Comparison of different methods for choosing the collocation points in the boundary collocation method for 2D-harmonic problems with special purpose Trefftz functions

2012 ◽  
Vol 36 (12) ◽  
pp. 1883-1893 ◽  
Author(s):  
M. Mierzwiczak ◽  
J.A. Kołodziej
Keyword(s):  
Collocation Method ◽  
Boundary Collocation Method ◽  
Collocation Points

scholarly journals Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

2020 ◽  
Vol 12 (4) ◽  
pp. 517-523
Author(s):  
G. Singh ◽  
I. Singh
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Hermite Polynomials ◽  
Electric Circuit ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Electric Engineering ◽  
Collocation Points ◽  
Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

scholarly journals Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 253 ◽  
Author(s):  
Aditya Kamath ◽  
Sergei Manzhos
Keyword(s):  
Schrödinger Equation ◽  
Vibrational Spectrum ◽  
Collocation Method ◽  
Schrodinger Equation ◽  
Basis Functions ◽  
Collocation Points ◽  
Basis Size ◽  
Six Dimensions ◽  
Gaussian Basis Functions

We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schrödinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions when using different numbers of width-optimized IMQ functions. The effects of the ratio of the number of collocation points to the number of basis functions and of the choice of the IMQ exponent are studied. We show that the IMQ basis can be used with parameters where the IMQ function is not integrable. We find that the quality of the spectrum with IMQ basis functions is somewhat lower that that with a Gaussian basis when the basis size is large, and for a range of IMQ exponents. The IMQ functions are; however, advantageous when a small number of functions is used or with a small number of collocation points (e.g., when using square collocation).

Application of boundary collocation method to two-dimensional wave propagation

2015 ◽  
Vol 29 (4) ◽  
pp. 579-587 ◽  
Author(s):  
Xue-ling Cao ◽  
Ya-ge You ◽  
Song-wei Sheng ◽  
Wen Peng ◽  
Yin Ye
Keyword(s):  
Wave Propagation ◽  
Collocation Method ◽  
Two Dimensional ◽  
Boundary Collocation Method ◽  
Dimensional Wave

scholarly journals A Nonclassical Radau Collocation Method for Nonlinear Initial-Value Problems with Applications to Lane-Emden Type Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mohammad Maleki ◽  
M. Tavassoli Kajani ◽  
I. Hashim ◽  
A. Kilicman ◽  
K. A. M. Atan
Keyword(s):  
Orthogonal Polynomials ◽  
Numerical Method ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Weighted Interpolation ◽  
Collocation Points ◽  
Present Solution ◽  
Nonlinear Algebraic Equations

We propose a numerical method for solving nonlinear initial-value problems of Lane-Emden type. The method is based upon nonclassical Gauss-Radau collocation points, and weighted interpolation. Nonclassical orthogonal polynomials, nonclassical Radau points and weighted interpolation are introduced on arbitrary intervals. Then they are utilized to reduce the computation of nonlinear initial-value problems to a system of nonlinear algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is very accurate.

The Over-Range Collocation Method for Nonlinear Analyses

2013 ◽  
Vol 300-301 ◽  
pp. 942-949
Author(s):  
Yong Ming Guo ◽  
Shunpei Kamitani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Collocation Method ◽  
Meshless Method ◽  
Collocation Methods ◽  
Nonlinear Analyses ◽  
Partial Differential ◽  
Boundary Points ◽  
Collocation Points

In this paper, a forging problem is analyzed by using the overrange collocation method (ORCM), which is a new meshless method. By introducing some collocation points, which are located out of domain of the analyzed body, unsatisfactory issue of the positivity conditions of boundary points in collocation methods can be avoided. Because the overrange points are used only in interpolating calculation, no overconstrain occurs in partial differential equations on the solved problems.

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