An adaptive approach for solving fourth-order partial differential equations: algorithm and applications to engineering models
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Date
2022
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Abstract
A novel numerical technique based on orthogonal Laguerre polynomials called the Laguerre matrix collocation method is proposed. The motivation of the study is to reduce the computational cost in mathematical models by adapting Laguerre polynomials directly without transforming them into the truncated Taylor polynomial basis. The new approach is suitable for solving fourth-order partial differential equations arising in physics and engineering. The algorithm and error analyses are presented in general form and applied to two physical models from solid mechanics. First, the technique is used to solve the governing equation for a plate deflection under a harmonically distributed static load. Second, the algorithm is applied to the bending model of a shear deformable plate under the harmonically distributed static load. The boundary conditions of the models are specified, and the bending responses of the models are obtained. The numerical results are compared with the exact results from the literature. The comparisons show that the new approach is suitable for numerical solutions of fourth-order partial differential equations which arise in physics and engineering. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.
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Keywords
Numerical methods , Partial differential equations , Polynomials , Adaptive approach , Collocation method , Differential equation algorithm , Engineering modelling , Fourth order partial differential equations , Laguerre's polynomials , Matrix methods , New approaches , Plate deflections , Static loads , Boundary conditions