A robust technique of cubic Hermite splines to study the non-linear reaction-diffusion equation with variable coefficients

The present study proposes a hybrid numerical technique to discuss the solution of non-linear reaction-diffusion equations with variable coefficients. The perturbation parameter was assumed to be time-dependent. The spatial domain was discretized using the cubic Hermite splines collocation method. These splines are smooth enough to interpolate the function as well as its tangent at the node points. The temporal domain was discretized using the Crank-Nicolson scheme, commonly known as the CN scheme. The cubic Hermite splines are convergent of order $h^4$, and the CN scheme is convergent of order $\mathrm{\Delta }{t}^2$. The technique is found to be convergent of order $O(h^2({\gamma }_2{\epsilon }_j\mathrm{\Delta }t+{\gamma }_0(1+\overline{\alpha })h^2)+\mathrm{\Delta }{t}^2)$. The step size in the space direction is taken to be $h$, and the step size in the time direction is $\mathrm{\Delta }t$. Stability of the proposed scheme was studied using the $L_2$ and $L_{\mathrm{\infty }}$ norms. The proposed scheme has been applied to different sets of problems and is found to be more efficient than existing schemes.