Fractional Duhamel principle
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Sabir Umarov
Abstract
The chapter discusses fractional generalizations of the well-known Duhamel principle. The Duhamel principle, introduced nearly 200 years ago, reduces the Cauchy problem for the inhomogeneous differential equation to the Cauchy problem for a corresponding homogeneous differential equation. Unlike the classical case, fractional versions of the Duhamel principle require a fractional derivative of the inhomogeneous term in the initial condition of the reduced equation. We present generalizations of the Duhamel principle to wide classes of single time-fractional and distributed order pseudo-differential equations, both containing Caputo-Djrbashian and Riemann-Liouville derivatives. The abstract case also presented to capture initialboundary value problems for equations given on bounded domains. Note that the fractional Duhamel principle for equations containing Caputo-Djrbashian derivatives and Riemann-Liouville derivatives significantly differ. A number of applications of the fractional Duhamel principle have been appeared recent years. Here, we discuss some of these applications, as well.
Abstract
The chapter discusses fractional generalizations of the well-known Duhamel principle. The Duhamel principle, introduced nearly 200 years ago, reduces the Cauchy problem for the inhomogeneous differential equation to the Cauchy problem for a corresponding homogeneous differential equation. Unlike the classical case, fractional versions of the Duhamel principle require a fractional derivative of the inhomogeneous term in the initial condition of the reduced equation. We present generalizations of the Duhamel principle to wide classes of single time-fractional and distributed order pseudo-differential equations, both containing Caputo-Djrbashian and Riemann-Liouville derivatives. The abstract case also presented to capture initialboundary value problems for equations given on bounded domains. Note that the fractional Duhamel principle for equations containing Caputo-Djrbashian derivatives and Riemann-Liouville derivatives significantly differ. A number of applications of the fractional Duhamel principle have been appeared recent years. Here, we discuss some of these applications, as well.
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
