Methods for solving singular perturbation problems arising in science and engineering

Mathematical and Computer Modelling

Volumn 54, Issue 1-2, 2011, Pages 556-575

  Kumar, Manoj ^a   Parul, ^a  

^a MOTILAL NEHRU NATIONAL INSTITUTE OF TECHNOLOGY   (India)

Author keywords

Asymptotic expansion method; Matched asymptotic expansion method; Multiple scale method; Poincare Lindstedt method; Singular perturbations; WKB method

Indexed keywords

ASYMPTOTIC EXPANSION METHOD; MATCHED ASYMPTOTIC EXPANSION METHOD; MULTIPLE SCALE METHOD; POINCARE; SINGULAR PERTURBATIONS; WKB METHOD;

ASYMPTOTIC ANALYSIS; CHEMICAL REACTORS; EXPANSION; MECHANICS; QUANTUM THEORY;

PERTURBATION TECHNIQUES;

EID: 79955417790     PISSN: 08957177     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mcm.2011.02.045     Document Type: Article

References (90)

1. Farrell P.A., Hegarty A.F., Miller J.J.H., O'Riordan E., Shishkin G.I. Robust Computational Techniques for Boundary Layers 2000, Chapman and Hall, CRC Press, London, Boca Raton, FL.
2. Roos H.G., Stynes M., Tobiska L. Numerical Methods for Singularly Perturbed Differential Equations 1996, Springer-Verlag, New York.
3. Miller J.J.H., O'Riordan E., Shishkin G.I. Fitted Numerical Methods for Singular Perturbation Problems 1996, World Scientific, Singapore.
4. Smith D.R. Singular-Perturbation Theory an Introduction with Applications 2009, Cambridge University Press, Cambridge.
5. Morton K.W. Numerical Solution of Convection-Diffusion Problems. Applied Mathematics and Mathematical Computation 1996, vol. 12. Chapman and Hall, London.
6. Zheng J., Sederberg T.W., Johnson R.W. Least square methods for solving differential equations using Bezier control points. Applied Numerical Mathematics 2004, 48:237-252.
7. Kumar M., Singh P., Mishra H.K. An initial-value technique for singularly perturbed boundary value problems via cubic spline. International Journal of Computational Methods in Engineering Science and Mechanics 2007, 8:1-9.
8. Kumar M., Mishra H.K., Singh P. A boundary value approach for singularly perturbed boundary value problems. Advances in Engineering Software 2009, 40(4):298-304.
9. Kumar M., Singh P., Mishra H.K. A recent survey on computational techniques for solving singularly perturbed boundary value problems. International Journal of Computer Mathematics 2007, 84:1-25.
10. Reddy Y.N., Chakravarthy P.P. An initial-value approach for solving singularly perturbed two-point boundary value problems. Applied Mathematics and Computation 2004, 155:95-110.
11. Mohanty R.K., Arora U. A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives. Applied Mathematics and Computation 2006, 172:531-544.
12. Holmes M.H. Introduction to Perturbation Methods 1995, Springer.
13. Hinch E.J. Perturbation Methods 1991, Cambridge University Press.
14. Bender C.M., Orszag S.A. Advanced Mathematical Methods for Scientists and Engineers, Asymptotic Methods and Perturbation Theory 1999, Springer-Verlag New York, Inc.
15. Owen M.R., Lewis M.A. How predation can slow, stop, or reverse a prey invasion. Bulletin of Mathematical Biology 2001, 63:655-684.
16. Nayfeh A.H. Perturbation Methods. Wiley Classics Library 2000, Wiley-Interscience.
17. http://en.wikipedia.org/wiki.
18. Johnson S. Singular Perturbation Theory, Mathematical and Analytical Techniques with Applications to Engineering 2005, Springer.
19. Ward M.J. Course notes on asymptotic methods http://www.math.ubc.ca/~ward/teaching/math550.html.
20. Zeytounian R.K. An asymptotic derivation of the initial condition for the incompressible and viscous external unsteady fluid flow problem. International Journal of Engineering Science 2000, 38:1983-1992.
