Revisiting the Mixed Problem for Equations of Compound and Hyperbolic Types of Order Four

Abstract

As is known, one of the methods of mathematical modeling of processes which occur in heterogeneous environments and sectionally homogeneous environments, and besides, under concentrated factors, is their reduction to studying of matching problems for equations in partial derivatives of order two, order three, and order four. Please note that the problems with nonlocal matching conditions for equations in partial derivatives are often used as an arithmetic model of the heat-transfer process in the compound system with various thermal and physical characteristics. At the same time, it should be mentioned that matching problems for equations in partial derivatives of order four have been studied rather marginally. As the degree of equation increases in partial derivatives, such matters as for example the correct statement of the problem, the model of solution of the problem, the number of hitching conditions and problem solvability are not always obvious and require a detailed study in every particular case. The relevance is stipulated by the proof of correctness of the matching problem for equations of compound and hyperbolic types of order four with minor terms. The authors of the article have proved unique solvability of the mixed problem in the rectangular region for equations of compound and hyperbolic types of order four using the methods of the theory of equations of mixed and mixed-compound types. The importance of the study of high-order equations of compound and hyperbolic types is pointed out in papers written by such authors as Bitsadze, A.V., Dzhuraev, T.D., Sopuev, A.S. et al. When the mixed problem is solved, the basic classical problems like Dirichlet problem and Goursat problem are commonly used. The solution of the mixed problem is split into two independent problems in respective subregions of regions under consideration. Auxiliary problems are equivalently reduced to integral Fredholm and Volterra equations of order two. Solvability conditions of obtained integral equations have been specified. Based on the solution of auxiliary problems, sufficient conditions of existence of a unique solution of the mixed problem have been found. The formula of the model of solution of the mixed problem has been obtained. The example of the problem with particular conditions on the portion of the boundary of the region under consideration has been set. Explicit hitching and matching conditions between these problems and auxiliary functions on the line of change of the type of equation have been presented. The formula of the model of solution of the problem in an explicit form has been obtained.