To the approximate solution of a singular integro-differential equation

M. G. Akhmadiev

Taurida Journal of Computer Science Theory and Mathematics, no. 1 (2024), pp. 13-20

Résumé

In this paper, we propose a method for approximate solution of the singular integro-differential equation of wing theory (Problem A) $$ B(t)\Gamma(t)-\frac{1}{\pi}\int_{-1}^1\frac{\Gamma'(\tau)}{\tau-t}d\tau=y(t) \,\,\,(-11), $$ under the initial condition $\Gamma(t_0 )=0$ $(-1 \leqslant t_0\leqslant1)$, where $B(t)$ and $y(t)$ are given functions, $\Gamma(t)$ is the desired function, and the singular integral is understood in the sense of the main value in the Cauchy sense. Many problems of hydromechanics and elasticity theory lead to the solution of problem A. In particular, this problem arises in connection with solving one problem of filtration theory, namely, when solving a flat problem, pressure filtration is established under a dam in an inhomogeneous isotropic permeable soil. Possible solutions of the equation of problem A are divided into the following classes according to N.\;I.\;Muskhelishvili $h_0$, $h(-1)$, $h(1)$, $h(-1,1)$ with respect to $\Gamma'(t)$. It follows that the derivative of the desired solution can be represented as $\Gamma'(t)=P(t)x(t)$, where $P(t)=1/{\sqrt{1-t^2}}$ for class $h_0$, $P(t)=\sqrt{(1+t)/(1-t)}$ and $P(t)=\sqrt{(1-t)/(1+t)}$ for classes $h(-1)$ and $h(1)$, respectively, $P(t)=\sqrt{1-t^2}$ for class $h(-1,1)$, and $x(t)$ is a new unknown function, quadratically summed by $[-1,1]$ with weight $P(t)$. It is possible to solve exactly such problems using analytical methods developed from a theoretical point of view in rare special cases. Therefore, it is necessary to resort to approximate solution methods that give an approximate analytical expression for the desired function, for which a general theory of approximate methods has been developed. In this work, computational schemes of the least squares method for solving problem A are constructed, based on approximations of the solution by polynomials, in various classes of solutions. The theoretical justification of the method is carried out on the spaces of quadratically-summable functions with weight on the basis of one of the variants of the general theory of approximate methods of functional analysis proposed by B. G. Gabdulkhaev. In particular, the unambiguous solvability of systems of linear algebraic equations with respect to unknown coefficients $\{\alpha_k\}$ of the approximate solution is proved, the convergence of approximate solutions to the exact solution is established, the convergence rate of the method is established.

Détail

Keywords: Singular integro-differential equations, filtration theory, quadratic summability, least squares method, approximate solution, convergence of the method.