21. Williams H.E. An asymptotic solution of the governing equation for the natural frequencies of a cantilevered, coupled beam model. Journal of Sound and Vibration 2008, 312:354-359.
22. Iosilevskii G. Asymptotic theory of an oscillating wing section in weak ground effect. European Journal of Mechanics, B/Fluids 2008, 27:477-490.
23. Carpinteri A., Paggi M. Asymptotic analysis in linear elasticity: from the pioneering studies by Wieghardt and Irwin until today. Engineering Fracture Mechanics 2009, 76:1771-1784.
24. Lebon F., Rizzoni R. Asymptotic analysis of a thin interface, the case involving similar rigidity. International Journal of Engineering Science 2010, 48:473-486.
25. Fernando M.P., Oliveira A.A.M., Fernando F.F. Asymptotic analysis of stationary adiabatic premixed flames in porous inert media. Combustion and Flame 2009, 156:152-165.
26. Nitsche L.C., Zhang W., Wedgewood L.E. Asymptotic basis of the L closure for finitely extensible dumbbells in suddenly started uniaxial extension. Journal of Non-Newtonian Fluid Mechanics 2006, 133:14-27.
27. Andrianov I.V., Danishevs'kyy Vladyslav V., Weichert D. Asymptotic determination of effective elastic properties of composite materials with fibrous square shaped inclusions. European Journal of Mechanics, A/Solids 2002, 21:1019-1036.
28. Steigmann D.J. Asymptotic finite-strain thin-plate theory for elastic solids. Computers and Mathematics with Applications 2007, 53:287-295.
29. Pelissou C., Lebon F. Asymptotic modeling of quasi-brittle interfaces. Computers and Structures 2009, 87:1216-1223.
30. Saintlos S., Mauss J. Asymptotic modelling for separating boundary layers in a channel. International Journal of Engineering Science 1996, 34(2):201-211.
31. Bunger A.P., Detournay E. Asymptotic solution for a penny-shaped near-surface hydraulic fracture. Engineering Fracture Mechanics 2005, 72:2468-2486.
32. Kelley W. Asymptotically singular boundary value problems. Mathematical and Computer Modelling 2000, 32:541-548.
33. Yip K.M. Model simplification by asymptotic order of magnitude reasoning. Artificial Intelligence 1996, 80:309-348.
34. Popovic N., Szmolyan P. Rigorous asymptotic expansions for Lagerstrom's model equation-a geometric approach. Nonlinear Analysis 2004, 59:531-565.
35. Verhulst F. Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics 2005, Springer.
36. Priede J., Gerbeth G. Matched asymptotic solution for the solute boundary layer in a converging axisymmetric stagnation point flow. International Journal of Heat and Mass Transfer 2007, 50:216-225.
37. Radulescu O., Olmsted P.D. Matched asymptotic solutions for the steady banded flow of the diffusive Johnson-Segalman model in various geometries. Journal of Non-Newtonian Fluid Mechanics 2000, 91:143-164.
38. Margerit J., Sero-Guillaume O. Study of the evaporation of a droplet in its stagnant vapor by asymptotic matching. International Journal of Heat and Mass Transfer 1996, 39(18):3887-3898.
39. K. Gersten, The method of matched asymptotic expansions, Von Karman Institute for Fluid Mechanics, Mathematical Methods in Fluid Mechanics, 1980.
40. Mauss J., Cousteix J. Uniformly valid approximation for singular perturbation problems and matching principle. Comptes Rendus (CR) Mecanique 2002, 330:697-702.
41. Kevorkian J., Cole J.D. Multiple Scale and Singular Perturbation Methods 1996, Springer.
42. Khanin R., Cartmella M., Gilbert A. A computerized implementation of the multiple scales perturbation method using Mathematica. Computers and Structures 2000, 76:565-575.
43. Dessi D., Mastroddi F., Morino L. A fifth-order multiple-scale solution for Hopf bifurcations. Computers and Structures 2004, 82:2723-2731.
44. Zanoosi A.A.P., Shooshtari A. A multiple times scale solution for non-linear vibration of mass grounded system. Applied Mathematical Modelling 2010, 34:1918-1929.
