Информационная система "Конференции"

Abstracts

Numerical solution of differential and integral equations

A new method of looking for optimum of several values function without assumption about it differentiability properties is presented. The method bases on the dynamic system of the mass point. They move by gravitational law: begin{equation} label{gr_alg} left{ begin{array}{l} vec{a}_i^{ n}=sum_{j=1,j neq i}^{N} frac{F(vec{r}_j^{ n})(vec{r}_j^{ n}-vec{r}_i^{ n})} {|vec{r}_j^{ n}-vec{r}_i^{ n}|^{k_1}} , vec{r}_i^{ n+1} = vec{r}_i^{ n} + lambda_i^nvec{a}_i^{ n}+biggl [frac{1}{2}lambda_i^nvec{a}_i^{ n-1}biggl ], vec{r}_i^{ 0}=vec{r}_{i_0}, vec{a}_i^{ 0}=0. end{array} right. end{equation} Also there are small probabilistic mutinies in the system. If mutinies are in the plane $Or_ur_v$ then we have $$vec r_i={z_{i_1},ldots,z_{i_{u-1}},x_i,z_{i_{u+1}},ldots,z_{i_{v-1}},y_i,z_{i_{v+1}},ldots,z_{i_{k}}}$$ and begin{equation} label{mal_vozm} left{ begin{array}{l} x_i^{n+1}=x_i^n + mu_i^nsum_{j=1,j neq i}^{N} frac{F(vec{r}_j^{ n})(y_j^n-y_i^n)}{|vec{r}_j^{ n}-vec{r}_i^{ n}|^{k_2}} , y_i^{n+1}=y_i^n - mu_i^nsum_{j=1,j neq i}^{N} frac{F(vec{r}_j^{ n})(x_j^n-x_i^n)}{|vec{r}_j^{ n}-vec{r}_i^{ n}|^{k_2}} , z_{i_s}^{n+1}=z_{i_{s}}^n, s neq u, v . end{array} right. end{equation} Condition for the fitness function is only continuity. In this case one was proofed that 'gravitation' method converges on probability to the extremum. The series of tests demonstrates satisfactory work of the presented method. With the help of this method some applied problems were solved. For example the method was used in the problems of optimal oil transportation and cluster analysis.

Note. Abstracts are published in author's edition

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