Sinc Collocation Method for Solving the Benjamin-Ono Equation

We propose a simple, though powerful, technique for numerical solutions of the Benjamin-Ono equation. This approach is based on a global collocation method using Sinc basis functions. Some properties of the Sinc collocation method required for our subsequent development are given and utilized to reduce the computation of the Benjamin-Ono equation to a system of ordinary differential equations. The propagation of one soliton and the interaction of two solitons are used to validate our numerical method. The method is easy to implement and yields accurate results.

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AN EFFECTIVE NUMERICAL METHOD FOR SOLVING THE NONLINEAR SINGULAR LANE-EMDEN TYPE EQUATIONS OF VARIOUS ORDERS

Jurnal Teknologi ◽

10.11113/jt.v79.8737 ◽

2016 ◽

Vol 79 (1) ◽

Author(s):

Kourosh Parand ◽

Mehdi Delkhosh

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Numerical Method ◽

Collocation Method ◽

Fractional Order ◽

Spectral Methods ◽

Mathematical Physics ◽

Infinite Domain ◽

Orthogonal Functions ◽

Non Linear

The Lane-Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. These equations are categorized as non-linear singular ordinary differential equations on the semi-infinite domain. In this paper, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) of the first kind have been introduced as a new basis for Spectral methods, and also presented an effective numerical method based on the GFCFs and the collocation method for solving the nonlinear singular Lane-Emden type equations of various orders. Obtained results have compared with other results to verify the accuracy and efficiency of the presented method.

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On eigenvalues of a matrix arising in energy-preserving/dissipative continuous-stage Runge-Kutta methods

Special Matrices ◽

10.1515/spma-2021-0101 ◽

2021 ◽

Vol 10 (1) ◽

pp. 34-39

Author(s):

Yusaku Yamamoto

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Numerical Method ◽

Collocation Method ◽

Short Note ◽

Complex Eigenvalues ◽

Hilbert Matrix ◽

Runge Kutta

Abstract In this short note, we define an s × s matrix Ks constructed from the Hilbert matrix H s = ( 1 i + j - 1 ) i , j = 1 s {H_s} = \left( {{1 \over {i + j - 1}}} \right)_{i,j = 1}^s and prove that it has at least one pair of complex eigenvalues when s ≥ 2. Ks is a matrix related to the AVF collocation method, which is an energy-preserving/dissipative numerical method for ordinary differential equations, and our result gives a matrix-theoretical proof that the method does not have large-grain parallelism when its order is larger than or equal to 4.

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Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

Information and Control Systems ◽

10.31799/1684-8853-2018-6-24-34 ◽

2018 ◽

pp. 24-34

Author(s):

V. F. Edneral ◽

O. D. Timofeevskaya

Keyword(s):

Normal Form ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Periodic Solutions ◽

Fourth Order ◽

Numerical Solutions ◽

Numerical Calculations ◽

Computational Time ◽

Resonant Normal Form ◽

Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

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A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

Numerical Mathematics Theory Methods and Applications ◽

10.4208/nmtma.2016.m1429 ◽

2016 ◽

Vol 9 (4) ◽

pp. 619-639 ◽

Cited By ~ 3

Author(s):

Zhong-Qing Wang ◽

Jun Mu

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Collocation Method ◽

Initial Value Problems ◽

Numerical Experiments ◽

High Order Accuracy ◽

Nonlinear Ordinary Differential Equations ◽

Initial Value ◽

Order Accuracy ◽

Long Time

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

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