Random Algorithm Research For Solving Hyperbolic Telegraph Equation

Monte Carlo (MC) method is also called stochastic simulation method, that is a kind of the simulation method depends on randomized trials. The convergence speed of Monte Carlo method in solving linear algebra system is not influence by the system dimension. Therefore, Monte Carlo method can effectively deal with high dimension problem. With the birth of the super-computer, randomized trial process becomes more quick and effective, and the advantage of Monte Carlo method becomes more outstanding.Hyperbolic partial differential equations in the industrial technology, fluid mechanics, financial, and many other fields have a wide range of applications. Telegraph equation is a kind of typical hyperbolic partial differential equation. It’s derived for the research on relationship between voltage and current on a transmission line. Telegraph equation can also be described as population dynamics, chemical diffusion problem, hyperbolic heat conduction and other physical phenomena. In practice, compared with the ordinary diffusion equa-tion, the telegraph equation is more suitable for depicting reaction diffusion problems in the field of physical, chemical and biological science.The central thought of this paper is:to discretizate hyperbolic partial d-ifferential equations becomes a linear algebraic system, then using the Monte Carlo stochastic simulation method to solve the linear algebra system. This paper propose using random search method to solve one dimensional second-order hyperbolic partial differential equation and through two numerical ex-ample shows the effectiveness of random search method. Using Importance Sampling method, Adaptive Importance Sampling method and Gibbs Sam-pling method to solve one dimensional second-order hyperbolic telegraph e-quation and comparison of three methods with Classical Markov Chain Monte Carlo method. Two numerical examples show the effectiveness of Importance Sampling method, Gibbs Sampling method and Adaptive Importance Sampling method in run time and accuracy.