Inverse problems of determining parameters of the fractional partial differential equations
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Zhiyuan Li
Abstract
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be directly measured, which requires one to discuss inverse problems of identifying these physical quantities from some indirectly observed information of solutions. Inverse problems in determining these 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. This chapter surveys works on such inverse problems for fractional diffusion equations.
Abstract
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be directly measured, which requires one to discuss inverse problems of identifying these physical quantities from some indirectly observed information of solutions. Inverse problems in determining these 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. This chapter surveys works on such inverse problems for fractional diffusion equations.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- General theory of Caputo-type fractional differential equations 1
- Problems of Sturm–Liouville type for differential equations with fractional derivatives 21
- Maps with power-law memory: direct introduction and Eulerian numbers, fractional maps, and fractional difference maps 47
- Symmetries and group invariant solutions of fractional ordinary differential equations 65
- Operational method for fractional ordinary differential equations 91
- Lyapunov-type inequalities for fractional boundary value problems 119
- Fractional-parabolic equations and systems. Cauchy problem 145
- Time fractional diffusion equations: solution concepts, regularity, and long-time behavior 159
- Layer potentials for the time-fractional diffusion equation 181
- Fractional-hyperbolic equations and systems. Cauchy problem 197
- Equations with general fractional time derivatives–Cauchy problem 223
- User’s guide to the fractional Laplacian and the method of semigroups 235
- Parametrix methods for equations with fractional Laplacians 267
- Maximum principle for the time-fractional PDEs 299
- Wave equation involving fractional derivatives of real and complex fractional order 327
- Symmetries, conservation laws and group invariant solutions of fractional PDEs 353
- Fractional Duhamel principle 383
- Inverse problems of determining sources of the fractional partial differential equations 411
- Inverse problems of determining parameters of the fractional partial differential equations 431
- Inverse problems of determining coefficients of the fractional partial differential equations 443
- Abstract linear fractional evolution equations 465
- Abstract nonlinear fractional evolution equations 499
- Index 515
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- General theory of Caputo-type fractional differential equations 1
- Problems of Sturm–Liouville type for differential equations with fractional derivatives 21
- Maps with power-law memory: direct introduction and Eulerian numbers, fractional maps, and fractional difference maps 47
- Symmetries and group invariant solutions of fractional ordinary differential equations 65
- Operational method for fractional ordinary differential equations 91
- Lyapunov-type inequalities for fractional boundary value problems 119
- Fractional-parabolic equations and systems. Cauchy problem 145
- Time fractional diffusion equations: solution concepts, regularity, and long-time behavior 159
- Layer potentials for the time-fractional diffusion equation 181
- Fractional-hyperbolic equations and systems. Cauchy problem 197
- Equations with general fractional time derivatives–Cauchy problem 223
- User’s guide to the fractional Laplacian and the method of semigroups 235
- Parametrix methods for equations with fractional Laplacians 267
- Maximum principle for the time-fractional PDEs 299
- Wave equation involving fractional derivatives of real and complex fractional order 327
- Symmetries, conservation laws and group invariant solutions of fractional PDEs 353
- Fractional Duhamel principle 383
- Inverse problems of determining sources of the fractional partial differential equations 411
- Inverse problems of determining parameters of the fractional partial differential equations 431
- Inverse problems of determining coefficients of the fractional partial differential equations 443
- Abstract linear fractional evolution equations 465
- Abstract nonlinear fractional evolution equations 499
- Index 515
