Application of cubic B-splines collocation method for solving nonlinear inverse diffusion problem

In this paper, we developed a collocation method based on cubic B-spline for solving nonlinear inverse parabolic partial differential equations as the following form \begin{align*} u_{t} &= [f(u)\,u_{x}]_{x} + \varphi(x,t,u,u_{x}),\,\quad\quad ۰ < x < ۱,\,\,\, ۰ \leq t \leq T, \end{align*} where f(u) and \varphi are smooth functions defined on \mathbb{R}. First, we obtained a time discrete scheme by approximating the first-order time derivative via forward finite difference formula, then we used cubic B-spline collocation method to approximate the spatial derivatives and Tikhonov regularization method for solving produced ill-posed system. It is proved that the proposed method has the order of convergence O(k+h^۲). The accuracy of the proposed method is demonstrated by applying it on three test problems. Figures and comparisons have been presented for clarity. The aim of this paper is to show that the collocation method based on cubic B-spline is also suitable for the treatment of the nonlinear inverse parabolic partial differential equations.

Keywords:

Cubic B-spline , Collocation method , Inverse problems , Convergence analysis , Stability of solution , Tikhonov regularization method , Ill-posed problems , Noisy data