Hybrid Chebyshev Polynomial Scheme for Solving Elliptic Partial Differential Equations

We propose hybrid Chebyshev polynomial scheme (HCPS), which couples the Chebyshev polynomial scheme and the method of fundamental solutions into a single matrix system. This hybrid formulation requires solving only one system of equations and opens up the possibilities for solving a large class of partial differential equations. In this work, we consider various boundary value problems and, in particular, the challenging Cauchy-Navier equation. The solution is approximated by the sum of the particular solution and the homogeneous solution. Chebyshev polynomials are used to approximate a particular solution of the given partial differential equation and the method of fundamental solutions is used to approximate the homogeneous solution. Numerical results show that our proposed approach is efficient, accurate, and stable

Hybrid Chebyshev Polynomial Scheme for the Numerical Solution of Partial Differential Equations

In the numerical solution of partial differential equations (PDEs), it is common to find situations where the best choice is to use more than one method to arrive at an accurate solution. In this dissertation, hybrid Chebyshev polynomial scheme (HCPS) is proposed which is applied in two-step approach and one-step approach. In the two-step approach, first, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then, the resulting homogeneous equation is solved by boundary type methods including the method of fundamental solution (MFS) and the equilibrated collocation Trefftz method. However, this scheme can be applied to solve PDEs with constant coefficients only. So, for solving a wide variety of PDEs, one-step hybrid Chebyshev polynomial scheme is proposed. This approach combines two matrix systems of two-step approach into a single matrix system. The solution is approximated by the sum of particular solution and homogeneous solution. The Laplacian or biharmonic operator is kept on the left hand side and all the other terms are moved to the right hand side and treated as the forcing term. Various boundary value problems governed by the Poisson equation in two and three dimensions are considered for the numerical experiments. HCPS is also applied to solve an inhomogeneous Cauchy-Navier equations of elasticity in two dimensions. Numerical results show that HCPS is direct, easy to implement, and highly accurate

Improving Numerical Accuracy of the Localized Oscillatory Radial Basis Functions Collocation Method for Solving Elliptic Partial Differential Equations in 2D

Recently, the localized oscillatory radial basis functions collocation method (L-ORBFs) has been introduced to solve elliptic partial differential equations in 2D with a large number of computational nodes. The research clearly shows that the L-ORBFs is very convenient and useful for solving large-scale problems, but this method is numerically less accurate. In this paper, we propose a numerical scheme to improve the accuracy of the L-ORBFs by adding low-degree polynomials in the localized collocation process. The numerical results validate that the proposed numerical scheme is highly accurate and clearly outperforms the results of the L-ORBFs