Solving the Cauchy Problem of the Nonlinear Steady-state Heat Equation Using Double Iteration Process

In this paper, the Cauchy inverse problem of the nonlinear steady-state heat equation is studied. The double iteration process is used to tackle this problem in which the outer loop is developed based on the residual norm based algorithm (RNBA) while the inner loop determines the evolution direction and the modified Tikhonov's regularization method (MTRM) developed by Liu (Liu, 2012) is adopted. For the conventional iteration processes, a fixed evolution direction such as F, B−1F, BTF or αF+(1-α)BTF is used where F is the residual vector, B is the Jacobian matrix, the superscript '-1' denotes the inverse, the superscript 'T' denotes the transpose of a matrix and α denotes the optimal coefficient. Unlike the conventional approaches, the current approach tries to find an appropriate direction from the initial guess BTF using the MTRM and the final evolution direction is determined once the value of a0 is less than the critical value ac. Since it may consume too much computation time for searching this appropriate evolution direction such that it makes this process computationally noneconomic, we terminate the inner iteration process as well as the whole process once the number of the iteration steps for the inner iteration exceeds a given maximum value, says Imax. Six examples are illustrated to show the validity of the current approach and results show that the proposed method is very efficient and accurate.