Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
New Applications of γ-Quasiconvexity Read chapter Read first chapter

2018 | OriginalPaper | Chapter

Qualitative, Approximate and Numerical Approaches for the Solution of Nonlinear Differential Equations

Authors : Eugenia N. Petropoulou, Michail A. Xenos

Published in: Applications of Nonlinear Analysis

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The differential equations that describe many realistic problems are nonlinear and most of these cannot be solved explicitly using standard analytic techniques. In such cases, qualitative, approximate or numerical techniques are employed, in order to obtain as much information as possible. The aim of the present chapter, is to give a description of the general ideas governing these techniques together with their advantages and limitations. This is achieved by implementing various methods to an initial value problem for a specific nonlinear ordinary differential equation, which combines both van der Pol and Duffing equations. This equation is solved using (a) the fourth order Runge-Kutta, the standard finite differences and the finite elements methods, (b) a nonstandard discretization technique based on functional analysis, (c) classical perturbation techniques and (d) the homotopy analysis method. Moreover, various results are given regarding the dynamic properties of its solution. Finally, this problem is connected with a Green function and this connection is again used for its numerical solution.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter On the Exact Solution of Nonlinear Integro-Differential Equations
next chapter On a Hilbert-Type Integral Inequality in the Whole Plane
Literature
1.
G. Adomian, A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135(3), 501–544 (1988)MathSciNetCrossRef
2.
G. Adomian, Solving Frontier Problems of Physics. The Decomposition Method (Springer, Berlin, 1994)CrossRef
3.
R.P. Agarwal, D. O’Regan, Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems (Springer Science+Business Media, LLC, New York, 2009)
4.
G.D. Akrivis, V.A. Dougalis, Numerical Methods for Ordinary Differential Equations (Crete University Press, Heraklion, 2006)
5.
G. Akrivis, Ch. Makridakis, Galerkin time-stepping method for nonlinear parabolic equations. M2AN Math. Model. Numer. Anal. 38, 261–289 (2004)
6.
V.I. Arnol’d, Ordinary Differential Equations (Springer, Berlin, 1992)
7.
J. Awreicewicz, On the occurence of chaos in van der Pol-Duffing’s oscillator. J. Sound Vib. 109(3), 519–522 (1986)
8.
C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. Asymptotic Methods and Perturbation Theory (Springer, New York, 1999)
9.
L.A. Bergman, J.E. Hyatt, Green functions for transversely vibrating uniform Euler-Bernoulli beams subject to constant axial preload. J. Sound Vib. 134(1), 175–180 (1989)CrossRef
10.
T.C. Bountis, L.B. Drossos, M. Lakshmanan, S. Parthasarathy, On the non-integrability of a family of Duffing-van der Pol oscillators. J. Phys. A Math. Gen. 26, 6927–6942 (1993)MathSciNetCrossRef
11.
J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edn. (Dover, New York, 2000)
12.
S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Method (Springer, New York, 1994)CrossRef
13.
J.C. Butcher, A stability property of implicit Runge-Kutta methods. BIT Numer. Math. 15, 358–361 (1975)CrossRef
14.
15.
V.K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems. J. Phys. A Math. Gen. 37, 4527–4534 (2004)MathSciNetCrossRef
16.
A. Chen, J. Jiang, Periodic solution of the Duffing-Van der Pol oscillator by homotopy perturbation method. Int. J. Comput. Math. 87(12), 2688–2696 (2010)MathSciNetCrossRef
17.
Y.M. Chen, J.K. Liu, Uniformly valid solution of limit cycle of the Duffing-van der Pol equation. Mech. Res. Commun. 36, 845–850 (2009)MathSciNetCrossRef
18.
A. Chudzik, Synchronisation and periodisation of Duffing oscillators coupled by elastic beam: finite element method approach. J. Theor. Appl. Mech. 48(2), 517–524 (2010)
19.
F. Dal, The method of multiple time scales and finite differences method for the van del Pol oscillator with small fractional damping. Asian Res. J. Math. 2(2), 1–11 (2017)CrossRef
20.
D.G. Duffy, Green’s Functions with Applications, 2nd edn. (Chapman and Hall/CRC, Boca Raton, 2017)
21.
