Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator

• Published in: Fractal and fractional Vol. 5; no. 3; p. 64
• Main Authors: Malyk, Igor V., Gorbatenko, Mykola, Chaudhary, Arun, Sharma, Shivani, Dubey, Ravi Shanker
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2021
• Subjects: Calculus; Diffusion; Exact solutions; existence and uniqueness; fractional derivative; Laplace transforms; Partial differential equations; α-homotopy analysis transform method
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract5030064

In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.