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4: Paper Source PDF document

Paper's Title:

Orthogonal Collocation on Finite Elements Using Quintic Hermite Basis


P. Singh, N. Parumasur and C. Bansilal

University of KwaZulu-Natal,
School of Mathematics Statistics and Computer Sciences,
Private Bag X54001,
Durban, 4000,
South Africa.


In this paper we consider the orthogonal collocation on finite elements (OCFE) method using quintic Hermite (second degree smooth) basis functions and use it to solve partial differential equations (PDEs). The method is particularly tailored to solve third order BVPS and PDEs and to handle their special solutions such as travelling waves and solitons, which typically is the case in the KdV equation. The use of quintic polynomials and collocation using Gauss points yields a stable high order superconvergent method. OCFE using quintic Hermite basis is optimal since it is computationally more efficient than collocation methods using (first degree smooth) piecewise-polynomials and more accurate than the (third degree smooth) B-splines basis. Various computational simulations are presented to demonstrate the computational efficiency and versatility of the OCFE method.

3: Paper Source PDF document

Paper's Title:

High Order Collocation Method for the Generalized Kuramoto-Sivashinsky Equation


Zanele Mkhize, Nabendra Parumasur and Pravin Singh

School of Mathematics, Statistics and Computer Sciences,
University of KwaZulu-Natal,
Private Bag X 54001,
Durban 4000.


In this paper, we derive the heptic Hermite basis functions and use them as basis functions in the orthogonal collocation on finite elements (OCFE) method. We apply the method to solve the generalized Kuramoto-Sivashinsky equation. Various numerical simulations are presented to justify the computational efficiency of the proposed method.

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