Flow in Straight Ducts of Arbitrary Cross Section

1979 ◽  
Vol 46 (2) ◽  
pp. 263-268 ◽  
Author(s):  
P. W. Duck
Keyword(s):  
Fourier Series ◽  
Conformal Mapping ◽  
Differential Equations ◽  
Cross Section ◽  
Coupled System ◽  
Arbitrary Cross Section ◽  
Numerical Results ◽  
Steady Flows ◽  
System Of Differential Equations ◽  
Oscillatory Flows

A method is presented for treating flows in straight ducts. The method involves the conformal mapping of the cross section on to a semicircle, and then solving the problem using Fourier series. For oscillatory flows (to which most of the paper is devoted) the method results in an infinite, coupled system of differential equations, although reliable numerical results may be obtained by truncation. For steady flows, however, the method yields a solution involving integrals of the Jacobian of the transformation.

A study of nonlinear biochemical reaction model

2016 ◽  
Vol 09 (05) ◽  
pp. 1650071 ◽  
Author(s):  
Muhammad Asad Iqbal ◽  
Syed Tauseef Mohyud-Din ◽  
Bandar Bin-Mohsin
Keyword(s):  
Differential Equations ◽  
Reaction Model ◽  
Numerical Results ◽  
Picard Iteration ◽  
System Of Differential Equations ◽  
Legendre Wavelets ◽  
Enzyme Substrate ◽  
Runge Kutta Method ◽  
Picard Method ◽  
Order Four

The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method (LWPM). Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge–Kutta method of order four (RK-4). Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.

scholarly journals Differential Equations of Motion for Naturally Curved and Twisted Composite Space Beams

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Ying Hao ◽  
Wei He ◽  
Yanke Shi
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Cross Sections ◽  
Design Principle ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Coupled System ◽  
Variable Coefficients ◽  
Curve Beam ◽  
Design Factor

The differential equations of motion for naturally curved and twisted elastic space beams made of anisotropic materials with noncircular cross sections, being a coupled system consisting of 14 second-order partial differential equations with variable coefficients, are derived theoretically. The warping deformation of beam’s cross section, as a new design factor, is incorporated into the differential equations in addition to the anisotropy of material, the curvatures of the rod axis, the initial twist of the cross section, the rotary inertia, and the shear and axial deformations. Numerical examples show that the effect of warping deformation on the natural frequencies of the beam is significant under certain geometric and boundary conditions. This study focuses on improving and consummating the traditional theories to build a general curve beam theory, thereby providing new scientific research reference and design principle for curve beam designers.

PERIODIC ORBITS IN THE NEWTON–LEIPNIK SYSTEM

2002 ◽  
Vol 12 (03) ◽  
pp. 511-523 ◽  
Author(s):  
BENJAMIN A. MARLIN
Keyword(s):  
Periodic Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Periodic Orbits ◽  
Closed Orbit ◽  
Numerical Results ◽  
Asymptotically Stable ◽  
System Of Differential Equations ◽  
Promising Candidate ◽  
Closed Orbits

This paper considers an autonomous nonlinear system of differential equations derived in [Leipnik, 1979]. A criterion for the existence of closed orbits in similar systems is presented. Numerical results are made rigorous by the use of interval analytic techniques in establishing the existence of a periodic solution which is not asymptotically stable. The limitations of the method of locating orbits are considered when a promising candidate for a closed orbit is shown not to intersect itself.

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