Mixed Initial-Boundary Value Problem for Telegraph Equation in Domain with Variable Borders

Abstract

Mixed initial-boundary value problem for telegraph equation in domain with variable borders is considered. On one part of domain’s border are the boundary conditions of the first type, on other part of the boundary are set boundary conditions of the second type. Besides, the sizes of area are variable. The solution of such problem demands development of special methods. With the help of consecutive application of procedure of construction waves reflected from borders of domain, it is possible to obtain the solution of this problem in quadratures. In addition, for construction of the waves reflected from mobile border, it is necessary to apply the procedure specially developed for these purposes.

Telegraph process in the bounded domain with absorbing lower boundary and reflecting with delay upper boundary

“…Substituting F 0 (0, A) = 0 and F 1 (0, A) = 0 into (23) and solving the system, we get Hence, in order to derive explicit formulae for ω 0 A , ψ 0 A , ω 1 A , ψ 1 A , ω 2 , ψ 2 , we need to find corresponding inverse Laplace transforms in (25), ( 26) and ( 27). After that ω and ψ can be found from (24) and φ can be found from (19). After that L 0 and L 1 can be found from (14), which gives the full description of the dynamics of the process.…”

“…Telegraph process has been applied to the number of problems in physics ( [17], [10], [9]) and economics ( [15], [3], [5], [21], [23], [22], [13]). A growing body of literature is dedicated to the properties of telegraph process: [18] and [6] derive the distribution of the symmetric telegraph process, its maximum and first passage time for unbounded domain, [16] solve telegrapher's equation with reflecting and partially reflecting boundaries, [12] solves telegrapher's equation by the Adomian decomposition method, [2] and [14] analyze asymmetric telegraph process on unbounded domain, [4] and [8] solve time-fractional telegrapher's equation including the case of bounded domain with different types of boundary conditions, [19] solves telegrapher's equation in the domain with variable borders.…”

Динамічне Поле Пружних Зміщень В Канаті, Який Намотується На Барабан, При Підйомі Вантажу

“…For construction of the solution (2.8) in [18] continuation of function −σ(t) on all axis t is executed as…”

“…For satisfaction it is necessary for this condition to build a wave v n10b , as the result of reection of a wave v n10 from the end x = 0. Construction of such reected wave is executed in [20] where it is shown, that…”

“…The author has developed methods for obtaining of exact solutions for the boundary-value problems with mobile borders for both the wave and telegraph equations [11][12][13][14][15][16][17][18][19][20]. They are based on maintenance of a continuity of the displacements in points of reection of waves.…”