Approximation of mild solutions of the linear and nonlinear elliptic equations
This work addresses the challenge of solving ill-posed elliptic Cauchy problems, but the contribution is incremental as it applies known techniques (separation of variables, modified method) to a specific class of operators.
The authors propose a modified method for approximating solutions to ill-posed Cauchy problems for linear and semi-linear elliptic equations, demonstrating its feasibility and efficiency through numerical examples including a modified Helmholtz equation and an elliptic sine-Gordon equation.
In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial t^{2}}u\left(t\right)=\mathcal{A}u\left(t\right)+f\left(t,u\left(t\right)\right),\quad t\in\left[0,T\right], \] where $\mathcal{A}$ is a positive-definite, self-adjoint operator with compact inverse. As we know, these problems are well-known to be ill-posed. On account of the orthonormal eigenbasis and the corresponding eigenvalues related to the operator, the method of separation of variables is used to show the solution in series representation. Thereby, we propose a modified method and show error estimations in many accepted cases. For illustration, two numerical examples, a modified Helmholtz equation and an elliptic sine-Gordon equation, are constructed to demonstrate the feasibility and efficiency of the proposed method.