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In this paper, two numerical methods are presented to solve an ill-posed inverse problem for Fisher's equation using noisy data. These two methods are the Haar basis and the Legendre wavelet methods combined with the Tikhonov regularization method. A sensor located at a point inside the body is used and u(x, t) at a point x = a, 0 < a < 1 is measured and these methods are applied to the inverse problem. We also show that an exponential rate of convergence of these methods. In fact, this work considers a comparative study between the Haar basis and the Legendre wavelet methods to solve some ill-posed inverse problems. Results show that an excellent estimation of the unknown function of the inverse problem which have been obtained within a couple of minutes CPU time at Pentium(R) Dual-Core CPU 2.20 GHz.

In this paper, we will first study the existence and uniqueness of the solution of a one-dimensional inverse problem for an inhomogeneous linear wave equation with initial and boundary conditions via an auxiliary problem. Then a stable numerical method consisting of zeroth-, first-, and second-order Tikhonov regularization to the matrix form of Duhamel's principle for solving this inverse problem is presented. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition method. Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data.