Boundary Value Problem for the Fifth Order Partial Differential Equations
Keywords:
fifth-order partial differential equation, boundary value problem, spectral parameter, Fourier method, eigenfunctions, regular parameter values, uniqueness, classical solvability, Samarskii–Ionkin type conditionsAbstract
We investigate a boundary value problem for a fifth-order partial differential equation with a real parameter in a rectangular domain. The equation contains third-order differentiation with respect to time and a mixed fifth-order derivative with respect to the temporal
and spatial variables. The problem is supplemented by homogeneous boundary conditions of
Samarskii–Ionkin type with respect to the spatial variable and by three nonstandard two-point
conditions with respect to time. The principal objective is to establish a constructive representation of the solution, determine the parameter values for which the problem is regular, and justify
the convergence of the resulting Fourier series.
The method of separation of variables reduces the spatial part of the problem to a Sturm–Liouville
spectral problem. Its eigenvalues and normalized eigenfunctions are obtained explicitly, and the
eigenfunctions form a complete orthonormal system in L2(0, 1). Expansion of the unknown solution and the right-hand side with respect to this system transforms the original partial differential
equation into a countable family of third-order ordinary differential equations. These equations
are solved by the method of variation of parameters. The three temporal boundary conditions
lead, for each Fourier mode, to a finite-dimensional linear algebraic system whose characteristic
determinant depends on the parameter ω.
The zeros of the corresponding characteristic expression determine an exceptional set of parameter values. Outside this set, the algebraic systems are uniquely solvable and an explicit Fourier
representation of the solution is obtained. It is proved that, for regular values of the parameter,
the boundary value problem can have at most one solution. Under additional smoothness and
compatibility assumptions on the boundary data and the right-hand side, estimates of the Fourier
coefficients together with Bessel’s inequality yield absolute and uniform convergence of the solution series and of the derivatives required by the differential equation. Consequently, the formal
Fourier representation defines a classical solution of the boundary value problem. The obtained
results clarify the role of the spectral parameter in the solvability of higher-order boundary value
problems and provide a constructive framework that can be adapted to related partial differential
equations with nonstandard temporal and spatial conditions.
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