A finite-difference and Haar wavelets hybrid collocation technique for non-linear inverse Cauchy problems
Muhammad Ahsan, Iltaf Hussain, Masood Ahmad
Abstract
In this research work, a finite-difference and Haar wavelet hybrid collocation scheme is introduced for the ill-posed non-linear inverse Cauchy problem with a source depending on space variable along with an unknown solution and unknown right side boundary. The first-order finite-difference approach is adopted to approximate the ∂u∂t part and two different Haar series are managed to approximate ∂2u∂x2 part and source term respectively. A simple linearization procedure is used to convert the non-linear problem into a linear form. In contradiction to various numerical schemes, the current introduced method generates a well-conditioned system of algebraic equations, therefore it is not required to apply a regularization approach. The results of the proposed method are stable and converge to the exact solution. Some numerical tests are also performed to confirm the accuracy, well-conditioning of the algebraic equations and easy applicability of the scheme on linear and non-linear cases.