• PDF / 3,416,081 Bytes
• 12 Pages / 594 x 792 pts Page_size
• 37 Downloads / 233 Views

DOWNLOAD

REPORT

---

BOUNDARY-VALUE PROBLEMS FOR THE FRACTIONAL WAVE EQUATION UDC 517.95

M. A. Kerefov and B. M. Kerefov

Abstract. A nonlocal wave equation with variable coefficients in a rectangular domain is considered. The first and third boundary-value problems in the differential form are examined; the method of lines in the difference form is applied for solving these problems. For the system of difference equations with constant coefficients arising from the method of lines, a solution is obtained. Keywords and phrases: nonlocal wave equation, fractional derivative, method of lines, a priori estimate. AMS Subject Classification: 35L99

1. Introduction. In mathematical modeling of continuous media with memory, a new type of wave motion arises; this motion occupies an intermediate position between ordinary diffusion and classical waves (see [7, 8]). Such motions are described by fractional differential equations, which allows one to construct mathematical models for a wide class of physical and chemical processes in media with fractal geometry. The monograph [9] contains a detailed bibliography on fractional partial differential equations, in particular, the diffusion-wave equation. This equation was studied in [3] by the method of separation of variables. A priori estimates for solutions of boundary-value problems for the diffusionwave equation with the Caputo fractional derivative were obtained in [1]. This paper is devoted to the study of boundary-value problems for the wave equation with the Riemann–Liouville fractional derivative. 2.

The third boundary value problem in the differential form.

Problem 1. In the domain QT = (0, ) × (0, T ], we consider the following boundary-value problem:   ∂ ∂u α+1 k(x, t) + f (x, t), 0 < x < l, 0 < t ≤ T, (1) D0t u = ∂x ∂x  k(0, t)ux (0, t) − β1 u(0, t) = 0, 0 ≤ t ≤ T, (2) k(l, t)ux (l, t) + β2 u(l, t) = 0,  α−1u(x,t)  α D0t u(x, t)t=0 = u0 (x), D0t = u1 (x), (3) t=0 γ is the Riemann–Liouville fractional differentiation operator of order γ. where 0 < α < 1 and D0t

A regular solution of Eq. (1) in the domain QT is a function u = u(x, t) from the class α−1 α ¯ T ), u(x, t), D0t u(x, t) ∈ C(Q D0t

α+1 uxx (x, t), D0t u(x, t) ∈ C(QT ),

which satisfies Eq. (1) at all points (x, t) ∈ QT (see [9, p. 103]). In the case where the coefficient of Eq. (1) is a constant, a solution of the problem (1)–(3) can be found by separation of variables [7, p. 221]. We assume that a regular solution of the problem (1)–(3) exists; then the following theorem holds. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018.

760

c 2020 Springer Science+Business Media, LLC 1072–3374/20/2505–0760 

Theorem 1. Let ¯ T ), kx (x, t), kt (x, t), f (x, t) ∈ C(Q β1 , β2 ≥ 0,

u0 (x), u1 (x) ∈ C[0, l],

0 < c1 ≤ k(x, t) ≤ c2 ,

β1 + β2 > 0,

0 < m1 ≤ kx (x, t) ≤ m2 ,

kt ≤ 0

¯ T and let the following condition hold: e