Информационная система "Конференции"

Abstracts

Numerical solution of differential and integral equations

Consider a boundary value problem: $$ varepsilonfrac{partial^2 u}{partial x^2}+ varepsilonfrac{partial^2 u}{partial y^2}- a(x,y)frac{partial u}{partial x}-b(x,y)u=f(x,y), (x,y)in D, $$ $$u(x,y)=phi(x,y), (x,y)in partial D, limlimits_{x rightarrow infty}u(x,y)=0 eqno{(1)} $$ for half-infinite strip $ D={0le x0, age alpha>0, bge beta>0, limlimits_{x rightarrow infty}phi_i(x)=0, limlimits_{x rightarrow infty}f(x,y)=0, $$ $$ limlimits_{x rightarrow infty}a(x,y)=a_0(y), limlimits_{x rightarrow infty}b(x,y)=b_0(y). $$ We use a method of lines to transform the problem to a system of ordinary differential equations on an infinite interval. Using lines, we take into account the boundary layers along a strip. Transformed problem has a form: $$ varepsilon {bf V}^{primeprime}-{bf a }(x){bf V}^{prime}-{bf M}(x){bf V}={bf F}(x), $$ $${bf V}(0)={bf A}, limlimits_{x rightarrow infty}{bf V}(x)={bf 0}, eqno{(2)} $$ where ${bf a }(x)$ - diagonal matrix, ${bf M}(x)$ - positive definite matrix. We use technique, connected with extraction of a set of solutions, that satisfy the limit conditions at the infinity to transform problem (2) to a finite interval. Then a problem can be solved by numerical method on an finite interval.

Note. Abstracts are published in author's edition

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