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Analytical and Numerical Solution of the Fractional Differential Equation

Submission Deadline: 30 December 2024 View: 239 Submit to Special Issue

Guest Editors

Dr. Ndolane Sene, Université Cheikh Anta Diop de Dakar, Senegal
A. Prof. Ameth Ndiaye, Cheikh Anta Diop University, Senegal

Summary

There exist many types of fractional differential equations, due to the fact there exist many types of fractional operators. We can cite the Caputo derivative, the Riemann-Liouville derivative, the Caputo-Fabrizio derivative, the Atangana-Baleanu derivative, and others. The variation in the fractional operators generates variations in the types of the solutions of the fractional differential equations. In other words, the form of the solutions changes, when the used fractional derivative is changed. The second remark is the complexity of the form of the differential equations makes it very hard to apply the Laplace transform which is the standard method in solving differential equations. Recently the Laplace transform has been combined with the homotopy method to get the solutions of the fractional differential equation, the process has had success but the inconvenience is the convergence and the stability of the solution obtained by this combination is not provided in many proposed research. Thus finding the solutions of the fractional differential equations is an open problem in the literature. Some researchers find alternatives in proposing numerical schemes. These numerical schemes are proposed using the numerical procedure of the fractional operators to give the graphics of the dynamics of the considered fractional differential equations. Many numerical schemes are proposed in the context of fractional differential equations, we can cite the Runge Kutta method in the context of fractional calculus, the Adams Basford, implicit and explicit schemes in the context of fractional operators, and others. Note that, writing the numerical schemes is not so complicated but the main problem in the numerical schemes is the implementation of the scheme in Matlab. The present issue is to collect methods stable and convergent methods utilized to give the solution of the fractional differential equations. It is a preferred method with simple implementation in Matlab. It is the preferred method where the application can be made in the fields of mathematical physics, mathematical modeling, and others. 


Keywords

1) Modeling fractional differential equations.
2) Finding the analytical method for solving fractional differential equations.
3) Proposing numerical schemes and their convergence to propose the solution of the fractional differential equations.
5) Modeling epidemic model using fractional differential equations.
6) Using numerical schemes or analytical solutions procedures for finding the solutions of the fluid and nanofluid models.
7) Local stability and the global stability of the equilibriums of the fractional differential equations.

Published Papers


  • Open Access

    ARTICLE

    An Efficient Technique for One-Dimensional Fractional Diffusion Equation Model for Cancer Tumor

    Daasara Keshavamurthy Archana, Doddabhadrappla Gowda Prakasha, Pundikala Veeresha, Kottakkaran Sooppy Nisar
    CMES-Computer Modeling in Engineering & Sciences, Vol.141, No.2, pp. 1347-1363, 2024, DOI:10.32604/cmes.2024.053916
    (This article belongs to the Special Issue: Analytical and Numerical Solution of the Fractional Differential Equation)
    Abstract This study intends to examine the analytical solutions to the resulting one-dimensional differential equation of a cancer tumor model in the frame of time-fractional order with the Caputo-fractional operator employing a highly efficient methodology called the -homotopy analysis transform method. So, the preferred approach effectively found the analytic series solution of the proposed model. The procured outcomes of the present framework demonstrated that this method is authentic for obtaining solutions to a time-fractional-order cancer model. The results achieved graphically specify that the concerned paradigm is dependent on arbitrary order and parameters and also disclose the More >

  • Open Access

    ARTICLE

    Modeling the Dynamics of Tuberculosis with Vaccination, Treatment, and Environmental Impact: Fractional Order Modeling

    Muhammad Altaf Khan, Mahmoud H. DarAssi, Irfan Ahmad, Noha Mohammad Seyam, Ebraheem Alzahrani
    CMES-Computer Modeling in Engineering & Sciences, Vol.141, No.2, pp. 1365-1394, 2024, DOI:10.32604/cmes.2024.053681
    (This article belongs to the Special Issue: Analytical and Numerical Solution of the Fractional Differential Equation)
    Abstract A mathematical model is designed to investigate Tuberculosis (TB) disease under the vaccination, treatment, and environmental impact with real cases. First, we introduce the model formulation in non-integer order derivative and then, extend the model into fractional order derivative. The fractional system’s existence, uniqueness, and other relevant properties are shown. Then, we study the stability analysis of the equilibrium points. The disease-free equilibrium (DFE) is locally asymptotically stable (LAS) when . Further, we show the global asymptotical stability (GAS) of the endemic equilibrium (EE) for and for . The existence of bifurcation analysis in the More >

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