The matching problem for fourth-order composite and hyperbolic equations with two lines of change of type

Abstract

Boundary value problems for higher-order equations of mixed and mixed-composite types play a significant role in the mathematical modelling of phenomena related to heat propagation, wave processes, and the motion of weakly viscous media. The relevance of this research stems from the need for rigorous analysis of such problems, particularly in the presence of type-change lines and complex boundary conditions. The aim of the study was to formulate and comprehensively investigate a boundary value problem for a fourth-order equation of composite and hyperbolic types in a domain divided into three subdomains with differing equation structures. The problem was reduced to three auxiliary subproblems posed in the corresponding subdomains. On the lines where the type of equation changes, conjugation conditions were imposed, involving the unknown function and its derivatives up to the second order. The investigation employed classical methods from the theory of boundary value problems, techniques for order reduction, and approaches from the theory of mixed-composite type equations. Each auxiliary problem was reduced to standard formulations – namely, Dirichlet, Goursat, and Darboux problems. On the type-change lines, second-order differential equations were obtained, for which boundary value problems were solved using explicitly constructed Green’s functions. The hyperbolic subproblems were reduced to Volterra and Fredholm integral equations of the second kind, and sufficient conditions for their unique solvability were derived via kernel estimates. As a result, explicit analytical expressions for the solutions in each subdomain were obtained. The results can be applied to the analysis of processes in inhomogeneous media and to the development of numerical models in mathematical physics problems