Analytical and numerical solutions of the convection-diffusion-reaction equations applying the differential transformation method and the Crank-Nicolson method, along with stability analysis and truncation error analysis

Habibah, Ummu; Rohman, Zalfa Camilla; Dewanti, Yashinta Novena; Hananti, Arfi Nadhifa

doi:10.22034/cmde.2025.64919.2958

Analytical and numerical solutions of the convection-diffusion-reaction equations applying the differential transformation method and the Crank-Nicolson method, along with stability analysis and truncation error analysis

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Science, Technology, and Mathematics, Brawijaya University, Indonesia.

10.22034/cmde.2025.64919.2958

Abstract

This study presents a unified approach for solving convection-diffusion-reaction equations by integrating the Differential Transformation Method (DTM) for analytical approximations with the Crank-Nicolson numerical scheme. The DTM is employed to derive an analytical solution, while the Crank-Nicolson method is used to compute the numerical solution. The results demonstrate that the analytical solution obtained via DTM is identical to the exact solution. Furthermore, the stability of the Crank-Nicolson numerical scheme is assessed using Von-Neumann stability analysis, confirming that the method is unconditionally stable. The local truncation error is determined via Taylor series expansion to establish its order of accuracy. This analysis reveals that the Crank-Nicolson scheme for the convection-diffusion-reaction equation exhibits a local truncation error of order $O(h^2+k^2)$, ensuring a second-order accurate scheme. Numerical simulations are conducted for various parameter values to examine their impact on the solution. The simulation results demonstrate the gradual transport of the substance from high to low concentration regions, observed through the diminishing displacement of material along the $x$-axis. Further numerical experiments investigate the effects of different values of $h$ and $k$. The results indicate a direct correlation between decreasing values of $h$ and $k$ and a reduction in the average error, underscoring the method’s accuracy and efficiency.

Keywords