Analysis and application of a hybrid tension-spline method for nonlinear time-fractional reaction–diffusion equations

Abstract

Nonlinear time-fractional reaction–diffusion equations are widely used to model real-world phenomena such as heat transfer, earthquakes, drug release from biomedical implants, volcanic eruptions, groundwater contamination, and brain cancer dynamics. Developing efficient and accurate numerical methods for these problems is challenging due to the combined effects of nonlinearity and fractional time derivatives. In this work, we proposed a hybrid numerical method that combines a tension-spline spatial discretization with the Crank–Nicolson time-stepping method, employing the Rubin–Graves linearization to effectively handle nonlinear reaction terms. The tension parameters provide a controllable smoothness that enhances stability and accuracy, while the Crank–Nicolson scheme ensures second-order temporal convergence. Moreover, the order of accuracy relies on the suitable choice of two tension-spline parameters that increment it from two to four, which is explained conceptually and numerically. Numerical experiments show that the proposed scheme achieves superior accuracy compared to various iterative and finite difference methods, making it a robust and practical tool for solving nonlinear time-fractional reaction–diffusion problems in real-world applications.