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An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion. (English) Zbl 1469.35251

Summary: This paper is concerned with the mathematical analysis of an inverse random source problem for the time fractional diffusion equation, where the source is driven by a fractional Brownian motion. Given the random source, the direct problem is to study the stochastic time fractional diffusion equation. The inverse problem is to determine the statistical properties of the source from the expectation and variance of the final time data. For the direct problem, we show that it is well-posed and has a unique mild solution under a certain condition. For the inverse problem, the uniqueness is proved and the instability is characterized. The major ingredients of the analysis are based on the properties of the Mittag-Leffler function and the stochastic integrals associated with the fractional Brownian motion.

MSC:

35R30 Inverse problems for PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35R11 Fractional partial differential equations
35R60 PDEs with randomness, stochastic partial differential equations
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs

Software:

Mittag-Leffler
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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