THE APPROXIMATE AND NUMERICAL SOLUTION OF ROMANOVSKIJ LINEAR PARTIAL INTEGRAL EQUATIONS

Abstract

The study of Markov chains with two-way coupling leads to the solution of linear partially integral equations of the second kind in the space of functions continuous on the square. A characteristic feature of the equations is the permutation of  variables for the unknown function under the integral sign and integration over part of the variables. Equations of such types are not Fredholm integral equations and for their study a well-developed theory of  Fredholm integral equations of the second kind can’t be directly applied. The  equations  considered in the article we call partially integral equations of Romanovskij, who first obtained them in the study of Markov chains with two-way coupling and studied these equations in the case of continuous kernels. An explicit solution of partially integral Romanovskij equations can be found in rare cases, and therefore the problem of studying approximate and numerical methods for solving such equations is vital. When using approximate and numerical methods, it should be taken into account that the linear partially integral operator in the Romanovskij equation is not completely continuous, and the direct application of methods associated with the complete continuity of operators for its solution requires justification. The justification of approximate and numerical methods for solving linear partially integral equations of Romanowskij is given in the annotated paper. The paper contains theorems on the solvability of equations, results on various approximate and numerical methods for their solution, the theorem on the solution of linear partially integral equations by Romanovskij, using the method of mechanical quadratures, together with an estimate of the rate of convergence of a numerical solution to an exact solution of this equation.