The inverse solution of the coupled nonlinear reaction–diffusion equations by the Haar wavelets

ABSTRACT

In this paper, a numerical method is proposed for the numerical solution of the coupled nonlinear reaction–diffusion equations with suitable initial and boundary conditions by using the Haar wavelet method to determine the unknown boundary conditions. More precisely, we apply the Haar wavelet method for discretizing the space derivatives and then use a quasilinearization technique to linearize the nonlinear term in the equations. This process generates an ill-posed linear system of equations. Hence, to regularize the resultant ill-posed linear system of equations, we employ the Tikhonov regularization method to obtain a stable numerical approximation to the solution. We also prove the convergence of order one (i.e. $O\left(1/M\right)$) and discuss the error estimation and stability computation for the proposed method. Finally, we report some numerical results which in compared with the finite difference method and the radial basis function method show the efficiency and capability of the proposed method.

Keywords:

• Ill-posed inverse problems
• Haar wavelets
• quasilinearization technique
• the Tikhonov regularization method
• convergence analysis
• stability analysis

2010 AMS Subject Classifications:

• 65M32
• 65T60
• 35K57

Disclosure statement

No potential conflict of interest was reported by the authors.