An LGDAE Method to Solve Nonlinear Cauchy Problem Without Initial Temperature

摘要

We recover an unknown initial temperature for a nonlinear heat conduction equation u(1)(x,t) = u(xx)(x,t) + H (x,t,u,u(x)), under the Cauchy boundary conditions specified on the left-boundary. The method in the present paper transforms the Cauchy problem into an inverse heat source problem to find F (x) in T-t(x, t) = T-xx(x,t) + H + F (x). By using the GL(N, R) Lie-group differential algebraic equations (LGDAE) algorithm to integrate the numerical method of lines discretized equations from sideways heat equation, we can fast recover the initial temperature and two boundary conditions on the right-boundary. The accuracy and efficiency are confirmed by comparing the exact solutions with the recovered results, where a large noisy disturbance is imposed on the Cauchy data.