C.J. Earle, R.S. Hamilton, A fixed point theorem for holomorphic mappings, In Global Analysis Proceedings Symposium Pure Mathematics, vol. XVI, Berkeley, CA, (1968) (American Mathematical Society, Providence, 1970), pp. 61–65
22.
23.
Z. Feng, G. Gao, J. Cui, Duffing-van der Pol-type oscillator system and its first integrals. Commun. Pure Appl. Anal. 10(5), 1377–1392 (2011)MathSciNetCrossRef
24.
C.A.J. Fletcher, Computational Techniques for Fluid Dynamics I (Spinger, Berlin, 1988)CrossRef
25.
C.A.J. Fletcher, Computational Techniques for Fluid Dynamics II (Spinger, Berlin, 1988)
26.
J. Gao, A.S. Selvarathinam, Y.J. Weitsman, Analysis of adhesively joined composite beams. J. Sandw. Struct. Mater. 1, 323–339 (1999)CrossRef
27.
I. Gohberg, S. Goldberg, Basic Operator Theory (Birkhäuser, Basel, 1980)
28.
D.H. Griffel, Applied Functional Analysis (Dover, New York, 2002)MATH
29.
M. Hatami, D.D. Ganji, M. Sheikholeslami, Differential Transformation Method for Mechanical Engineering Problems (Academic, Cambridge, 2016)
30.
31.
J.-H. He, Recent development of the homotopy perturbation method. Topol. Methods Nonlinear Anal. 31(2), 205–209 (2008)MathSciNetMATH
32.
P.J. Hilton, An Introduction to Homotopy Theory (Cambridge University Press, Cambridge, 1953)CrossRef
33.
A.J.T. Horvath, Periodic solutions of a combined Van der Pol-Duffing differential equation. Int. J. Mech. Sci. 17, 677–680 (1975)CrossRef
34.
P. Hou, K. Yuan, B. Chen, Study on the 3D Green’s functions of the fluid and piezoelectric bimaterials. Theor. Appl. Mech. Lett. 7, 105–116 (2017)
35.
E.K. Ifantis, Spectral theory of the difference equation f(n + 1) + f(n − 1) = [E − ϕ(n)]f(n). J. Math. Phys. 10(3), 421–425 (1969)MathSciNetCrossRef
36.
E.K. Ifantis, Solution of the Schrödinger equation in the Hardy–Lebesgue space. J. Math. Phys. 12, 1961–1965 (1971)MathSciNetCrossRef
37.
E.K. Ifantis, An existence theory for functional-differential equations and functional-differential systems. J. Differ. Equ. 29, 86–104 (1978)MathSciNetCrossRef
38.
E.K. Ifantis, Analytic solutions for nonlinear differential equations. J. Math. Anal. Appl. 124(2), 339–380 (1987)MathSciNetCrossRef
39.
E.K. Ifantis, On the convergence of power-series whose coefficients satisfy a Poincaré-type linear and nonlinear difference equation. Complex Var. 9, 63–80 (1987)MATH
40.
G. Iooss, D.D. Joseph, Elementary Stability and Bifurcation Theory, 2nd edn. (Springer, New York, 1990)CrossRef
41.
A. Iserles, A First Course in the Numerical Analysis of Differential Equations (Cambridge University Press, Cambridge, 1996)MATH
42.
Z. Jing, Z. Yang, T. Jiang, Complex dynamics in Duffing-Van der Pol equation. Chaos Solitons Fractals 27, 722–747 (2006)MathSciNetCrossRef
43.
D.W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations, 2nd edn. (Oxford University Press, Oxford, 1987)MATH
44.
A.Y.T. Leung, Q.C. Zhang, Complex normal form for strongly non-linear vibration systems exemplified by Duffing-van der Pol equation. J. Sound Vib. 213(5), 907–914 (1998)MathSciNetCrossRef
45.
R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Society for Industrial and Applied Mathematics, Philadelphia, 2007)CrossRef
46.
S. Liao, Beyond Perturbation. Introduction to the Homotopy Analysis Method (Chapman and Hall/CRC, Boca Raton, 2004)CrossRef
47.
S. Liao, Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14(4), 983–997 (2009)MathSciNetCrossRef
48.
Y. Liu, S.J. Liao, Z. Li, A Maple package of automated derivation of homotopy analysis solution for periodic nonlinear oscillations. J. Syst. Sci. Complex. 25(3), 594–616 (2012)MathSciNetCrossRef
49.
Y. Liu, S.J. Liao, Z. Li, Symbolic computation of strongly nonlinear periodic oscillations. J. Symb. Comput. 55, 72–95 (2013)MathSciNetCrossRef
50.
J. Logan, Applied Mathematics, 2nd edn. (Wiley, New York, 1997)MATH
51.
G.M. Mahmoud, A.A.M. Farghaly, Chaos control of chaotic limit cycles of real and complex van der Pol oscillators. Chaos Solitons Fractals 21, 915–924 (2004)MathSciNetCrossRef
52.
F.M. Moukam Kakmeni, S. Bowong, C. Tchawoua, E. Kaptouom, Strange attractors and chaos control in a Duffing-Van der Pol oscillator with two external periodic forces. J. Sound Vib. 277, 783–799 (2004)MathSciNetCrossRef
53.
R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations (World Scientific Publishing, River Edge, 1994)MATH
54.
R.E. Mickens, Nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl. 8(9), 823–847 (2002)MathSciNetCrossRef
