Информационная система "Конференции"

Abstracts

Numerical solution of differential and integral equations

Since many new applications of the Volterra integral equations have appeared for the last years, numerical methods for these equations are actively developing. This paper is dedicated to construction the optimal on the accuracy order methods of approximate solving the multidimensional Volterra and Abel-Volterra integral equations, based on the polynomial splines approximation of the exact solution. We consider the equations of the following types $$ x(t_1,ldots,t_l)+$$ $$ +intlimits_{0}^{t_l}cdotsintlimits_{0}^{t_1} h(t_1,ldots,t_l,tau_1,ldots,tau_l)g(t_1-tau_1,ldots,t_l-tau_l) x(tau_1,ldots,tau_l)dtau_1cdots dtau_l= $$ $$=f(t_1,ldots,t_l),$$ where $0le t_1,ldots,t_lle T,; h(t_1,ldots,t_l,tau_1,ldots,tau_l)$ and $f(t_1,ldots,t_l)$ are the functions, having partial derivatives till certain order $m$. Weakly singular kernels $g(t_1-tau_1,ldots,t_l-tau_l)$ may have the form $$ g(t_1,ldots,t_l)=t_1^{r_1+alpha_1}cdots t_l^{r_l+alpha_l}, ; 0 < alpha_i < 1 $$ or $$ g(t_1,ldots,t_l)=(t_1^2+cdots+t_l^2)^{r+alpha}, ; 0 < alpha < 1, $$ or $g(t_1,ldots,t_l)equiv 1$ in the regular equation case. Here we present the algorithm of parallelization of the constructed schemes to realize them on the computers with two and more processors. vspace{1cm} The work has been supported by Russian Humanitarian Scientific Fund (RHSF) (Research Grant Nr. 01-02-00147a)

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