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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.

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 ( τ 3 - 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.

The application of Levy processes including major movements or jumps over a small period of time has proved to be an effective technique in financial research to catch certain unusual or extreme cases in stock price dynamics. Models that follow the Levy process are The FMLS, Kobol, and CGMY models. These models gradually grow the interest for study among researchers because of some of the best choices them. Therefore the topic of approaching these three different models has drawn yet more attention. In the current paper, we present these models’ numerical method. At first, The Riemann-Liouville tempered fractional derivative (RLTFD) with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the fourth kind (SCPFK). In the second step, we gain the semi-discrete design to solve the tempered fractional B-S model (TFBSM) by applying finite difference approximation. We’re going to show that this system is stable and $$\mathcal {O}(\delta \tau )$$ is the convergence order. In fact, a fast stabilized method will obtain to reduce the time from processing and the computation time per repetition. Then to get the full scheme, we use SCPFK to approximate the spatial fractional derivative. Finally, two numerical examples are presented to demonstrate the accuracy and usefulness of the developed system.

The time-fractional Black-Scholes model governing European options in which the temporal derivative is focused on the Caputo fractional derivative with $$0

This paper develops the approximate solution of the two-dimensional space-time fractional diffusion equation. Firstly, the time-fractional derivative is discretized with a scheme of order $${\mathcal {O}}({\delta \tau }^{2-\alpha }),~ 0

This paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the$$(2-\alpha)$$(2-α)-order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.

The main purpose of this paper is to design a numerical method for solving the space–time fractional advection-diffusion equation (STFADE). First, a finite difference scheme is applied to obtain the semi-discrete in time variable with convergence order $$\mathcal {O}(\tau ^{2-\beta })$$ . In the next, to discrete the spatial fractional derivative, the Chebyshev collocation method of the fourth kind has been applied. This discrete scheme is based on the closed formula for the spatial fractional derivative. Besides, the time-discrete scheme has studied in the $$L_{2}$$ space by the energy method and we have proved the unconditional stability and convergence order. Finally, we solve three examples by the proposed method and the obtained results are compared with other numerical problems. The numerical results show that our method is much more accurate than existing techniques in the literature.

The fractional advection–diffusion equation, known as non-local diffusion, is a relationship utilized in groundwater hydrology as a reliable means of modeling the transport of passive tracers in porous media by fluid flow. The main target of this paper is to develop a numerical method for solving the space-time fractional advection–diffusion equation (STFADE) defined by Caputo sense. In this way, a finite difference formula with first-order accuracy is used to discretize the problem in the temporal direction, and also the Chebyshev collocation method of the third kind is applied to approximate the space variable. The stability and convergence of the fully discrete scheme are rigorously established in $$L^{2}$$ norm. The numerical results are presented and are compared with other methods to show the capability of the numerical scheme proposed here.

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.

Many researchers have expanded a new form of fractional diffusion models named the temporal-space fractional Bloch–Torrey equation (TSF-BTE) to evaluate the diffusion structure of human brain tissues as well as prepare additional insight into other studies of cells and tissues and the microenvironment. The main objective of this paper is to present an effective computational method of solving such models in two dimensions. The temporal and spatial directions are based on the Caputo and the Riemann–Liouville fractional derivative, respectively. The presented numerical scheme is derived from the following manners: at first, the semi-discrete is constructed in the temporal based on a quadratic interpolation with accuracy order $$\mathcal {O}(\tau ^{2-\alpha })$$ and secondly, the unconditional stability and convergence order are analyzed. For the constructed full-discrete scheme, the spatial derivative terms approximated with the helping of the collocation method based on the Legendre basis. Finally, to illustrate the high precision of the proposed design, we use some test problems. Furthermore, the obtained results are compared with some other techniques under which the suggested methodology is highly accurate and feasible.

This paper studies a new computational method for the approximate solution of the space fractional advection–dispersion equation in sense of Caputo derivatives. In the first method, a time discretization is accomplished via the compact finite difference, while the fourth kind shifted Chebyshev polynomials are used to discretize the spatial derivative. The unconditional stability and convergence order of the method are studied via the energy method. Three examples are given for illustrating the effectiveness and accuracy of the new scheme when compared with existing numerical methods reported in the literature.

The present paper is primarily aimed at obtaining the numerical solution of space fractional advection–diffusion equation including two fractional space derivatives of order. At the first stage, a difference approach with the second-order accuracy is formulated to obtain a semi-discrete scheme. Unconditional stability and convergence analysis has been analyzed. At the second stage, the spatial discretization is accomplished by means of the second kind shifted Chebyshev polynomials. Two examples are investigated, and numerical results are reported to confirm the theoretical results.

This paper develops a numerical method for approximating the space fractional diffusion equation in Caputo derivative sense. In this discretization process, firstly, the compact finite difference with convergence order $${\mathcal {O}}(\delta \tau ^{2})$$ is used to obtain the semi-discrete in time derivative. Afterward, the spatial fractional derivative is discretized by using the Chebyshev collocation method of the third-kind. This collocation scheme is based on the operational matrix. In addition, time-discrete stability and convergence are theoretically proved in detail. We solve two examples by the proposed method and the obtained results are compared with other numerical methods. The numerical results show that our method is much more accurate than existing methods.