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Research Papers

Fixed Point Analytical Method for Nonlinear Differential Equations

[+] Author and Article Information
Ding Xu

School of Aerospace,
State Key Laboratory for Strength and Vibration of Mechanical Structures
,Xi’an Jiaotong University,
No. 28, Xianning West Road,
Xi’an 710049, China
e-mail: dingxu@mail.xjtu.edu.cn

Xin Guo

Xi’an Research Institute of China
Coal Technology & Engineering Group Corp.,
Xi’an 710054, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 26, 2011; final manuscript received March 3, 2012; published online June 14, 2012. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 8(1), 011005 (Jun 14, 2012) (9 pages) Paper No: CND-11-1139; doi: 10.1115/1.4006337 History: Received August 26, 2011; Revised March 03, 2012

In this paper, a new analytical method, which is based on the fixed point concept in functional analysis (namely, the fixed point analytical method (FPM)), is proposed to acquire the explicit analytical solutions of nonlinear differential equations. The key idea of this approach is to construct a contractive map which replaces the nonlinear differential equation into a series of linear differential equations. Usually, the series of linear equations can be solved relatively easily and have explicit analytical solutions. The FPM is different from all existing analytical methods, such as the well-known perturbation technique applied in weakly nonlinear problems, because it is independent of any small physical or artificial parameters at all; thus, it can handle more nonlinear problems, including strongly nonlinear ones. Two typical cases are investigated by FPM in detail and the comparison with the numerical results shows that the present method is one of high accuracy and efficiency.

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References

Nayfeh, A. H., 1973, Perturbation Methods, Wiley Online Library, New York.
Nayfeh, A. H., 1981, Introduction to Perturbation Techniques, Wiley, New York.
Smith, D. R., 1985, Singular-Perturbation Theory: An Introduction With Applications, Cambridge University Press, Cambridge, England.
Bender, C. M., Milton, K. A., Pinsky, S. S., and Simmons, L. M., 1989, “A New Perturbative Approach to Nonlinear Problems,” J. Math. Phys., 30(7), pp. 1447–1455. [CrossRef]
Bender, C. M., 1991, “New Approach to the Solution of Nonlinear Problems,” Large Scale Structures in Nonlinear Physics, J.-D.Fournier and P.-L.Sulem, eds., Springer, Berlin, Heidelberg, pp. 190–210.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, Wiley Online Library, New York.
Zeidler, E., 1986, Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems, Springer-Verlag, Berlin.
Zeidler, E., 1995, Applied Functional Analysis: Applications to Mathematical Physics, Springer, New York.

Figures

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Fig. 1

λ-curves of yn(0). The valid convergence regions of λ correspond to the neighborhood of λ=0.045, (ɛ=0.001).

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Fig. 2

Comparison of the exact solution ye(x) with the nth-order approximation yn(x), (ɛ=0.001)

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Fig. 3

Relative error [yn(x)-ye(x)]/ye(x) in the whole region 0≤x≤1, (ɛ=0.001)

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Fig. 6

Relative error [yn(x)-ye(x)]/ye(x) in the whole region 0≤x≤1, (ɛ=1)

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Fig. 7

Comparison of the exact solution ye(x) with the nth-order approximation yn(x), (ɛ=1000)

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Fig. 8

Relative error [y1(x)-ye(x)]/ye(x) in the whole region 0≤x≤1, (ɛ=1000)

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Fig. 9

Comparison of the exact value ye(0) with the 2nd-order approximation y2(0) given by Eq. (28)

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Fig. 4

λ-curves of yn(0). The valid convergence regions of λ correspond to the neighborhood of λ=2.5, (ɛ=1).

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Fig. 5

Comparison of the exact solution ye(x) with the nth-order approximation yn(x), (ɛ=1)

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Fig. 10

β-curves of Ωn. The valid convergence regions of β correspond to the neighborhood of β=0.6, (ɛ=1)

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Fig. 11

Comparison of the numerical result with the 2nd-order approximation u2(t), (ɛ=1)

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Fig. 12

Relative error [u2(t)/unum(t)-1]; the maximum relative error is about 0.027% for u2(t), (ɛ=1)

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Fig. 15

Relative error [u2(t)/unum(t)-1]; the maximum relative error is about 0.34% for u2(t), (ɛ=1000)

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Fig. 16

Comparison of the exact period Te with the 2nd-order approximation T2 given by Eq. (48)

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Fig. 13

β-curves of Ωn, The valid convergence regions of β correspond to the neighborhood of β=0.0015, (ɛ=1000)

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Fig. 14

Comparison of the numerical result with the 2nd-order approximation u2(t), (ɛ=1000)

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