tailieunhanh - A numerical solution of wave equation arising in non-homogeneous cylindrical shells

A numerical solution of wave equation arising in non-homogeneous cylindrical shells is considered. Stable numerical schemes are developed. The stability estimates for the solution of these difference schemes and first and second order difference derivatives are presented. | Turk J Math 32 (2008) , 409 – 427. ¨ ITAK ˙ c TUB A Numerical Solution of Wave Equation Arising in Non-Homogeneous Cylindrical Shells∗ Allaberen Ashyralyev, Mehmet Emir K¨ oksal Abstract A numerical solution of wave equation arising in non-homogeneous cylindrical shells is considered. Stable numerical schemes are developed. The stability estimates for the solution of these difference schemes and first and second order difference derivatives are presented. Applying the difference schemes, the numerical methods are proposed for solving the the given initial-boundary value problem. Key Words: Hyperbolic equation; Difference schemes; Stability. 1. Introduction In various sciences and engineering, hyperbolic partial differential equations are used to formulate and solve problems that involve unknown functions of several variables, such as the propagation of sound, heat or wave, or more generally any process that is distributed in space, or distributed in space and time as in thermodynamics, elasticity and electromagnetic. In recent years, many applied problems in cylindrical and spherical coordinates by using method of characteristic in mechanics and engineering science, were formulated as the mathematical model of variable types. For instance, in [1], the dynamic response of layered composites consisting of N isotropic, elastic and functionally graded cylindrical layers (nonhomogeneous) were investigated. One-dimensional transient dynamic response AMS Mathematics Subject Classification: 65M12, 65J10. on Differential Equations and Its Applications, 8-10 Febr. 2007, METU, Ankara ∗ Workshop 409 ¨ ASHYRALYEV, KOKSAL of functionally graded spherical multilayered media was formulated as an initial-boundary value problem of which solutions were obtained by employing the method of characteristic in [2]. One-dimensional transient wave propagation in multilayered functionally graded media was investigated in papers [3] and [4]. . Ding et al. [5] has developed a solution .