Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems
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In this paper, the authors considered the initial value/boundary value problems for fractional diffusion-wave equation and established the unique existence of the weak solution and the asymptotic behavior as the time t goes to ∞ and the proofs are based on the eigenfunction expansions.About:
This article is published in Journal of Mathematical Analysis and Applications.The article was published on 2011-10-01 and is currently open access. It has received 965 citations till now. The article focuses on the topics: Boundary value problem & Initial value problem.read more
Citations
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Journal ArticleDOI
Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate
Masahiro Yamamoto,Ying Zhang +1 more
TL;DR: In this article, an inverse problem of determining a zeroth-order coefficient in a one-dimensional fractional diffusion equation of half-order in time is investigated under some assumptions on the regularity of the solutions and coefficients.
Journal ArticleDOI
Reconstruction of a time-dependent source term in a time-fractional diffusion equation
Ting Wei,Z.Q. Zhang +1 more
TL;DR: Based on the separation of variables and Duhamel's principle, the authors transform the inverse source problem into a first kind Volterra integral equation with the source term as the unknown function and then show the illposedness of the problem.
Book ChapterDOI
Inverse Problems of Determining Parameters of the Fractional Partial Differential Equations
TL;DR: Inverse problems in determining unknown parameters of the model are not only theoretically interesting, but also necessary for finding solutions to initial-boundary value problems and studying properties of solutions as discussed by the authors.
Journal ArticleDOI
Semi-implicit Galerkin–Legendre Spectral Schemes for Nonlinear Time-Space Fractional Diffusion–Reaction Equations with Smooth and Nonsmooth Solutions
TL;DR: The governing partial differential equation generalizes the Hodgkin–Huxley, the Allen–Cahn and the Fisher–Kolmogorov–Petrovskii–Piscounov equations, and guarantees the unconditional stability.
Determination of order in fractional diffusion equation
TL;DR: In this paper, the authors prove formulae of reconstructing the order of fractional derivative in time in the fractional diffusion equation by time history at one fixed spatial point, based on asymptotics of the solution as t! 0 or t! 1.
References
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Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
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Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
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Theory and Applications of Fractional Differential Equations
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.