Yuri Luchko

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

Fractional Schrödinger equation for a particle moving in a potential well

In this paper, the fractional Schrodinger equation that contains the quantum Riesz fractional derivative instead of the Laplace operator is revisited for the case of a particle moving in the infinite potential well. In the recent papers [M. Jeng, S.-L.-Y. Xu, E. Hawkins, and J. M. Schwarz, “On the nonlocality of the fractional Schrodinger equation,” J. Math. Phys. 51, 062102 (2010)10.1063/1.3430552] and [S. S. Bayin, “On the consistency of the solutions of the space fractional Schrodinger equation,” J. Math. Phys. 53, 042105 (2012)10.1063/1.4705268] published in this journal, controversial opinions regarding solutions to the fractional Schrodinger equation for a particle moving in the infinite potential well that were derived by Laskin [“Fractals and quantum mechanics,” Chaos 10, 780–790 (2000)10.1063/1.1050284] have been given. In this paper, a thorough mathematical treatment of these matters is provided. The problem under consideration is reformulated in terms of three integral equations with the power ...

Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density

AbstractIn this paper, the Cauchy problem for the spatially one-dimensional distributed order diffusion-wave equation

Fractional wave equation and damped waves

\int_0^2 {p(\beta )D_t^\beta u(x,t)d\beta } = \frac{{\partial ^2 }} {{\partial x^2 }}u(x,t)

Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

is considered. Here, the time-fractional derivative Dtβ is understood in the Caputo sense and p(β) is a non-negative weight function with support somewhere in the interval [0, 2]. By employing the technique of the Fourier and Laplace transforms, a representation of the fundamental solution of the Cauchy problem in the transform domain is obtained. The main focus is on the interpretation of the fundamental solution as a probability density function of the space variable x evolving in time t. In particular, the fundamental solution of the time-fractional distributed order wave equation (p(β) ≡ 0, 0 ≤ β < 1) is shown to be non-negative and normalized. In the proof, properties of the completely monotone functions, the Bernstein functions, and the Stieltjes functions are used.

Time-fractional diffusion equation in the fractional Sobolev spaces

In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of the same order α, 1 ⩽ α ⩽ 2, both in space and in time. We show that this feature is a decisive factor for inheriting some crucial characteristics of the wave equation like a constant propagation velocity of both the maximum of its fundamental solution and its gravity and “mass” centers. Moreover, the first, the second, and the Smith centrovelocities of the damped waves described by the fractional wave equation are constant and depend just on the equation order α. The fundamental solution of the fractional wave equation is determined and shown to be a spatial probability density function evolving in time all whose moments of order less than α are finite. To illustrate analytical findings, results of numerical calculations and plots are presented.

Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation

AbstractIn this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)