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Solvability of a system of dual integral equations of a mixed boundary value problem for the laplace equation

The aim of the present paper is to consider solvability and solution of a system of dual integral equations involving Fourier transforms occurring in mixed boundary value problems for Laplace equation. The uniqueness and existence theorems are proved. A method for reducing system of dual equations to a system of Fredholm integral equations of second kind is also proposed. | SOLVABILITY OF A SYSTEM OF DUAL INTEGRAL EQUATIONS OF A MIXED BOUNDARY VALUE PROBLEM FOR THE LAPLACE EQUATION Nguyen Thi Ngan1 and Nguyen Thi Minh, College of Education -TNU Summary. The aim of the present paper is to consider solvability and solution of a system of dual integral equations involving Fourier transforms occurring in mixed boundary value problems for Laplace equation. The uniqueness and existence theorems are proved. A method for reducing system of dual equations to a system of Fredholm integral equations of second kind is also proposed. Key words: Fourier transform, dual integral equations, Laplace equation, mixed boundary value problems. 1 Introduction Dual integral equations arise when integral transforms are used to solve mixed boundary value problems of mathematical physics and mechanics. Formal technique for solving such equations have been developed extensively, but their solvability problems have been considered comparatively weakly [1, 2]. The solvabilities of dual integral equations involving Fourier transforms and dual series equations involving orthogonal expansions of generalized functions were considered in [3, 4]. The solvability problems for systems of dual equations so far we know have been not considered. The aim of the present work is to consider existence and uniqueness problems for a system of dual integral equations involving Fourier transforms occurring in one mixed boundary value problem for the Laplace equation. A method for reducing system of dual equations to a system of Fredholm integral equations of second kind is also proposed. Consider the following problem: find a solution of the Laplace equation ∂2Φ ∂2Φ + = 0, ∂x2 ∂y 2 L1 (R)), direct and inverse Fourier transforms are defined by the formulas Z ∞ fˆ(ξ) = F [f ](ξ) = f (x)eixξ dx, (1.4) −∞ Z ∞ 1 f (x)e−ixξ dx. (1.5) f˘(ξ) = F −1 [f ](ξ) = 2π −∞ The Fourier transforms of tempered generalized functions can be found, for example, in [5,6]. The formulated problem .