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Burger and Fisher diffusion transfer properties and reactions from the characteristics are studied using a non-linear equation. The nonlinear fractional Burgers–Fisher equation (NFB-FE) appears in realistic physical situations such as ultra-slow kinetics, Brownian motion of particles, anomalous diffusion, polymerases of ribonucleic acid and deoxyribonucleic acid, continuous random movement, and formation of wave patterns. The present study focuses on the collocation scheme based on the shifted Chebyshev basis (SCB) and the compact finite difference method to obtain the numerical scheme of the NFB-FE. The simulation model is created in the two steps: Initially, a semi-discrete is formed in a temporal sense, applying a linear approximation with an accuracy order of two. Next, we examine the unconditional stability and the convergence order. In the second stage, the collocation approach based on the SCB of the fourth type is used to discretize the spatial derivative parts and generate the full-discrete scheme.

The computation of the nonlinear fractional Burgers–Fisher problem stated in the Caputo sense is the topic of this paper. The model depicts the issue of biological invasion and can be found in a variety of fields, including ecology, physiology, and basic stage transition situations. To produce the time discretization, the suggested methodology employs a one-order correct expression in the first process. To generate the full-discretization in the second level, the spectral collocation method approach that relies on the Legendre basis is presented. The theoretical investigation confirms the temporal discretized formulation’s stability and convergence, which are examined in relation to the associated norm. Three test examples demonstrate the computing capability and efficiency of the approach. We can use the provided approach to resolve more engineering and physics models and can also increase the convergence order of the method.