Институт вычислительной математики
и математической геофизики

The International Conference on Computational Mathematics


Numerical solution of differential and integral equations

The approximate solution of hypersingular intgeral equations.

Boikov I.V., Romanova E.G.

Penza State University (Penza)




Boikov I.V., Romanova E.G.

The approximate solution of hypersingular intgeral equations.


The paper is devoted to the approximate methods for solution of hypersingular intgeral equations of the form

$$ a(t_1,t_2)x(t_1,t_2)+frac{b(t_1,t_2)}{pi^2} intlimits^1_{-1}intlimits^1_{-1} frac{x( au_1, au_2)}{( au_1-t_1)^p ( au_2-t_2)^p}d au_1 d au_2+ $$

$$ +intlimits^1_{-1}intlimits^1_{-1} h(t_1,t_2, au_1, au_2)x( au_1, au_2)d au_1 d au_2=f(t_1,t_2), eqno (1) $$

where $p=1,2,3,cdots.$

In the case $p ge 2$ the integral is interpreted as Hadamard integral.

For equation (1) we present the spline-collocation type computing circuit

in the case $a,b,f in W^{r,r},$ $h in W^{r,r,r,r},$ $r > 2p.$

We also consider the equation

$$ frac{1}{pi}intlimits^{frac{pi}{2}}_{-frac{pi}{2}} left(frac{cos t}{sin au - sin t} +frac{1}{2} cth frac{ au-t}{2} ight) x( au)d au+ $$

$$ + frac{1}{pi}intlimits_{-frac{pi}{2}}^{frac{pi}{2}} sgn(c( au-t))(e^{-id| au-t|}-1)x( au)d au=f(t), quad t in left(-frac{pi}{2}; frac{pi}{2} ight) eqno (2) $$

in the class of complex-valued functions of $H_{alpha}(1),$ $0< alpha le 1.$

Here $c$ and $d-$ are the constants, having a certain physical meaning.

The equation (2) describes the diffusion of electromagnetic waves in the waveguide.

The numerical results verify the effectiveness of the suggested algorithms.


Note. Abstracts are published in author's edition

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