Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models.

Area of Science:

• Computational Mechanics
• Applied Mathematics
• Numerical Analysis

Background:

• Solving higher-order partial differential equations (PDEs) with variable coefficients, common in Euler-Bernoulli beam models, presents significant numerical challenges.
• Existing numerical methods may struggle with the complexity and accuracy requirements for these types of equations.

Purpose of the Study:

• To propose an efficient and robust numerical scheme for solving variable coefficients' fourth-order partial differential equations (FOPDEs).
• To address the difficulties in obtaining accurate numerical solutions for complex PDEs in engineering applications.

Main Methods:

• A hybrid numerical scheme combining second-order finite difference for temporal discretization and Haar wavelets for spatial approximation.
• Utilizing integration and Haar matrices to transform PDEs into a solvable system of linear equations.
• Stability analysis using the Lax-Richtmyer criterion and computational verification.

Main Results:

• The proposed scheme demonstrates computational convergence near order two.
• Stability of the scheme is theoretically derived and computationally verified.
• Numerical results for various test problems show good agreement with exact solutions and existing literature.

Conclusions:

• The developed hybrid Haar wavelet scheme is effective and robust for solving FOPDEs in Euler-Bernoulli beam models.
• The method provides accurate and reliable solutions, validating its applicability in computational mechanics and engineering simulations.