scholarly journals A Fourier Series Solution for Transient Three‐Dimensional Thermohaline Convection in Porous Enclosures

2020 ◽  
Vol 56 (11) ◽  
Author(s):  
Sara Tabrizinejadas ◽  
Marwan Fahs ◽  
Behzad Ataie‐Ashtiani ◽  
Craig T. Simmons ◽  
Raphaël Chiara Roupert ◽  
...  
Keyword(s):  
Fourier Series ◽  
Series Solution ◽  
Three Dimensional ◽  
Thermohaline Convection

Thermal Characterization of Electronic Packages Using a Three-Dimensional Fourier Series Solution

2000 ◽  
Vol 122 (3) ◽  
pp. 233-239 ◽  
Author(s):  
J. R. Culham ◽  
M. M. Yovanovich ◽  
T. F. Lemczyk
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Series Solution ◽  
Analytical Approach ◽  
Printed Circuit Boards ◽  
Three Dimensional ◽  
Thermal Characterization ◽  
Steady State Solution ◽  
Electronic Packages

The need to accurately predict component junction temperatures on fully operational printed circuit boards can lead to complex and time consuming simulations if component details are to be adequately resolved. An analytical approach for characterizing electronic packages is presented, based on the steady-state solution of the Laplace equation for general rectangular geometries, where boundary conditions are uniformly specified over specific regions of the package. The basis of the solution is a general three-dimensional Fourier series solution which satisfies the conduction equation within each layer of the package. The application of boundary conditions at the fluid-solid, package-board and layer-layer interfaces provides a means for obtaining a unique analytical solution for complex IC packages. Comparisons are made with published experimental data for both a plastic quad flat package and a multichip module to demonstrate that an analytical approach can offer an accurate and efficient solution procedure for the thermal characterization of electronic packages. [S1043-7398(00)01403-1]

scholarly journals Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Ming-Sheng Hu ◽  
Ravi P. Agarwal ◽  
Xiao-Jun Yang
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Series Solution ◽  
Differential Operators ◽  
Fractional Wave Equation ◽  
Local Fractional Calculus ◽  
Vibrating String ◽  
Fractional Differential ◽  
Fractional Differential Operators

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.

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

1967 ◽  
Vol 89 (2) ◽  
pp. 203-210 ◽  
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.

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

2014 ◽  
Vol 24 (05) ◽  
pp. 1450075 ◽  
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.

A series solution and numerical technique for wave diffraction by a three-dimensional canyon

Wave Motion ◽  
2004 ◽  
Vol 39 (2) ◽  
pp. 129-142 ◽  
Author(s):  
Wen-I Liao ◽  
Tsung-Jen Teng ◽  
Chau-Shioung Yeh
Keyword(s):  
Wave Diffraction ◽  
Series Solution ◽  
Numerical Technique ◽  
Three Dimensional

Inhomogeneous thermal conduction equations

1978 ◽  
Vol 56 (7) ◽  
pp. 928-935
Author(s):  
C. S. Lai
Keyword(s):  
Differential Equations ◽  
Thermal Conduction ◽  
Series Solution ◽  
Nonlinear Differential Equations ◽  
Three Dimensional ◽  
Incompressible Fluids ◽  
Self Similar Solution ◽  
Self Similar ◽  
A New Technique ◽  
The One

The method of self-similar solution of partial differential equations is applied to the one-, two-, and three-dimensional inhomogeneous thermal conduction equations with the thermometric conductivities χ ~ rmWn. Analytical solutions are obtained for the case that the total amount of heat is conserved. For the case that the temperature is maintained constant at r = 0, a new technique of the series solution about the point of intercept is proposed to solve the resultant nonlinear differential equations. The solutions obtained are useful in studying the thermal conduction characteristics of some incompressible fluids.

The phase problem and electron-density maps

2010 ◽  
Author(s):  
Jenny Pickworth Glusker ◽  
Kenneth N. Trueblood
Keyword(s):  
Crystal Structure ◽  
Fourier Series ◽  
Diffraction Pattern ◽  
Electron Density ◽  
Bragg Reflection ◽  
Three Dimensional ◽  
Density Maps ◽  
Structure Factors ◽  
Unit Cells ◽  
Phase Angles

In order to obtain an image of the material that has scattered X rays and given a diffraction pattern, which is the aim of these studies, one must perform a three-dimensional Fourier summation. The theorem of Jean Baptiste Joseph Fourier, a French mathematician and physicist, states that a continuous, periodic function can be represented by the summation of cosine and sine terms (Fourier, 1822). Such a set of terms, described as a Fourier series, can be used in diffraction analysis because the electron density in a crystal is a periodic distribution of scattering matter formed by the regular packing of approximately identical unit cells. The Fourier series that is used provides an equation that describes the electron density in the crystal under study. Each atom contains electrons; the higher its atomic number the greater the number of electrons in its nucleus, and therefore the higher its peak in an electrondensity map.We showed in Chapter 5 how a structure factor amplitude, |F (hkl)|, the measurable quantity in the X-ray diffraction pattern, can be determined if the arrangement of atoms in the crystal structure is known (Sommerfeld, 1921). Now we will show how we can calculate the electron density in a crystal structure if data on the structure factors, including their relative phase angles, are available. The Fourier series is described as a “synthesis” when it involves structure amplitudes and relative phases and builds up a picture of the electron density in the crystal. By contrast, a “Fourier analysis” leads to the components that make up this series. The term “relative” is used here because the phase of a Bragg reflection is described relative to that of an imaginary wave diffracted in the same direction at a chosen origin of the unit cell.

Initial design of low-thrust trajectories based on the Bezier curve-based shaping approach

2020 ◽  
Vol 234 (11) ◽  
pp. 1825-1835
Author(s):  
Zichen Fan ◽  
Mingying Huo ◽  
Naiming Qi ◽  
Ce Zhao ◽  
Ze Yu ◽  
...  
Keyword(s):  
Fourier Series ◽  
Three Dimensional ◽  
Initial Approximation ◽  
Mission Design ◽  
Bézier Curve ◽  
Design Stage ◽  
Bezier Curve ◽  
Series Method ◽  
Low Thrust ◽  
Fourier Series Method

This paper presents a method to use the Bezier curve to rapidly generate three-dimensional low-thrust trajectories, which can provide a suitable initial approximation to be used for more accurate trajectory optimal control tools. Two missions, from Earth to Mars and the asteroid Dionysus, are considered to evaluate the performance of the method. In order to verify the advantages of this method, it is compared with the finite Fourier series method. Numerical results show that the Bezier method can get better performance index in shorter computation time compared with the finite Fourier series method. The applicability of the solution obtained by Bezier method is evaluated by introducing the obtained solution into the Gauss pseudospectral method as an initial guess. The simulation results show that the Bezier method can rapidly generate a very suitable three-dimensional initial trajectory for the optimal solver. This is very important for rapid evaluation of the feasibility of a large number of low-thrust flight schemes in the preliminary mission design stage.

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