Solution of The Duffing Equation Using Exponential Time Differencing Method

To describe the spring stiffening effect that occurs in physics and engineering problems, Georg Duffing added the cubic stiffness term to the linear harmonic oscillator equation and is now known as the Duffing oscillator. Despite its simplicity, its dynamic behavior is very diverse. In this research, the Exponential Time Difference method is introduced to solve the Duffing oscillator numerically. To formulate the ETD method, we were using the integration factors. It is a function which, when multiplied by an ordinary differential equation, produces a differential equation that can be integrated. This method is an effective numerical method for solving complex differential equations, especially equations that have strong non-linearity The ETD method delivers highly accurate numerical solutions for the Duffing oscillator, with minimal discrepancy from the analytical results. Through parameter variation, the ETD method's applicability extends to diverse Duffing oscillator configurations

Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems

High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schr\"odinger Equation

In this work we introduce a new family of twelve-step linear multistep methods for the integration of the Schr\"odinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications.Comment: 36 pages, 6 figure