The International Conference on Computational Mathematics ICCM-2004

Abstracts

Numerical solution of differential and integral equations

The approximate solution of hypersingular intgeral equations.

Boikov I.V., Romanova E.G.

Penza State University (Penza)

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Boikov I.V., Romanova E.G.

The approximate solution of hypersingular intgeral equations.

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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.

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Note. Abstracts are published in author's edition

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