A nonlocal problem for loaded partial differential equations of fourth order

Abstract

A nonlocal problem for loaded partial differential equations of fourth order A nonlocal problem for the fourth order system of loaded partial differential equations is considered. The questions of a existence unique solution of the considered problem and ways of its construction are investigated. The nonlocal problem for the loaded partial differential equation of fourth order is reduced to a nonlocal problem for a system of loaded hyperbolic equations of second order with integral conditions by introducing new functions. As a result of solving nonlocal problem with integral conditions is applied a method of introduction functional parameters. The algorithms of finding the approximate solution to the nonlocal problem with integral conditions for the system of loaded hyperbolic equations are proposed and their convergence is proved. The conditions of the unique solvability of the nonlocal problem for the loaded hyperbolic equations are obtained in the terms of initial data. The results also formulated relative to the original problem.

Inverse Problem for a Hyperbolic Integro-Differential Equation with two Redefinition Conditions at the End of the Interval and Involution

“…Substituting the Fourier series (15) into this integro-differential equation, we obtain a countable system of second order ordinary differential equations…”

On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima

“…defined by the right-hand side of integral equation (11). Obviously, the fixed point of the operator Θ is the unique solution to the boundary value problem ( 1)-( 3).…”

“…Such differential equations are called differential equations with impulse effects [1][2][3][4][5][6][7][8]. As is known, in recent years the interest in the study of differential equations with nonlocal boundary conditions has increased (see, for example, [9][10][11][12][13][14][15][16][17]). In particular, in [15] a physical situation in which a non-metallic conductor is in contact with a perfect conductor is studied.…”

“…Substituting equality (15) into representation (11), we obtain the following new presentation of solution:…”

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“…When the boundary of a domain of a physical process is impossible to study, a nonlocal condition of integral form can be obtained as additional information sufficient for the unique solvability of the problem. Therefore, in recent years, research has intensified on the nonlocal direct and inverse boundary value problems for differential and integro-differential equations with integral conditions (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). The many problems of gas dynamics, the theory of elasticity, and the theory of plates and shells have been described by high-order partial differential equations.…”