55.
A. Okasha El-Nady, M.M.A. Lashin, Approximate solution of nonlinear Duffing oscillator using Taylor expansion. J. Mech. Eng. Autom. 6(5), 110–116 (2016)
56.
A.H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981)MATH
57.
V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations (Dover, New York, 1989)MATH
58.
E.N. Petropoulou, E.E. Tzirtzilakis, On the logistic equation in the complex plane. Numer. Funct. Anal. Optim. 34(7), 770–790 (2013)MathSciNetCrossRef
59.
E.N. Petropoulou, P.D. Siafarikas, E.E. Tzirtzilakis, A “discretization” technique for the solution of ODEs. J. Math. Anal. Appl. 331, 279–296 (2007)MathSciNetCrossRef
60.
E.N. Petropoulou, P.D. Siafarikas, E.E. Tzirtzilakis, A “discretization” technique for the solution of ODEs II. Numer. Funct. Anal. Optim. 30(5–6), 613–631 (2009)MathSciNetCrossRef
61.
Z.-H. Qin, Y.-S. Chen, Singularity analysis of Duffing-van der Pol system with two bifurcation parameters under multi-frequency excitations. Appl. Math. Mech. (English Ed.) 31(8), 1019–1026 (2010)MathSciNetCrossRef
62.
A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics (Springer, New York, 2000)MATH
63.
J.A. Rad, S. Kazem, K. Parand, A numerical solution of the nonlinear controlled Duffing oscillator by radial basis functions. Comput. Math. Appl. 64, 2049–2065 (2012)MathSciNetCrossRef
64.
S.S. Rao, Mechanical Vibrations, 5th edn. (Pearson Education, London, 2011)
65.
J. Rebenda, Z. Šmarda, A differential transformation approach for solving functional differential equations with multiple delays. Commun. Nonlinear Sci. Numer. Simul. 48, 246–257 (2017)MathSciNetCrossRef
66.
J. Rebenda, Z. Šmarda, Y. Khan, A new semi-analytical approach for numerical solving of Cauchy problem for differential equations with delay, Filomat 31(15), 4725–4733 (2017)MathSciNetCrossRef
67.
M. Sathyamoorthy, Nonlinear Analysis of Structures (CRC Press, Boca Raton, 1998)MATH
68.
A.S. Soomro, G.A. Tularam, M.M. Shaikh, A comparison of numerical methods for solving the unforced van der Pol’s equation. Math. Theory Model. 3(2), 66–77 (2013)
69.
W.-H. Steeb, N. Euler, Nonlinear Evolution Equations and Painlevé Test (World Scientific Publishing, Singapore, 1988)CrossRef
70.
W. Szemplińska-Stupnicka, J. Rudowski, The coexistence of periodic, almost-periodic and chaotic attractors in the van der Pol-Duffing oscillator. J. Sound Vib. 199(2), 165–175 (1997)MathSciNetCrossRef
71.
M.E. Taylor, Partial Differerential Equations I. Basic Theory, 2nd edn. (Springer Science+Business Media, LLC, New York, 2011)
72.
M.E. Taylor, Partial Differerential Equations II. Qualitative Studies of Linear Equations, 2nd edn. (Springer Science+Business Media, LLC, New York, 2011)
73.
M.E. Taylor, Partial Differerential Equations III. Nonlinear Equations, 2nd edn. (Springer Science+Business Media, LLC, New York, 2011)
74.
A. Venkatesan, M. Lakshmanan, Bifurcation and chaos in the double-well Duffing-van der Pol oscillator: numerical and analytical studies, Phys. Rev. E 56(6), 6321–6330 (1997)MathSciNetCrossRef
75.
D. Wu, L. Yang, Y. Gao, Three-dimensional Green’s functions of thermoporoelastic axisymmetric cones. Appl. Math. Model. 42, 315–329 (2017)MathSciNetCrossRef
76.
M.A. Xenos, An Euler-Lagrange approach for studying blood flow in an aneurysmal geometry. Proc. R. Soc. A 473, 20160774 (2017)MathSciNetCrossRef
77.
L. Xie, C. Zhang, C. Hwu, E. Pan, On novel explicit expressions of Green’s function and its derivatives for magnetoelectroelastic materials. Eur. J. Mech. A Solid 60, 134–144 (2016)MathSciNetCrossRef
78.
R. Yamapi, G. Filatrella, Strange attractors and synchronization dynamics of coupled Van der Pol-Duffing oscillators. Commun. Nonlinear Sci. Numer. Simul. 13, 1121–1130 (2008)MathSciNetCrossRef
79.
J. Yu, W.-Z. Pan, R.-B. Zhang, Period-doubling cascades and strange attractors in extended Duffing-van der pol oscillator. Commun. Theor. Phys. (Beijing, China) 51, 865–868 (2009)MathSciNetCrossRef
Metadata
Title
Qualitative, Approximate and Numerical Approaches for the Solution of Nonlinear Differential Equations
Authors
Eugenia N. Petropoulou
Michail A. Xenos
Copyright Year
2018
Book
Applications of Nonlinear Analysis

Print ISBN: 978-3-319-89814-8

Electronic ISBN: 978-3-319-89815-5

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-89815-5_22

Premium Partner

    Image Credits