Fourier series solution of Burger’s equation for nonlinear acoustics in relaxing media

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.

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A General Solution of Reynolds’ Equation for a Full Finite Journal Bearing

Journal of Lubrication Technology ◽

10.1115/1.3616949 ◽

1967 ◽

Vol 89 (2) ◽

pp. 203-210 ◽

Cited By ~ 2

Author(s):

R. R. Donaldson

Keyword(s):

Fourier Series ◽

Series Solution ◽

Reynolds Equation ◽

Journal Bearing ◽

Load Capacity ◽

Separation Of Variables ◽

Hill Equation ◽

Eigenvalues And Eigenvectors ◽

General Series ◽

Bearing Load

Reynolds’ equation for a full finite journal bearing lubricated by an incompressible fluid is solved by separation of variables to yield a general series solution. A resulting Hill equation is solved by Fourier series methods, and accurate eigenvalues and eigenvectors are calculated with a digital computer. The finite Sommerfeld problem is solved as an example, and precise values for the bearing load capacity are presented. Comparisons are made with the methods and numerical results of other authors.

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Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414500758 ◽

2014 ◽

Vol 24 (05) ◽

pp. 1450075 ◽

Cited By ~ 1

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Fourier Series ◽

Nonlinear Systems ◽

Analytical Solutions ◽

Nonlinear Oscillator ◽

Series Solution ◽

Parametric Oscillator ◽

Periodic Motions ◽

Parametric Oscillators ◽

Harmonic Term ◽

Nonlinear Behaviors

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

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