Exact solutions of multi-term fractional diffusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multi- term time-fractional diffusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial differential equations are converted into time-fractional ordinary differ- ential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-diffusion problems are given to validate the proposed analytical method.     

Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry

The paper is concerned with analysis of time-fractional diffusion-wave equation with Caputo fractional derivative in a half-space. Several examples of problems with Dirichlet and Neumann conditions at the boundary of a half-space are solved using integral transforms technique. For the first and second time-derivative terms, the obtained solutions reduce to the solutions of the ordinary diffusion and wave equations. Numerical results are presented graphically for various values of order of fractional derivative.     

Time-fractional radial diffusion in a sphere

The radial diffusion in a sphere of radius R is described using time-fractional diffusion equation. The Caputo fractional derivative of the order 0<α<2 is used. The Laplace and finite sin-Fourier transforms are employed. The solution is written in terms of the Mittag–Leffler functions. For the first and second time-derivative terms, the obtained solutions reduce to the solutions of the ordinary diffusion and wave equations. Several examples of signaling, source and Cauchy problems are presented. Numerical results are illustrated graphically.     

Exact solutions of multi-term fractional difusion-wave equations with Robin type boundary conditions

General exact solutions in terms of wavelet expansion are obtained for multiterm time-fractional difusion-wave equations with Robin type boundary conditions. By proposing a new method of integral transform for solving boundary value problems, such fractional partial diferential equations are converted into time-fractional ordinary diferential equations, which are further reduced to algebraic equations by using the Laplace transform. Then, with a wavelet-based exact formula of Laplace inversion, the resulting exact solutions in the Laplace transform domain are reversed to the time-space domain. Three examples of wave-difusion problems are given to validate the proposed analytical method.     

Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions

We give an analytical treatment of a time fractional diffusion equation with Caputo time-fractional derivative in a bounded domain with different boundary conditions. We use the Fourier method of separation of variables and Laplace transform method. The solution is obtained in terms of the Mittag-Leffler-type functions and complete set of eigenfunctions of the Sturm–Liouville problem. Such problems can be used in the context of anomalous diffusion in complex media, as well as for modeling voltammetric experiment in limiting diffusion space.     

Time-fractional radial diffusion in hollow geometries

The aim of this paper is to present the analytical solutions corresponding to the time-fractional radial diffusion in some hollow geometries. The Caputo fractional derivative is used. With the method of separation of variables and the Laplace transform, the solutions are presented in terms of the Mittag-Leffler functions. In the limitting cases, the similar solutions for the ordinary diffusion and wave equations are obtained. Furthermore, the numerical results are illustrated graphically.     

General solutions to a class of time fractional partial differential equations

A class of time fractional partial differential equations is considered, which in- cludes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier- Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine- Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.     

Coupling model for unsteady MHD flow of generalized Maxwell fluid with radiation thermal transform

This paper introduces a new model for the Fourier law of heat conduction with the time-fractional order to the generalized Maxwell fluid. The flow is influenced by magnetic field, radiation heat, and heat source. A fractional calculus approach is used to establish the constitutive relationship coupling model of a viscoelastic fluid. We use the Laplace transform and solve ordinary differential equations with a matrix form to obtain the velocity and temperature in the Laplace domain. To obtain solutions from the Laplace space back to the original space, the numerical inversion of the Laplace transform is used. According to the results and graphs, a new theory can be constructed. Comparisons of the associated parameters and the corresponding flow and heat transfer characteristics are presented and analyzed in detail.     

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress inten- sity factors （DSIFs） for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractional wave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation （ODE）. The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional dif- ferential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.     

Time and Space Fractional Diffusion in Finite Systems

Spatiotemporal nonlocal diffusion in a bounded system is addressed by considering fractional diffusion in a linear, composite system. By considering limiting conditions, solutions for combinations of Neumann and Dirichlet boundary conditions (either zero or nonzero) at the ends of a finite system are derived in terms of Mittag–Leffler functions by the Laplace transformation. Computational viability is demonstrated by inverting the solutions numerically and comparing resulting calculations with asymptotic solutions. Time and space fractional derivatives, defined by variables \(\alpha \) and \(\beta \), respectively, are employed in the Caputo sense; a single-sided, asymmetric space derivative is used. Inspection of the asymptotic solutions leads to insights on the structure of the solutions that may not be available otherwise; the resulting deductions are verified through the numerical inversions. For pure superdiffusion, characteristics of some of the solutions presented here are very similar to those of classical diffusion but combined effects for the corresponding situation result in power-law behaviors. Incidentally, to our knowledge, the pressure distribution for space fractional diffusion at long enough times in a finite system is derived based on first principles for the first time.