Time-fractional Diffusion-wave Equation Research Articles

Neumann boundary-value problems for a time-fractional diffusion-wave equation in a half-plane

The time-fractional diffusion-wave equation with the Caputo derivative of the order 0<α<2 is considered in a half-plane. Two types of Neumann boundary condition are examined: the mathematical condition with the prescribed boundary value of the normal derivative and the physical one with the prescribed boundary value of the matter flux.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited.In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Central symmetric solution to the Neumann problem for a time-fractional diffusion-wave equation in a sphere

In this paper, a time-fractional central symmetric diffusion-wave equation is investigated in a sphere. Two types of Neumann boundary condition are considered: the mathematical condition with the prescribed boundary value of the normal derivative and the physical condition with the prescribed boundary value of the matter flux. Several examples of problems are solved using the Laplace integral transform with respect to time and the finite sin-Fourier transform of the special type with respect to the spatial coordinate. Numerical results are illustrated graphically.

Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder

Abstract The time-fractional diffusion-wave equation is considered in an infinite cylinder in the case of three spatial coordinates r, ϕ and z. The Caputo fractional derivative of the order 0 &lt; α ≤ 2 is used. Several examples of problems with Dirichlet and Neumann boundary conditions at a surface of the cylinder are solved using the integral transforms technique. Numerical results are illustrated graphically.

Solutions to Diffusion-Wave Equation in a Body with a Spherical Cavity under Dirichlet Boundary Condition

Solutions to time-fractional diffusion-wave equation with a source term in spherical coordinates are obtained for an infinite medium with a spherical cavity. The solutions are found using the Laplace transform with respect to time t, the finite Fourier transform with respect to the angular coordinate Ï•, the Legendre transform with respect to the spatial coordinate μ, and the Weber transform of the order n+1/2 with respect to the radial coordinate r. In the central symmetric case with one spatial coordinate r the obtained resultscoincide with those studied earlier.

Solutions to Time-Fractional Diffusion-Wave Equation in Cylindrical Coordinates

Nonaxisymmetric solutions to time-fractional diffusion-wave equation with a source term in cylindrical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time , the Hankel transform with respect to the radial coordinate , the finite Fourier transform with respect to the angular coordinate , and the exponential Fourier transform with respect to the spatial coordinate . Numerical results are illustrated graphically.

Evolution of the initial box-signal for time-fractional diffusion–wave equation in a case of different spatial dimensions

In the case of time-fractional diffusion–wave equation considered in the spatial domain − ∞ < x < ∞ , evolution of initial box-signal was investigated by Mainardi [F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals 7 (1996) 1461–1477]. In the present paper, we supplement Mainardi’s results with additional numerical calculations illustrating the behavior of the solution and solve the corresponding problems for axisymmetric and central symmetric cases. The obtained results show an unusual behavior of solutions.

Microscopic formulation of fractional calculus theory of viscoelasticity based on lattice dynamics

A microscopic fractional calculus theory of viscoelasticity is developed on the basis of lattice dynamics by generalizing the standard model of a chain of coupled simple harmonic oscillators to a chain of coupled fractional oscillators by generalizing the integral equations of motion of a chain of simple harmonic oscillators into ones involving fractional integrals. This set of integral equations of motion pertaining to the chain of coupled fractional oscillators is solved by using Laplace transforms. In the continuum limit the time-fractional diffusion-wave equation in one dimension is obtained. The response of the system to the sinusoidal forcing in this limit consists of a transient part and an attenuated steady part. Expressions for the absorption coefficient and the specific dissipation function are derived and numerical applications are discussed.

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 distributed-order diffusion-wave equation. I. Volterra-type equation

A single-order time-fractional diffusion-wave equation is generalized by introducing a time distributed-order fractional derivative and forcing term, while a Laplacian is replaced by a general linear multi-dimensional spatial differential operator. The obtained equation is (in the case of the Laplacian) called a time distributed-order diffusion-wave equation. We analyse a Cauchy problem for such an equation by means of the theory of an abstract Volterra equation. The weight distribution, occurring in the distributed-order fractional derivative, is specified as the sum of the Dirac distributions and the existence and uniqueness of solutions to the Cauchy problem, and the corresponding Volterra-type equation were proven for a general linear spatial differential operator, as well as in the special case when the operator is Laplacian.

Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation

In this Letter we propose a new generalization of the two-dimensional differential transform method that will extend the application of the method to a diffusion-wave equation with space- and time-fractional derivatives. The new generalization is based on generalized Taylor's formula and Caputo fractional derivative. Theorems that are never existed before are introduced with their proofs. Several illustrative examples are given to demonstrate the effectiveness of the obtained results. The results reveal that the technique introduced here is very effective and convenient for solving partial differential equations of fractional order.

Exact solutions for time-fractional diffusion-wave equations by decomposition method

The time-fractional diffusion-wave equation is considered. The time-fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∊ (0,2]. The fractional derivative is described in the Caputo sense. This paper presents the analytical solutions of the fractional diffusion equations by an Adomian decomposition method. By using initial conditions, the explicit solutions of the equations have been presented in the closed form and then their numerical solutions have been represented graphically. Four examples are presented to show the application of the present technique. The present method performs extremely well in terms of efficiency and simplicity.

Multidimensional solutions of time-fractional diffusion-wave equations

Green's functions and propagator functions for the timefractional generalized diffusionwave equation in one and three dimensions are expressed in terms of Wright functions. In two dimensions they a...

FRACTIONAL DIFFUSIVE WAVES

By fractional diffusive waves we mean the solutions of the so-called time-fractional diffusion-wave equation. This equation is obtained from the classical D'Alembert wave equation by replacing the second-order time derivative with a fractional derivative of order β∈(0,2) and is expected to govern evolution processes intermediate between diffusion and wave propagation when β∈(1,2). Here it is shown to govern the propagation of stress waves in viscoelastic media which, by exhibiting a power law creep, are of relevance in acoustics and seismology since their quality factor turns out to be independent of frequency. The fundamental solutions for the Cauchy and signaling problems are expressed in terms of entire functions (of Wright type) in the similarity variable. Their behaviors turn out to be intermediate between those found in the limiting cases of a perfectly viscous fluid and a perfectly elastic solid. Furthermore, their scaling properties and the relations with some stable probability distributions are outlined.

Fractional Diffusive Waves

By fractional diffusive waves we mean the solutions of the so-called time-fractional diffusion-wave equation. This equation is obtained from the classical D'Alembert wave equation by replacing the second-order time derivative with a fractional derivative of order β∈(0,2) and is expected to govern evolution processes intermediate between diffusion and wave propagation when β∈(1,2). Here it is shown to govern the propagation of stress waves in viscoelastic media which, by exhibiting a power law creep, are of relevance in acoustics and seismology since their quality factor turns out to be independent of frequency. The fundamental solutions for the Cauchy and signaling problems are expressed in terms of entire functions (of Wright type) in the similarity variable. Their behaviors turn out to be intermediate between those found in the limiting cases of a perfectly viscous fluid and a perfectly elastic solid. Furthermore, their scaling properties and the relations with some stable probability distributions are outlined.

Wright functions as scale-invariant solutions of the diffusion-wave equation

The time-fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order α (0< α⩽2). Using the similarity method and the method of the Laplace transform, it is shown that the scale-invariant solutions of the mixed problem of signalling type for the time-fractional diffusion-wave equation are given in terms of the Wright function in the case 0< α<1 and in terms of the generalized Wright function in the case 1< α<2. The reduced equation for the scale-invariant solutions is given in terms of the Caputo-type modification of the Erdélyi–Kober fractional differential operator.

The fundamental solutions for the fractional diffusion-wave equation

The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0 < β ≤ 1 2 or 1 2 < β ≤ 1 , respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and Signalling problems can be expressed in terms of an auxiliary function M( z; β), where z = |x| t β is the similarity variable. Such function is proved to be an entire function of Wright type.

The time fractional diffusion-wave equation

The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0<β≤1/2 or 1/2<β≤1, respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and signalling problems can be expressed in terms of an auxiliary function M (z; β), where z is the similarity variable. Such function, which reduces to the well-known Gaussian function for β=1/2 (ordinary diffusion), is proved to be an entire function of Wright type.