by two powerful methods Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM). The given numerical examples showed the efficiency and accuracy of the introduced methods.</p> </abstract>

On a nonlinear integro-differential equation of Fredholm type

This work concerns an analytical and numerical study for a nonlinear Fredholm integro-differential equation, in this respect, we treat the conditions ensuring the existence and uniqueness of the solution for different varieties of kernels, thereafter, we construct a numerical approach based on the well known Nystrom method. The numerical examples illustrate the efficiency of the proposed method in connection with the existence and uniqueness conditions.

On the Analytical and Numerical Study for Nonlinear Fredholm Integro-Differential Equations

On the Analytical and Numerical Study for Nonlinear Fredholm Integro-Differential Equations