Abstract nonlinear fractional evolution equations
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Carlos Lizama
Abstract
In this paper, we review some of the main results in the field of abstract nonlinear fractional evolution equations. We study solutions of the semilinear Cauchy problem in the subdiffusive and superdiffusive cases, mainly with the Caputo and Riemann-Liouville fractional-order derivative, in the setting of the real semiaxis and real axis, and under various assumptions on the main data of the given equations. We consider in our analysis several kinds of perturbed systems, for example, delay, control, and stochastic properties, even with nonlocal conditions and impulses. We provide a complete description of the representation of mild solutions in terms of associated solution families of operators.
Abstract
In this paper, we review some of the main results in the field of abstract nonlinear fractional evolution equations. We study solutions of the semilinear Cauchy problem in the subdiffusive and superdiffusive cases, mainly with the Caputo and Riemann-Liouville fractional-order derivative, in the setting of the real semiaxis and real axis, and under various assumptions on the main data of the given equations. We consider in our analysis several kinds of perturbed systems, for example, delay, control, and stochastic properties, even with nonlocal conditions and impulses. We provide a complete description of the representation of mild solutions in terms of associated solution families of operators.
Kapitel in diesem Buch
- 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
Kapitel in diesem Buch
- 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
