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2018 | OriginalPaper | Buchkapitel

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

verfasst von : Eugenia N. Petropoulou, Michail A. Xenos

Erschienen in: Applications of Nonlinear Analysis

Verlag: Springer International Publishing

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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.

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Vorheriges Kapitel On the Exact Solution of Nonlinear Integro-Differential Equations

Nächstes Kapitel On a Hilbert-Type Integral Inequality in the Whole Plane

G. Adomian, A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135(3), 501–544 (1988)MathSciNetCrossRef

G. Adomian, Solving Frontier Problems of Physics. The Decomposition Method (Springer, Berlin, 1994)CrossRef

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)

G.D. Akrivis, V.A. Dougalis, Numerical Methods for Ordinary Differential Equations (Crete University Press, Heraklion, 2006)

G. Akrivis, Ch. Makridakis, Galerkin time-stepping method for nonlinear parabolic equations. M2AN Math. Model. Numer. Anal. 38, 261–289 (2004)

V.I. Arnol’d, Ordinary Differential Equations (Springer, Berlin, 1992)

J. Awreicewicz, On the occurence of chaos in van der Pol-Duffing’s oscillator. J. Sound Vib. 109(3), 519–522 (1986)

C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers. Asymptotic Methods and Perturbation Theory (Springer, New York, 1999)

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.

J.H.E. Cartwright, O. Piro, The dynamics of Runge–Kutta methods. Int. J. Bifurc. Chaos 2, 427–449 (1992)MathSciNetCrossRef

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.

Z. Feng, Duffing-van der Pol-type oscillator systems. Discret. Contin. Dyn. Syst. Ser. S 7(6), 1231–1257 (2014)MathSciNetCrossRef

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.

J.-H. He, Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178(3–4), 257–262 (1999)MathSciNetCrossRef

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

Titel: Qualitative, Approximate and Numerical Approaches for the Solution of Nonlinear Differential Equations
verfasst von: Eugenia N. Petropoulou
Michail A. Xenos
Verlag: Springer International Publishing
Buch: Applications of Nonlinear Analysis
Print ISBN: 978-3-319-89814-8

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

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-89815-5_22

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