Submission Deadline: 01 July 2025 (closed) View: 6861 Submit to Journal
Dr. Ndolane Sene, Université Cheikh Anta Diop de Dakar, Senegal
A. Prof. Ameth Ndiaye, Cheikh Anta Diop University, Senegal
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.
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