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Abstract Fractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings.
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An anomalous fractional diffusion operator
Abstract In this article, using that the fractional Laplacian can be factored into a product of the divergence operator, a Riesz potential operator and the gradient operator, we introduce an anomalous fractional diffusion operator, involving a matrixK(x), suitable when anomalous diffusion is being studied in a non homogeneous medium. For the case ofK(x) a constant, symmetric positive definite matrix we show that the fractional Poisson equation is well posed, and determine the regularity of the solution in terms of the regularity of the right hand side function.
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- Award ID(s):
- 2245097
- PAR ID:
- 10581220
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Fractional Calculus and Applied Analysis
- Volume:
- 28
- Issue:
- 3
- ISSN:
- 1311-0454
- Format(s):
- Medium: X Size: p. 1198-1228
- Size(s):
- p. 1198-1228
- Sponsoring Org:
- National Science Foundation
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