Operational method for fractional ordinary differential equations
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Yuri Luchko
Abstract
In this chapter, a survey of some applications of the operational calculi of Mikusiński type to fractional ordinary differential equations is presented. We start with constructing of an operational calculus for the general multiple Erdélyi-Kober fractional derivative that includes both differential operators of the hyper-Bessel type and the Riemann-Liouville fractional derivative, and then proceed with its modification for the Caputo fractional derivative. The operational calculi allow to easily treat the Cauchy initial value problems for the linear ordinary fractional differential equations with these fractional derivatives. In particular, the multi-term linear fractional differential equations with the Caputo fractional derivatives of arbitrary orders are considered. In all cases, the obtained solutions are expressed in terms the Mittag-Leffler-type functions.
Abstract
In this chapter, a survey of some applications of the operational calculi of Mikusiński type to fractional ordinary differential equations is presented. We start with constructing of an operational calculus for the general multiple Erdélyi-Kober fractional derivative that includes both differential operators of the hyper-Bessel type and the Riemann-Liouville fractional derivative, and then proceed with its modification for the Caputo fractional derivative. The operational calculi allow to easily treat the Cauchy initial value problems for the linear ordinary fractional differential equations with these fractional derivatives. In particular, the multi-term linear fractional differential equations with the Caputo fractional derivatives of arbitrary orders are considered. In all cases, the obtained solutions are expressed in terms the Mittag-Leffler-type functions.
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
