Abstract
In this paper, the existence and uniqueness of solution of the linear two dimensional Volterra integral equation of the second kind with Continuous Kernel are discussed and proved.RungeKutta method(R. KM)and Block by block method (B by BM) are used to solve this type of two dimensional Volterra integral equation of the second kind. Numerical examples are considered to illustrate the effectiveness of the proposed methods and the error is estimated.
Highlights
1.INTRODUCTION: Two- dimensional integral equations provide an important tool for modeling many problems in engineering and science [1]
From 1960to the present day, many new numerical methods have been developed for the solution of many types of integral equations, such as the Toeplitz matrix method, the product Nyström method, the Galerkin method; R
In [9], the authors solved the TD-NIE of the second kind using degenerate kernel method
Summary
Two- dimensional integral equations provide an important tool for modeling many problems in engineering and science [1]. In [10], obtained numerically the solution of two-dimensional nonlinear Volterra integral equation by collocation and iteration collocation methods. In [11], Guoqiang and Jiong analyzed the existence of asymptotic error expansion of the Nyström solution for two-dimensional nonlinear Fredholm integral equation of the second kind. Theorem 1: If the conditions (i), (ii) and (iii) are verified, equation (1) has auneque solution in the Banach space. Lemma 1: Underthe conditions (i)-(iii), the operator T defined by (3), maps the space C ([0, X] [0, Y]) into itself. Lemma 2: If the conditions (i) and (iii) are satisfied . the operator T is contractive in the Banach space C ([0, X] [0, Y]). The operator T has a unique fixed point which is the unique solution of equation (1)
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