Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method

Abstract

In this study, we analyze the performance of a numerical scheme based on 3-scale Haar wavelets for dynamic Euler-Bernoulli equation, which is a fourth order time dependent partial differential equation. This type of equations governs the behaviour of a vibrating beam and have many applications in elasticity. For its solution, we first rewrite the fourth order time dependent partial differential equation as a system of partial differential equations by introducing a new variable, and then use finite difference approximations to discretize in time, as well as 3-scale Haar wavelets to discretize in space. By doing so, we obtain a system of algebraic equations whose solution gives wavelet coefficients for constructing the numerical solution of the partial differential equation. To test the accuracy and reliability of the numerical scheme based on 3-scale Haar wavelets, we apply it to five test problems including variable and constant coefficient, as well as homogeneous and non-homogeneous partial differential equations. The obtained results are compared wherever possible with those from previous studies. Numerical results are tabulated and depicted graphically. In the applications of the proposed method, we achieve high accuracy even with small number of collocation points. 

Keywords

• 3-Scale Haar wavelets
• Vibrating beam
• Dynamic Euler-Bernoulli equation

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