Goursat Problem in the Non-Classical Treatment for a Sixth Order Pseudoparabolic Equation

Ilgar G Mamedov

doi:10.13189/ujcmj.2013.010103

Abstract

In this paper substantiated for a differential equation of pseudoparabolic type with discontinuous coefficients a Goursat problem with non-classical boundary conditions is considered, which requires no matching conditions. Equivalence of these conditions boundary condition is substantiated classical, in the case if the solution of the problem in the anisotropic S. L. Sobolev's space is found. The considered equation as a pseudoparabolic equation generalizes not only classic equations of mathematical physics (heat-conductivity equations, string vibration equation) and also many models differential equations (telegraph equation, Aller's equation , moisture transfer generalized equation, Manjeron equation, Boussinesq - Love equation and etc.). It is grounded that the Goursat boundary conditions in the classic and non-classic treatment are equivalent to each other, and such boundary conditions are demonstrated in geometric form. Even from geometric interpretation can see that the grounded non-classic treatment doesn't require any additional conditions of agreement type. Thus, namely in this paper, the non-classic problem with Goursat conditions is grounded for a pseudoparabolic equation of sixth order. For simplicity, this was demonstrated for one model case in one of S.L. Sobolev anisotropic space ( ) ( )

Highlights

Pseudoparabolic equations are attracted for sufficiently adequate description of a great deal of real processes occurring in the nature, engineering and etc
Many processes arising in the theory of fluid filtration in cracked media are described by discontinuous coefficient pseudoparabolic equations
The theme of the present paper, devoted to the investigation Goursat problem for sixth order differential equations of pseudoparabolic type, according to the above-stated is very actual for the solution of theoretical and practical problems

Summary

Introduction

Pseudoparabolic equations are attracted for sufficiently adequate description of a great deal of real processes occurring in the nature, engineering and etc. Urgency of investigations conducted in this field is explained by appearance of local and non-local problems for discontinuous coefficients equations connected with different applied problems Such type problems arise for example, while studying the problems of moisture, transfer in soils, heat transfer in heterogeneous media, diffusion of thermal neutrons in inhibitors, simulation of different biological processes, phenomena and etc. Here consider Goursat problem for sixth order equation with discontinuous coefficients The coefficients in this pseudoparabolic equation are not necessarily differentiable; there does not exist a formally adjoint differential equation making a certain sense. The theme of the present paper, devoted to the investigation Goursat problem for sixth order differential equations of pseudoparabolic type, according to the above-stated is very actual for the solution of theoretical and practical problems. From this point of view, the paper is devoted to the actual problems of mathematical physics and computational mathematics

Problem Statement

Methodology

Result

Discussion and Conclusions