Numerical Approximation for Nonlinear Gas Dynamic Equation

Laplace transform and new homotopy perturbation methods are adopted to study gas dynamic equation analytically. The solutions introduced in this study can be used to obtain the closed form of the solutions if they are required. The combined method needs less work in comparison with the other homotopy perturbation methods and decreases volume of calculations considerably. Results show that the new method is more effective and convenient to use, and is high accuracy evident. 1. Introduction Many phenomena in engineering, physics, chemistry, and other sciences can be described very successfully by nonlinear models. Except in a limited number of these problems, we have difficulty in finding their exact analytical solutions. Therefore, there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions [1]. In recent decades, numerical calculation methods were good means of analyzing the nonlinear equations, but as the numerical calculation methods improved, analytical methods did, too. Most scientists believe that the combination of numerical and analytical methods can also end with useful results. In recent years, several of such techniques have drawn special attention, such as homotopy perturbation method. In the recent years, an increasing interest of scientists and engineers has been devoted to the analytical asymptotic techniques for solving nonlinear problems. Many new numerical techniques have been widely applied to the nonlinear problems. Based on homotopy, which is a basic concept in topology, general analytical method, namely, the homotopy perturbation method (HPM) is established by He [2–8] in 1998 to obtain series solutions of nonlinear differential equations. The He’s HPM has been already used to solve various functional equations. In this method, the nonlinear problem is transferred to an infinite number of subproblems, and then the solution is approximated by the sum of the solutions of the first several subproblems. This simple method has been applied to solve linear and nonlinear equations of heat transfer [9–11], fluid mechanics [12], nonlinear Schrodinger equations [13], integral equations [14], boundary value problems [15], fractional KdV-Burgers equation [16], nonlinear system of second-order boundary value problems [17], and delay differential equations [18]. The equations of gas dynamics are mathematical expressions based on the physical laws of conservation, namely, the laws of conservation of mass, conservation of momentum, conservation of energy, and so forth. The