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Mathematics 2019, 7(1), 90; https://doi.org/10.3390/math7010090

An Efficient Spectral Method to Solve Multi-Dimensional Linear Partial Different Equations Using Chebyshev Polynomials

School of Aerospace and Mechanical Engineering, Korea Aerospace University, Goyang 10540, Korea
Received: 25 November 2018 / Revised: 30 December 2018 / Accepted: 11 January 2019 / Published: 16 January 2019
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Abstract

We present a new method to efficiently solve a multi-dimensional linear Partial Differential Equation (PDE) called the quasi-inverse matrix diagonalization method. In the proposed method, the Chebyshev-Galerkin method is used to solve multi-dimensional PDEs spectrally. Efficient calculations are conducted by converting dense equations of systems sparse using the quasi-inverse technique and by separating coupled spectral modes using the matrix diagonalization method. When we applied the proposed method to 2-D and 3-D Poisson equations and coupled Helmholtz equations in 2-D and a Stokes problem in 3-D, the proposed method showed higher efficiency in all cases than other current methods such as the quasi-inverse method and the matrix diagonalization method in solving the multi-dimensional PDEs. Due to this efficiency of the proposed method, we believe it can be applied in various fields where multi-dimensional PDEs must be solved. View Full-Text
Keywords: Chebyshev polynomials; Galerkin basis functions; partial differential equation; quasi-inverse technique; matrix diagonalization Chebyshev polynomials; Galerkin basis functions; partial differential equation; quasi-inverse technique; matrix diagonalization
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Oh, S. An Efficient Spectral Method to Solve Multi-Dimensional Linear Partial Different Equations Using Chebyshev Polynomials. Mathematics 2019, 7, 90.

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