Application of cubic Bsplines collocation method for solving nonlinear inverse diffusion problem

In this paper, we developed a collocation method based on cubic Bspline for solving nonlinear inverse parabolic partial differential equations as the following form lt;br / gt; begin{align*} lt;br / gt; u_{t} = [f(u),u_{x}]_{x} + varphi(x,t,u,u_{x}),,quadquad 0 lt; x lt; 1,,,, 0 leq t leq T, lt;br / gt; end{align*} lt;br / gt; where $f(u)$ and $varphi$ are smooth functions defined on $mathbb{R}$. First, we obtained a time discrete scheme by approximating the firstorder time derivative via forward finite difference formula, then we used cubic Bspline collocation method to approximate the spatial derivatives and Tikhonov regularization method for solving produced illposed system. It is proved that the proposed method has the order of convergence $O(k+h^2)$. 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 Bspline is also suitable for the treatment of the nonlinear inverse parabolic partial differential equations.