Convergence Analysis of ‎Numerical ‎solution ‎of ‎Secon-order ‎r‎eaction-‎d‎iffusion ‎e‎quation with boundary ‎conditions

Document Type : Original Paper


Department of Mathematics, Faculty of Science, Urmia University, P.O.Box 165, Urmia, Iran.


‎The aim of this work is to provide a specific process for solving a reaction-diffusion partial differential equation with boundary conditions ‎(‎RPDEs)‎‎. ‎We first convert this ‎R‎PDE problem to Volterra-Fredholm integral equation (VFIE)‎, ‎because of the good numerical stability properties of integral operators in compare to differential operator‎, ‎then apply the numerical Tau method to solve the obtained integral equation‎. ‎‎‎‎We present the convergence analysis and error estimation of the Tau method based on the proposed process‎. ‎Applying the Tau method yields a system of the ordinary differential equation such that this system is solved by piecewise polynomial collocation methods‎. ‎Intended to show advantages of converting ‎RPD‎E to an integral equation‎, ‎we consider two cases to solve the proposed examples‎. ‎In the first case‎, ‎we apply the Tau method to solve the ‎converted ‎‎R‎PDE problem ‎(‎integral ‎form‎‎‎) and in the second case‎, ‎we solve ‎the ‎R‎PDE ‎problem‎ ‎directly ‎(direct ‎form‎‎)‎ by Tau method‎. ‎Comparing the numerical results‎, ‎we observe that the results obtained from the ‎integral ‎form‎ ‎ are higher than which obtained from the ‎direct ‎form‎‎‎‎.


Main Subjects

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