Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density

DOI10.2478/s13540-013-0019-6zbMath1312.35179OpenAlexW2163467777MaRDI QIDQ2347307

Yuri Luchko, Rudolf Gorenflo, Mirjana Stojanovic

Publication date: 27 May 2015

Published in: Fractional Calculus \& Applied Analysis (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.2478/s13540-013-0019-6

zbMATH Keywords

Fourier transformLaplace transformprobability densityStieltjes functionsBernstein functionscompletely monotone functionscomplete Bernstein functionstime-fractional diffusion-wave equation of distributed order

Mathematics Subject Classification ID

Integro-partial differential equations (45K05) Mittag-Leffler functions and generalizations (33E12) Initial value problems for PDEs with pseudodifferential operators (35S10) Fractional partial differential equations (35R11)

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Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
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• Unnamed Item
• Unnamed Item
• Unnamed Item
• General fractional calculus, evolution equations, and renewal processes
• Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation
• Fractional relaxation-oscillation and fractional diffusion-wave phenomena.
• Some aspects of fractional diffusion equations of single and distributed order
• Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation
• The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type
• Fractals and fractional calculus in continuum mechanics
• Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
• Time-fractional telegraph equations and telegraph processes with Brownian time
• Integrodifferential equation which interpolates the heat equation and the wave equation
• Distributed order calculus and equations of ultraslow diffusion
• Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations. I
• Fractional diffusion and wave equations
• Time-fractional Diffusion of Distributed Order
• The Two Forms of Fractional Relaxation of Distributed Order
• Time distributed-order diffusion-wave equation. I. Volterra-type equation
• Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations
• Completely monotonic functions
• The fundamental solution of the space-time fractional diffusion equation
• Fractional Calculus: Integral and Differential Equations of Fractional Order
• Fractional diffusion in inhomogeneous media
• DISTRIBUTED ORDER FRACTIONAL SUB-DIFFUSION
• Bernstein functions. Theory and applications