604 research outputs found
Inverse Problems of Determining Coefficients of the Fractional Partial Differential Equations
When considering fractional diffusion equation as model equation in analyzing
anomalous diffusion processes, some important parameters in the model, for
example, the orders of the fractional derivative or the source term, are often
unknown, which requires one to discuss inverse problems to identify these
physical quantities from some additional information that can be observed or
measured practically. This chapter investigates several kinds of inverse
coefficient problems for the fractional diffusion equation
Numerical analysis of nonlinear subdiffusion equations
We present a general framework for the rigorous numerical analysis of
time-fractional nonlinear parabolic partial differential equations, with a
fractional derivative of order in time. The framework relies
on three technical tools: a fractional version of the discrete Gr\"onwall-type
inequality, discrete maximal regularity, and regularity theory of nonlinear
equations. We establish a general criterion for showing the fractional discrete
Gr\"onwall inequality, and verify it for the L1 scheme and convolution
quadrature generated by BDFs. Further, we provide a complete solution theory,
e.g., existence, uniqueness and regularity, for a time-fractional diffusion
equation with a Lipschitz nonlinear source term. Together with the known
results of discrete maximal regularity, we derive pointwise norm
error estimates for semidiscrete Galerkin finite element solutions and fully
discrete solutions, which are of order (up to a logarithmic factor)
and , respectively, without any extra regularity assumption on
the solution or compatibility condition on the problem data. The sharpness of
the convergence rates is supported by the numerical experiments
A note on semilinear fractional elliptic equation: analysis and discretization
In this paper we study existence, regularity, and approximation of solution
to a fractional semilinear elliptic equation of order . We
identify minimal conditions on the nonlinear term and the source which leads to
existence of weak solutions and uniform -bound on the solutions. Next
we realize the fractional Laplacian as a Dirichlet-to-Neumann map via the
Caffarelli-Silvestre extension. We introduce a first-degree tensor product
finite elements space to approximate the truncated problem. We derive a priori
error estimates and conclude with an illustrative numerical example
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
Inverse Problems of Determining Sources of the Fractional Partial Differential Equations
In this chapter, we mainly review theoretical results on inverse source
problems for diffusion equations with the Caputo time-fractional derivatives of
order . Our survey covers the following types of inverse
problems: 1. determination of time-dependent functions in interior source terms
2. determination of space-dependent functions in interior source terms 3.
determination of time-dependent functions appearing in boundary condition
Recommended from our members
Computational Inverse Problems for Partial Differential Equations (hybrid meeting)
Inverse problems in partial differential equations (PDEs) consist in reconstructing
some part of a PDE such as a coefficient, a boundary condition, an initial condition, the shape
of a domain, or a singularity from partial knowledge of solutions to the PDE.
This has numerous applications in nondestructive testing, medical imaging, seismology, and optical
imaging. Whereas classically mostly boundary or far field data of solutions to deterministic PDEs were considered,
more recently also statistical properties of solutions to random PDEs have been studied.
The study of numerical reconstruction methods of inverse problems in PDEs is at the interface of
numerical analysis, PDE theory, functional analysis, statistics, optimization, and differential geometry.
This workshop has mainly addressed five related topics of current interest:
model reduction, control-based techniques in inverse problems,
imaging with correlation data of waves, fractional diffusion,
and model-based approaches using machine learning
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