Two-dimensional Nonlinear Volterra Integral Equations Research Articles

Numerical solution of two-dimensional nonlinear Volterra integral equations of the first kind using operational matrices of hybrid functions

In this paper, operational matrices of two-dimensional hybrid of block-pulse functions and Legendre polynomials are employed to approximate the solutions of two-dimensional nonlinear Volterra integral equations of the first kind (2DNVIEF). To this aim, we first convert 2DNVIEF to the two-dimensional linear Volterra integral equations of the second kind (2DLVIES). Then the operational matrices of integration, together with the product operational matrix, can be applied to reduce the 2DLVIES to a system of algebraic equations. The convergence of the proposed method is studied, and an error estimation under the -norm is provided. Finally, some numerical examples are prepared to clarify the accuracy and efficiency of the numerical method especially in dealing with functions that are not continuous in the whole interval.

Solving a Class of Two-dimensional Nonlinear Volterra Integral Equations As a Generalized Problem of Moments

It will be shown that solve an equation two-dimensional Volterra nonlinear can be solved numerically applying the techniques of inverse generalized moments problem in two steps writing the Volterra's equation as a Klein-Gordon equation of the form wtt – wxx = H(x, t), which H(x, t) it is unknown. In a first step, H(x, t) is numerically approximate, and in a second step we numerically approximate the solution of Klein-Gordon equation using the H(x, t) previously approximated. The method is illustrated with examples.

Solving Two-Dimensional Nonlinear Volterra Integral Equations Using Rationalized Haar Functions in the Complex Plane

In this work, two-dimensional rational Haar wavelet method has been used to solve the twodimensional Volterra integral equations. By using fixed point Banach theorem we achieved the order of convergence and the rate of convergence is O(n(2q)n). Numerical solutions of three examples are presented by applying a simple and efficient computational algorithm.

Two-step collocation methods for two-dimensional Volterra integral equations of the second kind

Abstract In this paper, we develop two-step collocation (2-SC) methods to solve two-dimensional nonlinear Volterra integral equations (2D-NVIEs) of the second kind. Here we convert a 2D-NVIE of the second kind to a one-dimensional case, and then we solve the resulting equation numerically by two-step collocation methods. We also study the convergence and stability analysis of the method. At the end, the accuracy and efficiency of the method is verified by solving two test equations which are stiff. In examples, we use the well-known differential transform method to obtain starting values.

A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations

This article describes a numerical scheme to solve two-dimensional nonlinear Volterra integral equations of the second kind. The method estimates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin method results from the numerical integration of all integrals associated with the scheme. In the current work, we employ the composite Gauss-Legendre integration rule to approximate the integrals appearing in the method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The algorithm of the described scheme is computationally attractive and easy to implement on computers. The error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates.

Application of hybrid functions operational matrices in the numerical solution of two-dimensional nonlinear integral equations

In this paper, an effective numerical technique based on the two-dimensional hybrid of Block-pulse functions and Legendre polynomials is presented to approximate the solution of a class of two-dimensional nonlinear Volterra integral equations of the second kind. The main idea of this work is to develop the operational matrices of hybrid functions into two-dimensional mode. These operational matrices are then applied to convert two-dimensional nonlinear integral equations into a system of nonlinear algebraic equations that can be solved numerically by Newton's method. An approximation of the error bound for the proposed method will be presented by proving some theorems. Finally, some numerical examples are considered to confirm the applicability and efficiency of the method, especially in cases where the solution is not sufficiently smooth.

A new method for solving of Darboux problem with Haar Wavelet

In this paper we have introduced a computational method for a class of Darboux problem that change to two-dimensional nonlinear Volterra integral equations, based on the expansion of the solution as a series of Haar functions. Also, by using the Banach fixed point theorem, we get an upper bound for the error of our method. Since our examples in this article are selected from different references, so the numerical results obtained here can be compared with other numerical methods.

Existence of solution for some nonlinear two-dimensional Volterra integral equations via measures of noncompactness

In this paper, we analyze the existence of solution for two-dimensional nonlinear Volterra integral equations (2DVIE) by using the techniques of measures of noncompactness and Petryshyn fixed point theorem which contains as particular cases a lot of integral and functional-integral equations that arise in nonlinear analysis. Also some illustrative examples are given.

Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials

In this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations is discussed. The properties of two-dimensional shifted Legendre functions are presented. The operational matrices of integration and product together with the collocation points are utilized to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. Some results concerning the error analysis are obtained. We also consider the application of the method to the solution of certain partial differential equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

NUMERICAL SOLUTION OF A CLASS OF TWO-DIMENSIONAL NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND

In this work, we investigate solving two-dimensional nonlinear Volterra integral equations of the first kind (2DNVIEF). Here we convert 2DNVIEF to the two-dimensional linear Volterra integral equations of the first kind (2DLVIEF) and then we solve it by using operational approach of the Tau method. But for solving the 2DLVIEF we convert it to an equivalent equation of the second kind and then by giving some theorems we formulate the operational Tau method with standard base for solving the equation of the second kind. Finally, some numerical examples are given to clarify the efficiency and accuracy of presented method.

Extrapolation method of iterated collocation solution for two-dimensional nonlinear Volterra integral equations

In this paper, we study the numerical solution of two-dimensional nonlinear Volterra integral equation by collocation and iterated collocation method. Asymptotic error expansion of iterated collocation solution is obtained. Using Richardson's extrapolation, we show that the accuracy of numerical solution in the iterated collocation method can be improved greatly. Some numerical examples are given to illustrate this theory.