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

Document Type : Research Paper


1 Faculty of Engineering, Sabzevar University of New Technology, Sabzevar, Iran

2 School of Mathematics and Computer Science, Damghan University, P. O. Box 36715-364, Damghan, Iran


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
u_{t} &= [f(u)\,u_{x}]_{x} + \varphi(x,t,u,u_{x}),\,\quad\quad 0 < x < 1,\,\,\, 0 \leq t \leq T,
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 $O(k+h^2)$. 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 B-spline is also suitable for the treatment of the nonlinear inverse parabolic partial differential equations.


Volume 7, Issue 3
July 2019
Pages 434-453
  • Receive Date: 04 November 2017
  • Revise Date: 03 September 2018
  • Accept Date: 22 September 2018
  • First Publish Date: 01 July 2019