Investigation of an approximate solution of the integral
equation of the exterior Dirichlet boundary value problem
for the Helmholtz equation in the two-dimensional space
Abstract:
The substantiation of the collocation method for the integral equation of the external Dirichlet boundary value problem for the Helmholtz equation in two-dimensional space is given. A new method for constructing a quadrature formula for the potentials of the simple and double layers is proposed, which makes it possible to determine the rate of convergence of these quadrature formulas, on the basis of which the considered integral equation is replaced by a system of algebraic equations, while establishing the existence and uniqueness of a solution to this system. The convergence of the solution of the system of algebraic equations to the value of the exact solution of the integral equation at the reference points is proved, and the rate of convergence of the method is indicated. In addition, a sequence is constructed that converges to an exact solution of the exterior Dirichlet boundary value problem for the Helmholtz equation in two-dimensional space.
Keywords:exterior Dirichlet boundary value problem, Helmholtz equation, potentials of simple and double layers, Hankel function, quadrature formulas, collocation method.
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