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In the present article‎, ‎to find the answer to the mobile-immobile advection-dispersion model of temporal fractional order $0< \beta \leq 1$ (MI-ADM-TF)‎, ‎which can be applied to model the solute forwarding in watershed catchment and flood‎, ‎the effective high-order numerical process is gonna be built‎.‎To do this‎, ‎the temporal-fractional derivative of the MI-ADM-TF is discretized by using the linear interpolation‎, ‎and the temporal-first derivative by applying the first-order precision of the finite-difference method‎. ‎On the other hand‎, ‎After obtaining a semi-discrete form‎, ‎to obtain the full-discrete technique‎, ‎the space derivative is approximated utilizing a collocation approach based on the Legendre basis‎.‎The convergence order of the implicit numerical design for MI-ADM-TF is discussed in that is linear‎.‎Moreover‎, ‎the temporal-discretized structure of stability is also discussed theoretically in general in the article‎.‎Eventually‎, ‎two models are offered to demonstrate the quality and authenticity of the established process‎.

In this paper, the space-time fractional advection-diffusion equation (STFADE) is considered in the finite domain that the time and space derivatives are the Caputo fractional derivative. At first, a quadratic interpolation with convergence order O(τ3-α) is applied to obtain the semi-discrete in time variable. Then, the Chebyshev collocation method of the fourth kind has been used to approximate the spatial fractional derivative. In addition, the energy method has been employed to show the unconditional stability and gained convergence order of the time-discrete scheme. Finally, the accuracy of the numerical method is analyzed and showed that our method is much more accurate than existing techniques.

In this paper, we present a well-organized method to estimate the one-dimensional fractional Rayleigh–Stokes model using the construction of orthogonal Gegenbauer polynomials (GBPs) and Lagrange square interpolation to estimate the time derivative. Therefore, we design an authentic and fast numerical calculation approach based on the elaborated convergence rate recovery method. The matrix of the derivative operation of an orthogonal GBP is gained by employing the characteristic of this type of polynomial. The privilege of the numerical method is the orthogonality of the GBP and operational matrices, which reduces time computation and increases speed. Eventually, we propose three numerical examples to check the validity and numerical studies to illustrate the precision and efficiency of the new approach.

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.