Haar basis method to solve some inverse problems for two-dimensional parabolic and hyperbolic equations

A numerical method consists of combining Haar basis method and Tikhonov regularization method. We apply the method to solve some inverse problems for twodimensional parabolic and hyperbolic equations using noisy data. In this paper, a stable numerical solution of these problems is presented. This method uses a sensor located at a point inside the body and measures the u(x; y; t) at a point x = a; 0 < a < 1. We also show that the rate of convergence of the method is as exponential. Numerical results show that a good estimation on the unknown functions of the inverse problems can be obtained within a couple of minutes CPU time at Pentium IV-2.53 GHz PC.Publisher's Versio

Numerical solution of the inverse Gardner equation

In this paper, the numerical solution of the inverse Gardner equation will be considered. The Haar wavelet collocation method (HWCM) will be used to determine the unknown boundary condition which is estimated from an over-specified condition at a boundary. In this regard, we apply the HWCM for discretizing the space derivatives and then use a quasilinearization technique to linearize the nonlinear term in the equations. It is proved that the proposed method has the order of convergence O(∆x). The efficiency and robustness of the proposed approach for solving the inverse Gardner equation are demonstrated by one numerical example.Publisher's Versio