The need for the fractional operators

AbstractIn this review paper, I focus on presenting the reasons of extending the partial differential equations to space-time fractional differential equations. I believe that extending any partial differential equations or any system of equations to fractional systems without giving concrete reasons has no sense. The experiments agrees with the theoretical studies on extending the first order-time derivative to time-fractional derivative. The simulations of some processes also agrees with the theory of continuous time random walks for extending the second-order space fractional derivative to the Riesz–Feller fractional operators. For this aim, I give a condense review the theory of Brownian motion, Langevin equations, diffusion processes and the continuous time random walk. Some partial differential equations that are successfully extended to space-time-fractional differential equations are also presented.

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The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

Journal of Physics Conference Series ◽

10.1088/1742-6596/2090/1/012031 ◽

2021 ◽

Vol 2090 (1) ◽

pp. 012031

Author(s):

E.A. Abdel-Rehim

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fractional Differential Equations ◽

Differential Operators ◽

Space Time ◽

Partial Differential ◽

Stochastic Problems ◽

Fractional Differential ◽

Fractional Differential Operators ◽

Time Fractional Differential Equations

Abstract The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

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Continuous time random walks and space-time fractional differential equations

Basic Theory ◽

10.1515/9783110571622-016 ◽

2019 ◽

pp. 385-406

Author(s):

Mark Meerschaert ◽

Hans Peter Scheffler

Keyword(s):

Differential Equations ◽

Random Walks ◽

Fractional Differential Equations ◽

Continuous Time ◽

Space Time ◽

Continuous Time Random Walks ◽

Fractional Differential ◽

Time Fractional Differential Equations

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Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0019 ◽

2018 ◽

Vol 21 (2) ◽

pp. 312-335 ◽

Cited By ~ 10

Author(s):

Xiao-Li Ding ◽

Juan J. Nieto

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Fractional Differential Equations ◽

Analytical Solutions ◽

Laplacian Operator ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Time Space ◽

Fractional Differential ◽

Time Fractional Differential Equations

AbstractIn this paper, we consider the analytical solutions of multi-term time-space fractional partial differential equations with nonlocal damping terms for general mixed Robin boundary conditions on a finite domain. Firstly, method of reduction to integral equations is used to obtain the analytical solutions of multi-term time fractional differential equations with integral terms. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time-space fractional partial differential equations with nonlocal damping terms to the multi-term time fractional differential equations with integral terms. By applying the obtained analytical solutions to the resulting multi-term time fractional differential equations with integral terms, the desired analytical solutions of the multi-term time-space fractional partial differential equations with nonlocal damping terms are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

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