Method of approximate solution of partial derivative equations

T. V. Zhukovskaya ; E. S. Zhukovskiy ; M. A. Rybakov ; A. S. Trofimova

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 147, pp. 255-267

Résumé

The article considers a partial differential equation of the form $$\frac{\partial u}{\partial t}=f\big(t,x,y, u, \frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial^2 u}{\partial x^2},\frac{\partial^2 u}{\partial y^2}, \frac{\partial^2 u}{\partial x \partial y}\big), \ \ (x,y)\in D \subset \mathbb{R}^2, \ \ t\geq 0,$$ with respect to an unknown function $u,$ defined in a domain $D$ of spatial variables $x,y$ and for $t\geq 0.$ A method for finding an approximate solution is proposed. The equation under consideration is replaced by an approximate one by introducing the shift operator $S:D\to D ,$ which allows replacing at each step of the calculations the unknown values of the function $u(x,y,t)$ on the right side with the values $u(S(x,y),t),$ obtained at the previous step. The idea of the proposed method goes back to the idea of the Tonelli method, known for differential equations with respect to functions of one variable (with ordinary, not partial derivatives). The advantages of the proposed method are the simplicity of the obtained iteration relation and the possibility of application to a wide class of equations and boundary conditions. In the article, iteration formulas are obtained for solving a boundary value problem with the Dirichlet condition for spatial variables and with an initial or boundary condition for the variable $t.$ Based on the proposed method, an approximate solution is obtained for a specific initial-boundary value problem for the heat conductivity equation in a square domain.

Détail

Keywords: partial differential equation, boundary value problem, approximate analytical solution, heat equation