Solutions of the Main Boundary Value Problems for the Time-Fractional Telegraph Equation by the Green Function Method

Abstract

The inhomogenous time-fractional telegraph equation with Caputo derevatives with constant coefficients is considered. For considered equation the general representation of regular solution in rectangular domain is obtained, and the fundamental solution is presented. Using this representation and the properties of fundamental solution, the Cauchy problem and the basic problems in half-strip and rectangular domains are studied. For Cauchy problem the theorems of existence and uniqueness of solution are proved, and the explicit form of solution is constructed. The solutions of the investigated problems are constructed in terms of the appropriate Green functions, which are also constructed an explicit form.MSC 2010 : Primary 33R11; Secondary 35A08, 35A09, 35C05, 35C15, 35E05, 34A08.

Fundamental Solution of the Multi-Dimensional Time Fractional Telegraph Equation

“…In [15], Mamchuev considered the inhomogeneous time-fractional telegraph equation with Caputo derivatives, and obtained a general representation of regular solution in rectangular domain in terms of fundamental solution and appropriate Green functions. Regarding the multidimensional case, in [6] the authors discussed and derived the solution of the time-fractional telegraph equation in R n × R + with three kinds of nonhomogeneous boundary conditions, by the method of separation of variables.…”

Green Functions of the First Boundary-Value Problem for a Fractional Diffusion—Wave Equation in Multidimensional Domains

“…Proof. It follows from Corollary 1 and Lemma 2 that W(x, y, s) is a regular solution of (7), and, moreover, (30) holds for any x, s ∈ S. By (22) and 31, we get…”

“…Active research in this direction began with works [1][2][3][4]. In the study of equations with fractional diffusion-wave operators in the main part, a wide range of methods and approaches were used, such as: integral transforms [5][6][7][8][9]; group analysis [10][11][12][13]; Fourier methods [14][15][16][17][18]; maximum principles [19,20]; functional methods [21]; asymptotic behavior [22]; Green function methods and method of potentials [23][24][25][26][27][28][29][30][31][32]; parametrix methods [33]; inverse problems and regularization methods [34]; reduction methods [35,36]; etc. We also mention the monographs [37][38][39], which reflect many of these approaches and contain vast bibliographies concerning the issue.…”

Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model

“…In a number of works various problems for the two-term time-fractional diffusion-wave equation are studied, e.g., [25], [26] (Chapter 6) and [27][28][29][30][31][32]. In this work, we study the Stokes' first problem following the technique developed in [33] for the multi-term time-fractional diffusion-wave equation.…”