Hybrid Chebyshev polynomial scheme for solving elliptic partial differential equations

Abstract

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.

A Robust and Effective Method for Solving Two-Point BVP in Modelling Viscoelastic Flows

“…In recent years, a lot of attention has been devoted to the study of Chebyshev methods to investigate various scientific models. Using these methods made it possible to solve Troeschs problem [12], twelfth-order boundary-value problems [13], high-order nonlinear ordinary differential equations [14], linear integrodifferential equations [15], fourth-order Sturm-Liouville problems [16], Generalized Sturm-Liouville problems [17], the parabolic inverse problem [18], twodimensional heat equation [19], fractional diffusion equation [20], elliptic partial differential equations [21], integral and integro-differential equations of the third kind [22], the constant mobility Cahn-Hilliard equation in a square domain [23], Lane-Emden problem [24]. Recently, has been made numerical comparison of sinc-collocation and Chebychev-collocation methods for determining the eigenvalues of Sturm-Liouville problems with parameter-dependent boundary conditions by El-Gamel [25].…”

A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations

“…Although many types of "basis functions" are available, a good choice for most of all applications is the Fourier series [39][40][41][42][43]. Another popular used "basis function" is the well-known Chebyshev series, which is just a Fourier cosine expansion with a change of variable [25,40,42,44,45]. Once the particular solutions have been obtained, the solution of the original problem can then be converted to a homogeneous one which can be solved by using the BEM/MFS-based methods [11,13,14,19,[46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64].…”

A localized Fourier collocation method for 2D and 3D elliptic partial differential equations: Theory and MATLAB code

“…It should be noted that most of the existing Fourier-based collocation methods adopt global basis functions in which x ϕ ( ) n are a set of trigonometric polynomials over the entire computational domain. 12,14,[17][18][19][20][21] Such methods produce nonsymmetric and dense algebraic equations that may present a major problem for large-scale simulations. In addition, when higher accuracy is desired, one may be forced to increase the degree of the polynomials, which can lead to illconditioned and numerically unstable systems of equations.…”

“…Although various types of basis functions $${\phi }_{n}({\boldsymbol{x}})$$ are available, a good choice may be the Fourier series, 11–17 since it can be, theoretically, used to represent any solution with desired accuracy by taking the degree of trigonometric polynomials to be sufficiently large. It should be noted that most of the existing Fourier‐based collocation methods adopt global basis functions in which $${\phi }_{n}({\boldsymbol{x}})$$ are a set of trigonometric polynomials over the entire computational domain 12,14,17–21 . Such methods produce nonsymmetric and dense algebraic equations that may present a major problem for large‐scale simulations.…”