Cubic spline Numerov type approach for solution of Helmholtz equation

Subject Areas : Difference and functional equations

J. Rashidinia ¹ , H. S. Shekarabi ² , M. Aghamohamadi ³

1 - Department of Mathematics,College of basic science, Islamic Azad University, Alborz, Iran
2 - Department of Mathematics,College of basic science, Islamic Azad University, Alborz, Iran
3 - Department of Mathematics,College of basic science, Islamic Azad University, Alborz, Iran

Received: 2014-05-14 Accepted : 2014-08-28 Published : 2014-03-01

Keywords:

References:

[1] Carlos J. S. Alves, Svilen S. Valtchev,Numerical simulation of acoustic wave scattering using a meshfree plane waves method,International Workshop on MeshFree Methods( 2003),1-6.

 

[2] K. Atkinson, W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Springer,(2005).

 

[3] R. J. Astley, P. Gamallo, Special short elements for flow acoustics, Comput. Method Appl. Mech. Engrg. 194 (2005), 341-353.

[4] R. K. Beatson, J. B. Cherrie, C. T. Mouat, Fast fitting of radial basis functions: method based on preconditioned GMRES iteration, Adv. Comput. Math. 11 (1999), 253-270.

 

[5] R. K. Beatson, W. A. Light, S. Billings, Fast solution of the radial basis function interpolation equations: domain decomposition methods, SIAM J. Sci. Comput. 5 (2000),1717-1740.

 

[6] A. I. Bouhamid, A. Le Mhaut, Spline curves and surfaces under tension, (1994),51-58.

 

[7] A. I. Bouhamid, A. Le Mhaut, Multivariate interpolating (m;s)-spline, Adv. Comput. Math. 11 (1999), 287-314.

 

[8] G. M. L. Gladwell, N. B. Willms, On the mode shape of the Helmholtz equation, J. Sound Vib. 188(1995), 419-433.

 

[9] Charles I. Goldstein, A Finite Element Method for Solving Helmholtz,Type Equationsin Waveguides and Other Unbounded Domains, mathematics of computation,39 (160) (1982), 309-324. 

 

[10] F. Ihlenburg, I. Babusk. Finite element solution of the Helmholtz equation with high wave number part I: the hversion of the FEM. Computers Mathematics with Applications, 30 (9) (1995), 9-37.

 

[11] F. Ihlenburg, I. Babuska. Finite element solution of the Helmholtz equation with high wave number part II: the hp version of the FEM. SIAM Journal of Numerical Analysis, 34 (1) (1997), 315-358.

 

[12] M. K. Jain, Numerical Solution of Dierential Equations, 2nd edn. Wiley, New Delhi (1984).

 

[13] E. J. Kansa, A scattered data approximation scheme with applications to computational fluid dynamics. I. Surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (8,9) (1990),127-145.

 

[14] E. J. Kansa, Multiquadrics a scattered data approximation scheme with applications to computational fluid dynamics. II. Solutions to parabolic, hyperbolic partial differential equations, Comput. Math. Appl. 19 (8,9) (1990), 127-145.

 

[15] Y. C. Hon,C. S. Chen, Numerical comparisons of two meshless methods using radial basis functions engineering analysis with boundary elements. 26 (2002), 205-225.

 

[16] R. K. Mohanty, Stability interval for explicit difference schemes for multi-dimensional second order hyperbolic equations with significant first order space derivative terms, Appl. Math. Comput. 190 (2007),1683-1690.

 

[17] R. K. Mohanty, Venu Gopal, High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations,Applied Mathematics and Computation 218 (2011), 4234-4244.

 

[18] C. C. Paige, M. A. Saunders, LSQR: an algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Softw. 8 (1982) ,43-71.

 

[19] J. Rashidinia, R. Jalilian, V. Kazemi, Spline methods for the solutions of hyperbolic equations, Appl. Math. Comput. 190 (2007), 882-886.

 

[20] A. S. Wood, G. E. Tupholme, M. I. H. Bhatti, P. J. Heggs, Steady-state heattransfer through extended plane surfaces, Int. Commun. Heat Mass Transfer 22 (1995), 99-109.