Shahid Chamran University of AhvazJournal of Advanced Mathematical Modeling2251-808812220220622Convergence Analysis of Numerical solution of Secon-order reaction-diffusion equation with boundary conditionsConvergence Analysis of Numerical solution of Secon-order reaction-diffusion equation with boundary conditions2893031770610.22055/jamm.2022.39941.2008FAShokriShokriDepartment of Mathematics, Faculty of Science, Urmia University, P.O.Box 165, Urmia, Iran.Journal Article20220202The 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.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.https://jamm.scu.ac.ir/article_17706_4593d0d8e6ba619d6faa60dc9ddcec9a.pdf