The fundamental solution of the space-time fractional diffusion equation

By Francesco Mainardi, Yuri Luchko, Gianni Pagnini. 2007

880 citations recorded by the source corpus.

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order � ∈ (0,2] and skewness � (|�| ≤ min {�,2 − �}), and the first-order time derivative with a Caputo derivative of order � ∈ (0,2]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We review the particular cases of space-fractional diffusion {0 < � ≤ 2, � = 1}, time-fractional diffusion {� = 2, 0 < � ≤ 2}, and neutral-fractional diffusion {0 < � = � ≤ 2}, for which the fundamental solution can be interpreted as a spatial probability density function evolving

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/e9a7209531e4f0632397aad8d98ea4adf2fe0360

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In Applied Mathematics

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6. Juan Trujillo
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In Modeling and Simulation

1. Dumitru Băleanu
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3. Ian L Turner
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6. Yong Zhou
7. Juan Trujillo
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In Numerical Analysis

1. Fawang Liu
2. Ian L Turner
3. Vo Anh
4. Dumitru Băleanu
5. Bashir Ahmad
6. Yong Zhou
7. Changpin Li
8. Chang-Ming Chen

Citations in the corpus

• Numerical approximation of Lévy-Feller diffusion equation and its probability interpretation (2007, background): “In Mainardi et al. [ 23 ] the fundamental solution for the problem (2) was expressed as the convolution of the Green function and initial condition.”
• Inverse problem for a three-parameter space-time fractional diffusion equation (2018, background): “While most of these works have been dedicated to fractional derivatives only in the time variable [3, 10, 13, 14, 17, 18, 22, 23, 24], space-time fractional derivatives were considered in [20, 21], similarly as in this article.”
• SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations (2009, background): “[35]) that the density fD 1 (τ) of D1 is infinitely differentiable on (0,∞), with the following asymptotics at zero and infinity [31, 18]:”
• A UNIQUENESS RESULT FOR AN INVERSE PROBLEM IN A SPACE-TIME FRACTIONAL DIFFUSION EQUATION (2013, background): “Important references documenting the recent interest on inverse problems with fractional derivatives are [4, 6, 9, 10, 13, 14, 16, 17, 18].”
• Thermoelasticity that uses fractional heat conduction equation (2009, background): “The Riesz form of the fractional derivative is a symmetric operator with respect to x [22, 42 ] (we consider this operator for 0 42 , 63]”
• A fully discrete difference scheme for a diffusion-wave system (2006, methodology): “[14] presented the fundamental solution (Green function) for the space–time fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order α ∈ (0,2] and skewness θ (|θ | min{α,2 − α}), and t”