scholarly journals Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation

2019 ◽  
Vol 98 ◽  
pp. 127-147 ◽  
Author(s):  
Yu. Luchko
Keyword(s):  
Wave Equation ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Space Time ◽  
Diffusion Wave ◽  
Dimensional Space Time

scholarly journals Existence of Solution of Space–Time Fractional Diffusion-Wave Equation in Weighted Sobolev Space

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Kangqun Zhang
Keyword(s):  
Wave Equation ◽  
Sobolev Space ◽  
Fractional Derivatives ◽  
Weighted Sobolev Space ◽  
Fractional Diffusion ◽  
Space Time ◽  
Existence Of Solution ◽  
Multiplier Theorem ◽  
Diffusion Wave ◽  
Loss Of Regularity

In this paper, we consider Cauchy problem of space-time fractional diffusion-wave equation. Applying Laplace transform and Fourier transform, we establish the existence of solution in terms of Mittag-Leffler function and prove its uniqueness in weighted Sobolev space by use of Mikhlin multiplier theorem. The estimate of solution also shows the connections between the loss of regularity and the order of fractional derivatives in space or in time.

scholarly journals On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equatio

2017 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Open Problem ◽  
Special Functions ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Two Dimensional ◽  
Diffusion Wave ◽  
Space And Time ◽  
Mellin Convolution

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

scholarly journals Subordination Approach to Space-Time Fractional Diffusion

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 415 ◽  
Author(s):  
Emilia Bazhlekova ◽  
Ivan Bazhlekov
Keyword(s):  
Fundamental Solution ◽  
Diffusion Equation ◽  
Numerical Evaluation ◽  
Dimensional Space ◽  
Gaussian Function ◽  
Fractional Diffusion ◽  
Space Time ◽  
Integral Representations ◽  
Dimensional Space Time ◽  
Time Fractional Diffusion Equation

The fundamental solution to the multi-dimensional space-time fractional diffusion equation is studied by applying the subordination principle, which provides a relation to the classical Gaussian function. Integral representations in terms of Mittag-Leffler functions are derived for the fundamental solution and the subordination kernel. The obtained integral representations are used for numerical evaluation of the fundamental solution for different values of the parameters.

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