When integration sparsification fails: Banded Galerkin discretizations for Hermite functions, rational Chebyshev functions and sinh-mapped Fourier functions on an infinite domain, and Chebyshev methods for solutions with C∞ endpoint singularities

2019 ◽  
Vol 160 ◽  
pp. 82-102
Author(s):  
Zhu Huang ◽  
John P. Boyd
Keyword(s):  
Hermite Functions ◽  
Infinite Domain ◽  
Rational Chebyshev Functions ◽  
Rational Chebyshev ◽  
Chebyshev Functions

scholarly journals A Comparison Study of Numerical Techniques for Solving Ordinary Differential Equations Defined on a Semi-Infinite Domain Using Rational Chebyshev Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Mohamed A. Ramadan ◽  
Taha Radwan ◽  
Mahmoud A. Nassar ◽  
Mohamed A. Abd El Salam
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Operational Matrices ◽  
Comparison Study ◽  
Infinite Domain ◽  
Linear Ordinary Differential Equations ◽  
Infinite Domains ◽  
Collocation Technique ◽  
Rational Chebyshev Functions ◽  
Rational Chebyshev

A rational Chebyshev (RC) spectral collocation technique is considered in this paper to solve high-order linear ordinary differential equations (ODEs) defined on a semi-infinite domain. Two definitions of the derivative of the RC functions are introduced as operational matrices. Also, a theoretical study carried on the RC functions shows that the RC approximation has an exponential convergence. Due to the two definitions, two schemes are presented for solving the proposed linear ODEs on the semi-infinite interval with the collocation approach. According to the convergence of the RC functions at the infinity, the proposed technique deals with the boundary value problem which is defined on semi-infinite domains easily. The main goal of this paper is to present a comparison study for differential equations defined on semi-infinite intervals using the proposed two schemes. To demonstrate the validity of the comparisons, three numerical examples are provided. The obtained numerical results are compared with the exact solutions of the proposed problems.

scholarly journals Application of Rational Second Kind Chebyshev Functions for System of Integrodifferential Equations on Semi-Infinite Intervals

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
M. Tavassoli Kajani ◽  
S. Vahdati ◽  
Zulkifly Abbas ◽  
Mohammad Maleki
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Galerkin Method ◽  
High Accuracy ◽  
Integrodifferential Equations ◽  
Expansion Test ◽  
Infinite Intervals ◽  
System Of Integrodifferential Equations ◽  
Rational Chebyshev ◽  
Chebyshev Functions

Rational Chebyshev bases and Galerkin method are used to obtain the approximate solution of a system of high-order integro-differential equations on the interval [0,∞). This method is based on replacement of the unknown functions by their truncated series of rational Chebyshev expansion. Test examples are considered to show the high accuracy, simplicity, and efficiency of this method.

A Sylvester-Based IMEX Method via Differentiation Matrices for Solving Nonlinear Parabolic Equations

2013 ◽  
Vol 14 (4) ◽  
pp. 1001-1026 ◽  
Author(s):  
Francisco de la Hoz ◽  
Fernando Vadillo
Keyword(s):  
Chebyshev Polynomials ◽  
Three Dimensional ◽  
Nonlinear Parabolic Equations ◽  
Basis Functions ◽  
Hermite Functions ◽  
Nonlinear Parabolic ◽  
Sinc Functions ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral ◽  
Rational Chebyshev

AbstractIn this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains, taking Hermite functions, sinc functions, and rational Chebyshev polynomials as basis functions. The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme, being of particular interest the treatment of three-dimensional Sylvester equations that we make. The resulting method is easy to understand and express, and can be implemented in a transparent way by means of a few lines of code. We test numerically the three choices of basis functions, showing the convenience of this new approach, especially when rational Chebyshev polynomials are considered.

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