45. Zhu S., Yu P. Computation of the normal forms for general M-DOF systems using multiple time scales. Part I: autonomous systems. Communications in Nonlinear Science and Numerical Simulation 2005, 10:869-905.
46. Coulaud O. Multiple time scales and perturbation methods for high frequency electromagnetic-hydrodynamic coupling in the treatment of liquid metals. Nonlinear Analysis: Theory, Methods & Applications 1997, 30(6):3637-3643.
47. Munz C.D., Fortenbach R., Dumbser M. Computational Aero Acoustics: from acoustic sources modeling to farfield radiated noise prediction multiple scale modelling of acoustic sources in low Mach-number flow. Comptes Rendus (CR) Mecanique 2005, 333:706-712.
48. Van Horssen W.T. On integrating vectors and multiple scales for singularly perturbed ordinary differential equations. Nonlinear Analysis 2001, 46:19-43.
49. Lakrad F., Belhaq M. Periodic solutions of strongly non-linear oscillators by the multiple scales method. Journal of Sound and Vibration 2002, 258(4):677-700.
50. Belhaq M., Lakrad F. The elliptic multiple scales method for a class of autonomous strongly non-linear oscillators. Journal of Sound and Vibration 2000, 234(3):547-553.
51. Filippi Paul Acoustics: Basic Physics, Theory and Methods 1999, Academic Press, 171.
52. Gosse L., Mauser N.J. Multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice-III. From ab initio models to WKB for Schrodinger-Poisson. Journal of Computational Physics 2006, 211:326-346.
53. Wazwaz A.M. A comparative study on a singular perturbation problem with two singular boundary points. Applied Mathematics and Computation 1999, 99:179-193.
54. Abdallah N.B., Pinaud O. Multiscale simulation of transport in an open quantum system-resonances and WKB interpolation. Journal of Computational Physics 2006, 213:288-310.
55. Gill A.E. Atmosphere Ocean Dynamics. International Geophysics Series 1982, vol. 30. Academic Press.
56. Spigler R., Vianello M. A survey on the Liouville-Green (WKB) approximation for linear difference equations of the second order. Advances in Difference Equations: Proceedings of the Second International Conference on Difference Equations: Veszprém, Hungary 1995, 567. CRC Press. S. Elaydi, I. Gyori, G.E. Ladas (Eds.).
57. Krivec R., Mandelzweig V.B. Quasilinearization method and WKB. Computer Physics Communications 2006, 174:119-126.
58. Drazin P.G. Nonlinear Systems 1992, Cambridge University Press, pp. 181-186.
59. Poincare H. Les Méthodes Nouvelles de la Mécanique Célèste 1957, vol. II. Dover Publication, New York, 123-128.
60. Lindstedt A. Beitrag zur Integration der Differentialgleichungen der Storungstheorie. Abh. K. Akad. Wiss. St. Petersburg 1882, 31.
61. Logan J.D. Applied Mathematics 1997, John Wiley and Sons. second ed.
62. Yanga C.H., Zhu S.M., Chen S.H. A modified elliptic Lindstedt-Poincare method for certain strongly non-linear oscillators. Journal of Sound and Vibration 2004, 273:921-932.
63. Liu H.M. Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method. Chaos, Solitons & Fractals 2005, 23:577-579.
64. Hu H., Xiong Z.G. Comparison of two Lindstedt-Poincare type perturbation methods. Journal of Sound and Vibration 2004, 278:437-444.
65. Khrustalev O., Vernov S. Construction of doubly periodic solutions via the Poincare-Lindstedt method in the case of massless φ4 theory. Mathematics and Computers in Simulation 2001, 57:239-252.
66. Ozis T., Yildirim A. Determination of periodic solution for a u1/3 force by He's modified Lindstedt-Poincare method. Journal of Sound and Vibration 2007, 301:415-419.
