Fractional diffusion-wave phenomena
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Yuri Luchko
Abstract
In this chapter, basic properties of the fundamental solutions to the initialvalue problems for the fractional diffusion-wave equations with the time-fractional Caputo derivative and the Riesz-Feller space-fractional derivative or the Riesz derivative (fractional Laplacian) are discussed. We start with the Mellin-Barnes representations of the fundamental solution to the one-dimensional diffusion-wave equation with the Riesz-Feller space-fractional derivative and continue with a discussion of its properties. In the multidimensional case, we restrict ourselves to analysis of the diffusion-wave equation with the fractional Laplacian. For its fundamental solution, we provide both its Mellin-Barnes representation and several important results that follow from this representation including some closed-form formulas for its particular cases and connection between solutions in different dimensions. The main focus of presentation is on both probabilistic and physical interpretations of solutions to the initial-value problems for the fractional diffusion-wave equations. In particular, their interpretations as anomalous diffusion processes or diffusive waves, respectively, are discussed.
Abstract
In this chapter, basic properties of the fundamental solutions to the initialvalue problems for the fractional diffusion-wave equations with the time-fractional Caputo derivative and the Riesz-Feller space-fractional derivative or the Riesz derivative (fractional Laplacian) are discussed. We start with the Mellin-Barnes representations of the fundamental solution to the one-dimensional diffusion-wave equation with the Riesz-Feller space-fractional derivative and continue with a discussion of its properties. In the multidimensional case, we restrict ourselves to analysis of the diffusion-wave equation with the fractional Laplacian. For its fundamental solution, we provide both its Mellin-Barnes representation and several important results that follow from this representation including some closed-form formulas for its particular cases and connection between solutions in different dimensions. The main focus of presentation is on both probabilistic and physical interpretations of solutions to the initial-value problems for the fractional diffusion-wave equations. In particular, their interpretations as anomalous diffusion processes or diffusive waves, respectively, are discussed.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- Fractional electromagnetics 1
- Fractional electrodynamics with spatial dispersion 25
- Fractional-calculus tools applied to study the nonexponential relaxation in dielectrics 53
- Fractional diffusion-wave phenomena 71
- Fractional diffusion and parametric subordination 99
- The fractional advection-dispersion equation for contaminant transport 129
- Anomalous diffusion in interstellar medium 151
- Fractional kinetics in random/complex media 183
- Nonlocal quantum mechanics: fractional calculus approach 207
- Fractional quantum fields 237
- Fractional quantum mechanics of open quantum systems 257
- Fractional quantum mechanics with topological constraint 279
- Fractional time quantum mechanics 299
- Index 317
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
- Fractional electromagnetics 1
- Fractional electrodynamics with spatial dispersion 25
- Fractional-calculus tools applied to study the nonexponential relaxation in dielectrics 53
- Fractional diffusion-wave phenomena 71
- Fractional diffusion and parametric subordination 99
- The fractional advection-dispersion equation for contaminant transport 129
- Anomalous diffusion in interstellar medium 151
- Fractional kinetics in random/complex media 183
- Nonlocal quantum mechanics: fractional calculus approach 207
- Fractional quantum fields 237
- Fractional quantum mechanics of open quantum systems 257
- Fractional quantum mechanics with topological constraint 279
- Fractional time quantum mechanics 299
- Index 317
