12 The heat equation with fractional diffusion

2021 ◽  
pp. 373-432
Keyword(s):  
Heat Equation ◽  
Fractional Diffusion

scholarly journals Determination of source term for fractional heat equation on the sphere

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 361-370
Author(s):  
Nguyen Phuong ◽  
Tran Binh ◽  
Nguyen Luc ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Fractional Heat Equation ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

In this work, we study a truncation method to solve a time fractional diffusion equation on the sphere of an inverse source problem which is ill-posed in the sense of Hadamard. Through some priori assumption, we present the error estimates between the regularized and exact solutions.

Fractional diffusion and fractional heat equation

2000 ◽  
Vol 32 (4) ◽  
pp. 1077-1099 ◽  
Author(s):  
J. M. Angulo ◽  
M. D. Ruiz-Medina ◽  
V. V. Anh ◽  
W. Grecksch
Keyword(s):  
Heat Equation ◽  
Fractional Diffusion ◽  
Fractional Heat Equation

scholarly journals Asymptotic properties of solutions of the fractional diffusion-wave equation

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Anatoly Kochubei
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Parabolic Equations ◽  
Hyperbolic Equations ◽  
Asymptotic Properties ◽  
Fractional Diffusion ◽  
Time Variable ◽  
Diffusion Wave

AbstractFor the fractional diffusion-wave equation with the Caputo-Djrbashian fractional derivative of order α ∈ (1, 2) with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).

scholarly journals Two equivalent Stefan’s problems for the time fractional diffusion equation

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Sabrina Roscani ◽  
Eduardo Marcus
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Classical Solutions ◽  
Constant Condition ◽  
Flux Condition ◽  
Time Fractional Diffusion Equation ◽  
Fractional Equation

AbstractTwo Stefan’s problems for the diffusion fractional equation are solved, where the fractional derivative of order α ∈ (0, 1) is taken in the Caputo sense. The first one has a constant condition on x = 0 and the second presents a flux condition T x(0,t) = q/t α/2. An equivalence between these problems is proved and the convergence to the classical solutions is analyzed when α ↗ 1 recovering the heat equation with its respective Stefan’s condition.

scholarly journals A new equivalence of Stefan’s problems for the time fractional diffusion equation

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Sabrina Roscani ◽  
Eduardo Marcus
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Classical Solutions ◽  
Constant Condition ◽  
Convective Condition ◽  
Flux Condition ◽  
Time Fractional Diffusion Equation

AbstractA fractional Stefan’s problem with a boundary convective condition is solved, where the fractional derivative of order α ∈ (0, 1) is taken in the Caputo sense. Then an equivalence with other two fractional Stefan’s problems (the first one with a constant condition on x = 0 and the second with a flux condition) is proved and the convergence to the classical solutions is analyzed when α ↗ 1 recovering the heat equation with its respective Stefan’s condition.

scholarly journals Determination of source term for fractional heat equation on the sphere

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 361-370
Author(s):  
Nguyen Phuong ◽  
Tran Binh ◽  
Nguyen Luc ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Fractional Heat Equation ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

In this work, we study a truncation method to solve a time fractional diffusion equation on the sphere of an inverse source problem which is ill-posed in the sense of Hadamard. Through some priori assumption, we present the error estimates between the regularized and exact solutions.

Fractional diffusion and fractional heat equation

2000 ◽  
Vol 32 (4) ◽  
pp. 1077-1099 ◽  
Author(s):  
J. M. Angulo ◽  
M. D. Ruiz-Medina ◽  
V. V. Anh ◽  
W. Grecksch
Keyword(s):  
Fourier Transform ◽  
Green Function ◽  
Heat Equation ◽  
Fractional Diffusion ◽  
Long Range Dependence ◽  
Riesz Potentials ◽  
Diffusion Operator ◽  
Fractional Heat Equation ◽  
Sample Paths ◽  
The Fourier Transform

This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

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