3,792 research outputs found
High-Order Numerical Solution of Second-Order One-Dimensional Hyperbolic Telegraph Equation Using a Shifted Gegenbauer Pseudospectral Method
We present a high-order shifted Gegenbauer pseudospectral method (SGPM) to
solve numerically the second-order one-dimensional hyperbolic telegraph
equation provided with some initial and Dirichlet boundary conditions. The
framework of the numerical scheme involves the recast of the problem into its
integral formulation followed by its discretization into a system of
well-conditioned linear algebraic equations. The integral operators are
numerically approximated using some novel shifted Gegenbauer operational
matrices of integration. We derive the error formula of the associated
numerical quadratures. We also present a method to optimize the constructed
operational matrix of integration by minimizing the associated quadrature error
in some optimality sense. We study the error bounds and convergence of the
optimal shifted Gegenbauer operational matrix of integration. Moreover, we
construct the relation between the operational matrices of integration of the
shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive
the global collocation matrix of the SGPM, and construct an efficient
computational algorithm for the solution of the collocation equations. We
present a study on the computational cost of the developed computational
algorithm, and a rigorous convergence and error analysis of the introduced
method. Four numerical test examples have been carried out in order to verify
the effectiveness, the accuracy, and the exponential convergence of the method.
The SGPM is a robust technique, which can be extended to solve a wide range of
problems arising in numerous applications.Comment: 36 pages, articl
Spectral methods in general relativistic astrophysics
We present spectral methods developed in our group to solve three-dimensional
partial differential equations. The emphasis is put on equations arising from
astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures,
submitted to Journal of Computational & Applied Mathematic
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