The truncation regularization method for identifying the initial value of heat equation on a spherical symmetric domain
The inverse problem of identifying the initial value for the heat equation with inhomogeneous source on a spherical symmetric domain is investigated. The authors consider an inhomogeneous heat equation on a symmetric domain as follows: \[ \begin{cases} u_t- \frac{2}{r} u_r - u_{rr}=f(r,t), \ \ &0<t<T, \ 0<r<r_0,\\ u(r,0)= \varphi(r), \ \ &0 \leq r \leq r_0,\\ u(r_0,t)=0, \ \ &0 \leq t \leq T,\\ u(r,t) \text{ is bounded},\ \ &0<t<T,\ 0<r< r_0. \end{cases} \] The additional condition $u(r, T)=g(r)$ is used to determine the initial condition $\varphi(r)$. \par The truncation regularization method (the projection regularization method) is used for solving the inverse problem. The convergence of the approximate solutions to the exact solution under a priori and a posteriori regularization parameter choice rules is proved. A numerical example is presented to validate the effectiveness of this method.
scientific article; zbMATH DE number 6996321
scientific article