The solution to a boundary value problem for a third-order equation with variable coefficients

Yu. P. Apakov ; R. A. Umarov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 28 (2024) no. 1, pp. 171-185.

Résumé

In a rectangular domain, the second boundary value problem for a non-homogeneous third-order partial differential equation with multiple characteristics and variable coefficients is considered. The uniqueness of the solution to the given problem is proved using the energy integral method. For the case where the conditions of the uniqueness theorem are violated, a counterexample is constructed. The solution to the problem is sought in the form of a product of two functions $X(x)$ and $Y(y)$ using the separation of variables method. An ordinary differential equation of the second order with two boundary conditions on the boundaries of the segment $[0, q]$ is obtained for determining $Y(y)$. For this problem, the eigenvalues and the corresponding eigenfunctions are found for $n=0$ and $n\in \mathbb N$. An ordinary differential equation of the third order with three boundary conditions on the boundaries of the segment $[0, p]$ is obtained for determining $X(x)$. The solution to this problem is constructed using the Green's function method. A separate Green's function was built for $n=0$ and another for the case when $n$ is a natural number. It is verified that the found Green's functions satisfy the boundary conditions and properties of the Green's function. The solution for $X(x)$ is expressed through the constructed Green's function. After some transformations, a Fredholm integral equation of the second kind is obtained, and its solution is written in terms of the resolvent. Estimates for the resolvent and the Green's function are derived. Uniform convergence of the solution is proven, along with the possibility of term-by-term differentiation under certain conditions on the given functions. The convergence of the third-order derivative of the solution with respect to the variable $x$ is established using Cauchy–Schwarz and Bessel inequalities. In justifying the uniform convergence of the solution, the absence of a “small denominator” is proven.

Détail

Keywords: differential equation, third order, multiple characteristics, second boundary value problem, regular solution, uniqueness, Green's function
Mots-clés : existence

Bibliographie

[1] Apakov Yu. P., Umarov R. A., “Solution of the first boundary problem for a third order equation with minor terms, a method for constructing the Green's function”, Bulletin of Osh State University, 2022, no. 1, 73–92 (In Russian) | DOI

[2] Apakov Yu. P., Hamitov A. A., “On solvability of the boundary value problem posed for an equation with the third order multiple characteristics in a semi-bounded domain in three dimensional space”, Journal of Osh State University. Mathematics. Physics. Technical Sciences, 1:2, 13–23 (In Russian) | DOI