A Semi-Analytical Solution to a Generalized Nonlinear Van Der Pol Equation in Plasma By MsDTM (Contemporary Mathematics, 2024)
This article examines the effectiveness of the multistage differential transform method (MsDTM) in solving equations with a very strong nonlinear term. It introduces MsDTM as a method for solving the generalized nonlinear Van der Pol equation, which features strong nonlinearity. The generalized nonlinear Van der Pol equation arises in plasma and describes the propagation of various nonlinear phenomena, such as wave propagation in astrophysical plasma. MsDTM demonstrates greater accuracy compared to other analytical and numerical methods, such as the 4th-order Runge-Kutta Method (4thRKM), due to its ability to enhance accuracy through two factors: the number of iterations and the time step size. Most numerical methods rely solely on reducing the time step size to improve accuracy, but for some types of equations, this requires an impractically small time step size, causing the method to fail. In contrast, MsDTM offers an additional means of improving accuracy by increasing the number of iterations. The paper successfully applies MsDTM to solve the Van der Pol equation and presents the results, demonstrating that the method is highly effective for equations with very strong nonlinearity.
Keywords: van der pol equation; plasma; multistage differential transform method; numerical simulation