Novel simulation of the time fractional Burgers–Fisher equations using a non-polynomial spline fractional continuity method

Abstract

In a recent study, we investigate the Burgers–Fisher equation through a developed scheme, namely, the non-polynomial spline fractional continuity method. The proposed models represent nonlinear optics, chemical physics, gas dynamics, and heat conduction. The basic concept of the new approach is constructing a non-polynomial spline with a fractional continuity equation instead of a natural derivative. Furthermore, the truncation error is analyzed to determine the order of convergence for the proposed scheme, and we presented theoretically the stability of the developed scheme using the von Neumann method. One might easily conclude that the new scheme is quite successful and effective in obtaining the numerical solutions of the time partial/fractional partial differential equations. In addition, we plotted contour, 2D, and 3D graphs for some reported solutions to compare the presented solution with an exact solution. The investigated method was tested in some examples and compared to previous solutions for showing the applicability and effectiveness of the developed numerical scheme. The absolute and norm errors L2 and L∞ has calculated to validate the accuracy and efficiency of the presented scheme. To our knowledge, all obtained solutions in this research paper are novel and not published beforehand.