67. Pusenjak R.R. Extended Lindstedt-Poincare method for non-stationary resonances of dynamical systems with cubic nonlinearities. Journal of Sound and Vibration 2008, 314:194-216.
68. Amore P., Aranda A. Improved Lindstedt-Poincare method for the solution of nonlinear problems. Journal of Sound and Vibration 2005, 283:1115-1136.
69. Abbasbandy S. Iterated He's homotopy perturbation method for quadratic Riccati differential equation. Applied Mathematics and Computation 2006, 175:581-589.
70. Benbachir S. Lindstedt-Poincare method and periodic families of the Barbanis contopoulos Hamiltonian system. Mathematics and Computers in Simulation 2000, 51:579-596.
71. Marasco A. Lindstedt-Poincare method and mathematica applied to the motion of a solid with a fixed point. Computers and Mathematics with Applications 2000, 40:333-343.
72. Abd G.M., Latif E.I. On a problem of modified Lindstedt-Poincare for certain strongly non-linear oscillators. Applied Mathematics and Computation 2004, 152:821-836.
73. He J.H. Modified Lindstedt Poincare methods for some strongly non-linear oscillations part II: a new transformation. International Journal of Non-Linear Mechanics 2002, 37:315-320.
74. Chen S.H., Huang J.L., Szeb K.Y. Multidimensional Lindstedt-Poincare method for nonlinear vibration of axially moving beams. Journal of Sound and Vibration 2007, 306:1-11.
75. Navarro J.F. On the implementation of the Poincare-Lindstedt technique. Applied Mathematics and Computation 2008, 195:183-189.
76. Chen S.H., Yang X.M., Cheung Y.K. Periodic solutions of strongly quadratic nonlinear oscillators by the elliptic Lindstedt Poincare method. Journal of Sound and Vibration 1999, 227(5):1109-1118.
77. He J.H. The homotopy perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics and Computation 2004, 151:287-292.
78. Yuce H., Wang C.Y. A boundary perturbation method for circularly periodic plates with a core. Journal of Sound and Vibration 2009, 328:345-368.
79. Couchouron J.F., Kamenski M., Precup R. A nonlinear periodic averaging principle. Nonlinear Analysis 2003, 54:1439-1467.
80. Kamenskii M., Nistri P. An averaging method for singularly perturbed systems of semi linear differential inclusions with analytic semigroups. Nonlinear Analysis 2003, 53:467-480.
81. Akulenko L.D. Higher-order averaging schemes in systems with fast and slow phases. Journal of Applied Mathematics and Mechanics5 2002, 66(2):153-163.
82. Gine J., Llibre J. Limit cycles of cubic polynomial vector fields via the averaging theory. Nonlinear Analysis 2007, 66:1707-1721.
83. Mahmoud G.M., Aly S.A. Periodic attractors of complex damped non-linear systems. International Journal of Non-Linear Mechanics 2000, 35:309-323.
84. Marinca V., Herisanu N. Periodic solutions for some strongly nonlinear oscillations by He's variational iteration method. Computers and Mathematics with Applications 2007, 54:1188-1196.
85. Lin G. Periodic solutions for van der Pol equation with time delay. Applied Mathematics and Computation 2007, 187:1187-1198.
86. Lakrad F., Belhaq M. Periodic solutions of strongly non-linear oscillators by the multiple scales method. Journal of Sound and Vibration 2002, 258(4):677-700.
87. Wang Z.H., Hub H.Y., Wang H.L. Robust stabilization to non-linear delayed systems via delayed state feedback-the averaging method. Journal of Sound and Vibration 2005, 279:937-953.
88. Dohnal F. Suppressing self-excited vibrations by synchronous and time-periodic stiffness and damping variation. Journal of Sound and Vibration 2007, 306:136-152.
89. Ozis T., Yildirim A. Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method. Nonlinear Analysis. Real World Applications 2009, 10:1984-1989.
90. Sanders J., Verhulst F., Murdock J. Averaging Methods in Nonlinear Dynamical Systems 2007, Springer, New York.