Sinc-Galerkin method for solving hyperbolic partial differential equations
Sinc-Galerkin method for solving hyperbolic partial differential equations
In this work, we consider the hyperbolic equations to determine the approximatesolutions via Sinc-Galerkin Method (SGM). Without any numerical integration,the partial differential equation transformed to an algebraic equationsystem. For the numerical calculations, Maple is used. Several numericalexamples are investigated and the results determined from the method arecompared with the exact solutions. The results are illustrated both in tableand graphically.
___
- Stenger, F. (1979). A Sinc-Galerkin Method of Solution
of Boundary Value Problems. Mathematics of
Computation, 33 85-109.
- Stenger, F. (2000). Summary of Sinc numerical methods.
Journal of Computational and Applied Mathematics,
121 379-420.
- Alonso, N., Bowers, K.L. (2009). An AlternatingDirection
Sinc-Galerkin Method for Elliptic Problems.
Journal of Complexity, 25, 237–252.
- Bowers, K.L., Lund, J. (1987). Numerical Solution
of Singular Poisson Problems via the Sinc-Galerkin
Method. SIAM Journal on Numerical Analysis, 24(1)
36-51.
- Koonprasert, S. (2003). The Sinc-Galerkin Method
for Problems in Oceanography. PhD thesis, Montana
State University, Bozeman, Montana.
- Lewis, D.L., Lund, J., Bowers, K. L. (1987).
The Space-Time Sinc-Galerkin Method for Parabolic
Problems. International Journal for Numerical Methods
in Engineering, 24 1629-1644.
- Lund, J. (1986). Symmetrization of the Sinc-Galerkin
Method for Boundary Value Problems. Mathematics
of Computation, 47, 571-588.
- Lund, J., Bowers, K.L., McArthur, K. M., (1989).
Symmetrization of the Sinc-Galerkin Method with
Block Techniques for Elliptic Equations. IMA Journal
of Numerical Analysis, 9, 29-46.
- Lund, J., Bowers, K.L. (1992). Sinc Methods
for Quadrature and Differential Equations. SIAM:
Philadelphia.
- Lybeck, N.J. (1994). Sinc Domain Decomposition
Methods for Elliptic Problems. PhD thesis, Montana
State University, Bozeman, Montana.
- Lybeck, N.J., Bowers, K.L. (1996). Domain Decomposition
in Conjunction with Sinc Methods for Poisson’s
Equation. Numerical Methods for Partial Differential
Equations, 12, 461-487.
- McArthur, K.M., Bowers, K.L., Lund, J. (1987). Numerical
Implementation of the Sinc-Galerkin Method
for Second-Order Hyperbolic Equations. Numerical
Methods for Partial Differential Equations, 3, 169-185.
- McArthur, K.M., Bowers, K.L., Lund, J. (1990). The
Sinc Method in Multiple Space Dimensions: Model
Problems. NumerischeMathematik, 56, 789-816.
- Morlet, A.C., Lybeck, N.J., Bowers, K.L. (1997).
The Schwarz Alternating Sinc Domain Decomposition
Method. Applied Numerical Mathematics, 25, 461-483.
- Morlet, A.C., Lybeck, N.J., Bowers, K.L. (1999). Convergence
of the Sinc Overlapping Domain Decomposition
Method. Applied Mathematics and Computation,
98, 209-227.
- Ng, M. (1999). Fast Iterative Methods for Symmetric
Sinc-Galerkin Systems. IMA Journal of Numerical
Analysis, 19 357-373.
- Ng, M., Bai, Z. (2003). A Hybrid Preconditioner
of Banded Matrix Approximation and AlternatingDirection
Implicit Iteration for Symmetric SincGalerkin
Linear Systems, Linear Systems, Linear Algebra
and its Applications, 366 317-335.
- Stenger, F. (1981). Numerical Methods Based on
Whittaker Cardinal, or Sinc Functions. SIAM Review,
23 165-224.
- Stenger, F. (1993). Numerical Methods Based on Sinc
and Analytic Functions, Springer-Verlag, New York.
- Whittaker, E.T. (1915). On the Functions which are
Represented by the Expansions of the Interpolation
Theory. Proceedings of the Royal Society of Edinburg,
35, 181-194.
- Whittaker, J.M. (1961). Interpolation Function Theory,
Cambridge Tracts in Mathematics and Mathematical
Physics. No. 33, Cambridge University Press,
London.
- Stenger, F. (1976). Approximations via Whittaker’s
Cardinal Function. Journal of Approximation Theory,
17, 222-240.
- Stenger, F., O’Reilly, M. J. (1998). Computing Solutions
to Medical Problems via Sinc Convolution. IEEE
Transactions on Automatic Control, 43, 843.
- Narasimhan, S., Majdalani, J., Stenger, F. (2002). A
First Step In Applying The Sinc Collocation Method
To The Nonlinear Navier Stokes Equations. Numerical
Heat Transfer,41, 447-462.
- Mueller, J.L., Shores, T.S. (2004). A New SincGalerkin
Method for Convection-Diffusion Equations
with Mixed Boundary Conditions. Computers and
Mathematics with Applications, 47, 803-822.
- El-Gamel, M., Behiry, S.H., Hashish, H. (2003). Numerical
method for the solution of special nonlinear
fourth-order boundary value problems. Applied Mathematics
and Computation, 145, 717–734.
- Alkan, S., Secer, A. (2015). Solving nonlinear boundary
value problems by the Galerkin method with sinc
functions. Open Physics, 13, 1, 389-394.
- Zamani, N.G. (1987). A finite element based collocation
method for eigenvalue calculations. Journal of the
Franklin Institute, 324, 205-217.
- Alkan, S., Secer, A. (2015). Application of SincGalerkin
method for solving space-fractional boundary
value problems. Mathematical Problems in Engineering,
2015, 1-10.
- Lybeck, N.J., Bowers, K.L. (1996). Sinc methods
for domain decomposition. Applied Mathematics and
Computation, 75, 4-13.
- Secer, A., Alkan, S., Akinlar, M.A., Bayram, M.
(2013). Sinc-Galerkin method for approximate solutions
of fractional order boundary value problems.
Boundary Value Problems, 2013, 1, 281.
- Secer, A., Kurulay M., Bayram, M., Akinlar, M.A.
(2013). An efficient computer application of the sincGalerkin
approximation for nonlinear boundary value
problems, Boundary Value Problems, 2012, 1, 117.