A differential quadrature algorithm for simulations of nonlinear Schrödinger equation

Computers and Mathematics with Applications

Volumn 56, Issue 9, 2008, Pages 2222-2234

  Korkmaz, Alper ^a   Daǧ, Idris ^b  

^a Anadolu Guzel Sanatlar Lisesi   (Turkey)

^b ESKISEHIR OSMANGAZI UNIVERSITY   (Turkey)

Author keywords

Birth of soliton; Bound state; Differential quadrature method; Nonlinear Schr dinger equation; Soliton

Indexed keywords

COMPUTER SIMULATION; DIFFERENCE EQUATIONS; NONLINEAR EQUATIONS; SOLITONS;

BIRTH OF SOLITON; BOUND STATE; BOUND STATE SOLUTIONS; CONSERVED QUANTITIES; DIFFERENTIAL QUADRATURE; DIFFERENTIAL QUADRATURE METHOD; DINGER EQUATION; MAXIMUM ERROR; NUMERICAL SIMULATIONS; SOLITON;

DIFFERENTIATION (CALCULUS);

EID: 52049124456     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2008.03.047     Document Type: Article

References (23)

1. Karpman V.I., and Krushkal E.M. Modulated waves in non-linear dispersive media. Soviet Phys. JEPT 28 (1969) 277
2. Scott A.C., Chu F.Y.F., and Mclaughlin D.W. The soliton: A new concept in applied science. Procc. IEEE 61 (1973) 1443
3. Zakharov V.E., and Shabat A.B. Exact theory of two dimensional self focusing and one dimensional self waves in non-linear media. Soviet Phys. JEPT 34 (1972) 62
4. Delfour M., Fortin M., and Payne G. Finite-difference solutions of a non-linear Schrödinger equation. J. Comput. Phys. 44 (1981) 277-288
5. Taha T.R., and Ablowitz M.J. Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equations. J. Comput. Phys. 55 (1984) 203-230
6. Argyris J., and Haase M. An engineer's guide to soliton phenomena: Application of the finite element method. Comput. Methods Appl. Mech. Engrg. 61 (1987) 71-122
7. Twizell E.H., Bratsos A.G., and Newby J.C. A finite-difference method for solving the cubic Schrödinger equation. Math. Comput. Simulation 43 (1997) 67-75
8. Daǧ İ. A quartic B-spline finite element method for solving nonlinear Schrödinger equation. Comput. Methods Appl. Mech. Engrg. 174 (1999) 247-258
9. Bellman R., Kashef B.G., and Casti J. Differential quadrature: A tecnique for the rapid solution of nonlinear differential equations. J. Comput. Phys. 10 (1972) 40-52
10. Quan J.R., and Chang C.T. New sightings in involving distributed system equations by the quadrature methods-I. Comput. Chem. Engrg. 13 (1989) 779-788
11. Quan J.R., and Chang C.T. New sightings in involving distributed system equations by the quadrature methods-II. Comput. Chem. Engrg. 13 (1989) 71017-71024
12. Bellman R., Kashef B., Lee E.S., and Vasudevan R. Differential Quadrature and Splines, Computers and Mathematics with Applications (1976), Pergamon, Oxford pp. 371-376
13. Shu C., and Richards BE. Application of generalized differential quadrature to solve two dimensional incompressible Navier Stokes equations. Internat. J. Numer. Methods Fluids 15 (1992) 791-798
14. Shu C., and Wu Y.L. Integrated radial basis functions-based differential quadrature method and its performance. Internat. J. Numer. Methods Fluids 53 (2007) 969-984
15. Shu C., and Xue H. Explicit computation of weighting coefficients in the harmonic differential quadrature. Journal of Sound and Vibration 204 3 (1997) 549-555
16. Striz A.G., Wang X., and Bert C.W. Harmonic differential quadrature method and applications to analysis of structural components. Acta Mech. 111 (1995) 85-94
17. Chang S. Differential Quadrature and its Application in Engineering (2000), Springer-Verlag London Limited
18. Whitham G.B. Linear and Nonlinear Waves (1974), Wiley/Interscience, New York
19. Herbst B.M., Li J., Morris, and Mitchel A.R. Numerical experince with the nonlinear Schrödinger equation. J. Comput. Phys. 60 (1985) 282-305
20. Gardner L.R.T., Gardner G.A., Zaki S.I., and Sharawi Z.El. B-spline finite element studies of the non-linear Schrödinger equation. Comput. Methods Appl. Mech. Engrg. 108 (1993) 303-318
21. Gardner L.R.T., Gardner G.A., Zaki S.I., and Sharawi Z.El. A leapfrog algorithm and stability studies for the non-linear Schrödinger equation. Arabian J. Sci. Engrg. (1993) 23-32
22. Miles J.M. An envolope soliton problems. SIAM J. Appl. Math. 41 (1981) 227
23. Delfour M., Fortin M., and Payne G. Finite-difference solutions of non-linear Schrödinger equation. J. Comput. Phys. 44 (1981) 277-288