Browsing by Author "Van Gorder, Robert A."

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A Jacobi rational pseudospectral method for Lane-Emden initial value problems arising in astrophysics on a semi-infinite interval
(SPRINGER HEIDELBERG, 2014) Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.; Van Gorder, Robert A.
We derive an operational matrix representation for the differentiation of Jacobi rational functions, which is used to create a new Jacobi rational pseudo spectral method based on the operational matrix of Jacobi rational functions. This Jacobi rational pseudospectral method is implemented to approximate solutions to Lane-Emden type equations on semi-infinite intervals. The advantages of using the Jacobi rational pseudospectral method over other techniques are discussed. Indeed, through several numerical examples, including the Lane-Emden problems of first and second kind, we evaluate the accuracy and performance of the proposed method. We also compare our method to other approaches in the literature. The results suggest that the Jacobi rational pseudospectral method is a useful tool for studying Lane-Emden initial value problems, as well as related problems which have regular singular points and are nonlinear.
Jacobi rational-Gauss collocation method for Lane-Emden equations of astrophysical significance
(INST MATHEMATICS & INFORMATICS, 2014) Doha, Eid H.; Bhrawy, Ali H.; Hafez, Ramy M.; Van Gorder, Robert A.
In this paper, a new spectral collocation method is applied to solve Lane-Emden equations on a semi-infinite domain. The method allows us to overcome difficulty in both the nonlinearity and the singularity inherent in such problems. This Jacobi rational-Gauss method, based on Jacobi rational functions and Gauss quadrature integration, is implemented for the nonlinear Lane-Emden equation. Once we have developed the method, numerical results are provided to demonstrate the method. Physically interesting examples include Lane-Emden equations of both first and second kind. In the examples given, by selecting relatively few Jacobi rational-Gauss collocation points, we are able to get very accurate approximations, and we are thus able to demonstrate the utility of our approach over other analytical or numerical methods. In this way, the numerical examples provided demonstrate the accuracy, efficiency, and versatility of the method.
A new Jacobi spectral collocation method for solving 1+1 fractional Schrodinger equations and fractional coupled Schrodinger systems
(SPRINGER HEIDELBERG, 2014) Bhrawy, A. H.; Doha, E. H.; Ezz-Eldien, S. S.; Van Gorder, Robert A.
The Jacobi spectral collocation method (JSCM) is constructed and used in combination with the operational matrix of fractional derivatives (described in the Caputo sense) for the numerical solution of the time-fractional Schrodinger equation (T-FSE) and the space-fractional Schrodinger equation (S-FSE). The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, the presented approach is also applied to solve the time-fractional coupled Schrodinger system (T-FCSS). In order to demonstrate the validity and accuracy of the numerical scheme proposed, several numerical examples with their approximate solutions are presented with comparisons between our numerical results and those obtained by other methods.