A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

“…4 and 5, it can be observed that, as expected, the alternating iterative algorithm II applied to Example 1 is more sensitive to noise added into the temperature u| 1 than to perturbations of the normal heat flux q| 1 . Table II tabulates the values of the optimal iteration number, k opt , according to the stopping criterion (38), the corresponding accuracy errors given by Eqs. (34a) and (34b), and the CPU time, obtained using the alternating iterative algorithm II, various levels of noise added into the Neumann data on 1 and various values of the relaxation parameter, ω ∈ (0, 2), for the Cauchy i.…”

“…4(a, b) can be drawn from Fig. 5(a,b), which present the numerical values for the temperature and normal heat flux obtained on the underspecified boundary 2 , in comparison with their analytical counterparts, using the alternating iterative algorithm II, the regularizing stopping criterion (38), ω = 1.50 and various amounts of noise added into the normal heat flux q| 1 , i.e., p q ∈ {1%, 2%, 3%}, for Example 1. By comparing Figs.…”

“…The values of the optimal iteration number, k opt , the corresponding accuracy errors, e u (k opt ) and e q (k opt ), and the CPU time, obtained using the alternating iterative algorithm II, the stopping criterion (38), various levels of noise added into the Dirichlet data on 1 and various values of the relaxation parameter, ω ∈ (0, 2), for the Cauchy problem given by Example 1, are presented in Table I. The following major conclusions can be drawn from this table: i.…”

“…However, even in this case the errors E, e u and e q have a similar behavior and the error E may be used to stop the iterative process at the point where the rate of convergence is very small and no substantial improvement in the numerical solution is obtained even if the iterative process is continued. Therefore, it can be concluded that the regularizing stopping criterion (38) proposed for the alternating iterative algorithms with relaxation I and II is very efficient in locating the point where the errors start increasing and the iterative process should be ceased. Figure 4(a, b) illustrate the analytical and numerical values for the temperature, u, and normal heat flux, q, respectively, on the under-specified boundary 2 , obtained based on the stopping criterion (38) described in Section VIC and using the alternating iterative algorithm II, ω = 1.50 and various levels of noise added into the temperature data on the over-specified boundary 1 , for Example 1.…”

“…), have been used increasingly over the last decade for the numerical solution of inverse problems. For example, the Cauchy problem associated with the heat conduction equation [38][39][40][41][42][43][44][45][46][47][48], linear elasticity [49,50], steady-state heat conduction in functionally graded materials [51], Helmholtz-type equations [19][20][21]52], Stokes problems [53], the biharmonic equation [54], etc., have been successfully addressed by using the MFS.…”