Boundary value problem for a fractional order wave equation



1. Introduction

The mathematical framework of fractional-order integro-differentiation provides a powerful tool for modeling systems in which nonlocal properties in both time and space are significant. Fractional derivatives allow for the inclusion of memory effects (temporal nonlocality) and spatial correlations (spatial nonlocality), leading to their widespread application in the natural sciences.

Equations with fractional derivatives describe the evolution of physical systems with inherent losses. The order of the derivative reflects the degree to which the system's states are retained over time. Such «residual memory» systems occupy an intermediate position between systems with complete memory and memoryless Markov systems.

The classical treatment of compressible fluid flow and conservation laws in multiple spatial variables is comprehensively presented in [5], which serves as a foundational reference for this area of mathematical physics.

Fractional calculus is a branch of mathematical analysis concerned with derivatives and integrals of arbitrary (including real) order. One of its key applications is the study and solution of differential equations involving fractional derivatives.

In modeling continuous media with memory, equations emerge that describe a new class of wave phenomena that lie between classical diffusion and conventional wave propagation [1], [2].

Fractional differentiation is understood as a combination of differentiation and integration. In this paper, we address a boundary value problem for the wave equation involving the Riemann–Liouville fractional derivative in the upper half-plane. The initial conditions are also expressed in terms of fractional derivatives. The solution involves applying the Fourier transform with respect to the spatial variable, followed by the Laplace transform in time, and then performing the respective inverse transforms. Finally, two limiting cases of the fractional order are analyzed and discussed.

2. Preliminaries

This section will present the definition of fractional derivative in the Riemann-Liouville sense, the Mittag-Leffler function, the Laplace and Fourier transforms for fractional derivative, which will be used in solving the problem.

2.1. Riemann-Liouville Fractional Derivatives

This expression is the best known definition of the fractional derivative, usually called the fractional derivative in the Riemann-Liouville sense [3].

2.2. Mittag-Leffler type functions

A two-parameter function of Mittag-Leffler type is defined by the expansion of the series [3]

2.3. Boundary value problem for the wave equation. Green's function

In [6], the author investigated a fractional wave equation involving fractional derivatives of the same order

(where ) in both space and time. It was demonstrated that the fundamental solution of this equation describes damped waves propagating at different constant velocities, which depend on the value of the fractional order [7].

When the boundary conditions are satisfied and in the domain D

Where are given functions.

The function is the solution of the equation

and satisfy

Let us call the Green's function of the first boundary value problem for the wave equation. These properties are also satisfied by the function

where

3. Problem Formulation and Formal Solution

In the domain

find the solution of equation:

(4)

(5)

isthe Riemann-Liouville Fractional Derivative (1) of order .

Theorem. Function

where

and

is the Wright function is the solution of problem (4)–(5) under the condition .

Proof :

We apply the Fourier transform to the equation (4)-(5) with respect to the variable :

(6)

(7)

- the transforms of the input data in the problem (4)-(5)

We apply the Laplace transform to equation (6) with respect to the variable t, taking into account conditions (7). We obtain:

, (8)

where is the image of the function

We apply the inverse Laplace transform with respect to the variable to equation (5):

(9)

where is the Mittag-Leffler function (2).

Now, we apply the inverse Fourier transform with respect to the variable to equation (6) (see [3], pp. 141–142):

Applying the inverse Laplace transform with respect to

, using the formula:

(14)

with , we obtain

. (15)

Applying the inverse Laplace transform again, using the formula (14) with :

. (16)

Substituting equations (15) and (16) into the equation (10), we obtain the solution of the original problem (4)–(5).

Limiting cases of the fractional derivative order

Let us now consider the limiting cases of the fractional derivative order for и .

Case .

The problem (4)-(5) takes the form:

(17)

(18)

(19)

The solution to the problem (17)–(18) in domain given by:

(20)

where the Green's function (3), is given by formula 3.471 (2) [4]

. (21)

It is shown that for , solution (10) coincides with (20), considering expressions (15) and (16).

Case

Then the problem (4)–(5) becomes:

The solution is given by [5]:

(22)

Thus, the final expression becomes:

It is shown that for , solution (10) coincides with (22), considering relations (15) and (16). The theorem is proven.

References:

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2. Olemskoy, A. I., Flat, A. Ya. Application of the Fractal Concept in Condensed Matter Physics / Olemskoy, A. I., Flat, A. Ya. — 1st ed. — Moscow: Uspekhi Fizicheskikh Nauk, 1993. — Vol. 163, No. 12. — 50 c.
3. Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications / Podlubny, I. — 1st ed. — San Diego: Academic Press, 1999. — 340 c. — (Mathematics in Science and Engineering; Vol. 198).
4. Gradshteyn, I. S., Ryzhik, I. M. Table of Integrals, Series, and Products / Gradshteyn, I. S., Ryzhik, I. M. — 4th ed. — Moscow: Fizmatgiz, 1963. — 1100 c.
5. Majda, A. J. Compressible Fluid Flow and Systems of Conservation Laws in Several Variables / Majda, A. J. — 1st ed. — New York: Springer-Verlag, 1984. — 551 c.
6. Luchko, Yu. Fractional Wave Equation and Damped Waves / Luchko, Yu. — 1st ed. — New York: Journal of Mathematical Physics, 2013. — Vol. 54, 031505.
7. Pskhu, A. V. Boundary Value Problems for Partial Differential Equations of Fractional and Continuous Order / Pskhu, A. V. — Doctoral dissertation. — Moscow: Moscow State University, 2009. — 320 p.