scholarly journals Cauchy problem for inhomogeneous fractional nonclassical diffusion equation on the sphere

2021 ◽  
Vol 25 (04) ◽  
pp. 303-311
Author(s):  
L. D. Long
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Nonclassical Diffusion Equation

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

2021 ◽  
pp. 2140011
Author(s):  
Tomás Caraballo ◽  
Tran Bao Ngoc ◽  
Tran Ngoc Thach ◽  
Nguyen Huy Tuan
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Stochastic Integrals ◽  
Well Posedness ◽  
Time Data ◽  
Final Time ◽  
Fractional Brownian Motions ◽  
Nonclassical Diffusion Equation ◽  
Definition Of ◽  
Well Posed

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

scholarly journals Existence of Positive Solution to the Cauchy Problem for a Fractional Diffusion Equation with a Singular Nonlinearity

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Ailing Shi ◽  
Shuqin Zhang
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Positive Solution ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Successive Approximation Method ◽  
Fractional Diffusion Equations ◽  
Singular Nonlinearity ◽  
Analysis Technique ◽  
The Cauchy Problem

Fractional diffusion equations describe an anomalous diffusion on fractals. In this paper, by means of the successive approximation method and other analysis technique, we present a local positive solution to Cauchy problem for a fractional diffusion equation with singular nonlinearity. The fractional derivative is described in the Caputo sense.

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