Abstract
In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the Green function G3(t) of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag–Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag–Leffler function, the hypergeometric function 1F1, and the exponential functions are compared numerically. The Grünwald–Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller–Riesz space-fractional operator. The explicit difference scheme is numerically studied, and the simulations of the approximate solutions are plotted for different values of the fractional orders.
Highlights
Introduction and Important DefinitionsReceived: 14 August 2021The classical intermediate diffusion wave equation, the multiterm wave equation, can be written as: Accepted: 13 September 2021Published: 17 September 2021 ∂2 u( x, t) ∂u( x, t) +k= L FP (u( x, t)), −∞ < x < ∞, t ≥ 0, ∂t (1)Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Licensee MDPI, Basel, Switzerland.where the right-hand side of this equation is the known Fokker–Planck operator; see [1]
To computationally prove that the analytic solutions in terms of the Mittag–Leffler function are convergent, we give a brief review of its asymptotic behaviors; see [35] and the references therein
We studied the classical wave equation with the damped term and associated with the stochastic Fokker–Planck operator
Summary
The classical intermediate diffusion wave equation, the multiterm wave equation, can be written as: Accepted: 13 September 2021. Published: 17 September 2021 ∂2 u( x, t) ∂u( x, t) +k. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Where the right-hand side of this equation is the known Fokker–Planck operator; see [1]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
