A Bifurcation Problem for a Fourth Order Nonlinear Ordinary Differential Equation

1974 ◽  
Vol 27 (1) ◽  
pp. 93-101 ◽  
Author(s):  
R. W. Dickey
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fourth Order ◽  
Nonlinear Ordinary Differential Equation ◽  
Bifurcation Problem

scholarly journals Iterative Methods for Solving the Fractional Form of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
A. A. Hemeda ◽  
E. E. Eladdad
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Reynolds Number ◽  
Fractional Order ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Axisymmetric Flow ◽  
Nonlinear Ordinary Differential Equation ◽  
Absolute Residual

An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is proposed with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations of motion is reduced to a single fourth-order nonlinear ordinary differential equation. By using the basic definitions of fractional calculus, we introduced the fractional order form of the fourth-order nonlinear ordinary differential equation. The resulting boundary value fractional problems are solved by the new iterative and Picard methods. Convergence of the considered methods is confirmed by obtaining absolute residual errors for approximate solutions for various Reynolds number. The comparisons of the solutions for various Reynolds number and various values of the fractional order confirm that the two methods are identical and therefore are suitable for solving this kind of problems. Finally, the effects of various Reynolds number on the solution are also studied graphically.

Some Properties of Laminar Boundary Layers on Curved Surfaces

1968 ◽  
Vol 90 (2) ◽  
pp. 301-312 ◽  
Author(s):  
B. S. Massey ◽  
B. R. Clayton
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Ordinary Differential Equation ◽  
Fourth Order ◽  
Nonlinear Ordinary Differential Equation ◽  
Curved Surfaces ◽  
Two Dimensional ◽  
Laminar Boundary Layers ◽  
Displacement Thickness ◽  
Similar Solutions

Equations have been developed [1] which describe the flow in a steady, two-dimensional, incompressible, laminar boundary layer on a curved surface. The method of “similar solutions” yields a fourth-order, nonlinear, ordinary differential equation which may be solved on a digital computer. Account is taken of the effects of both surface curvature and displacement thickness, and in this paper attention is given to the influence of these effects on other boundary-layer properties up to and including the position of flow separation.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

Design of evolutionary cubic spline intelligent solver for nonlinear Painlevé-I transcendent

2021 ◽  
Author(s):  
Sharafat Ali ◽  
Iftikhar Ahmad ◽  
Muhammad Asif Zahoor Raja ◽  
Siraj ul Islam Ahmad ◽  
Muhammad Shoaib
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Genetic Algorithms ◽  
Statistical Analysis ◽  
Quadratic Programming ◽  
Sequential Quadratic Programming ◽  
Process Variation ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Derivative ◽  
Search Technique

In this research paper, an innovative bio-inspired algorithm based on evolutionary cubic splines method (CSM) has been utilized to estimate the numerical results of nonlinear ordinary differential equation Painlevé-I. The computational mechanism is used to support the proposed technique CSM and optimize the obtained results with global search technique genetic algorithms (GAs) hybridized with sequential quadratic programming (SQP) for quick refinement. Painlevé-I is solved by the proposed technique CSM-GASQP. In this process, variation of splines is implemented for various scenarios. The CSM-GASQP produces an interpolated function that is continuous upto its second derivative. Also, splines proved to be stable than a single polynomial fitted to all points, and reduce wiggles between the tabulated points. This method provides a reliable and excellent procedure for adaptation of unknown coefficients of splines by searching globally exploiting the performance of GA-SQP algorithms. The convergence, exactness and accuracy of the proposed scheme are examined through the statistical analysis for the several independent runs.

Approximated Solutions for Axisymmetric Wrinkled Inflated Membranes

2015 ◽  
Vol 82 (11) ◽  
Author(s):  
Riccardo Barsotti
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element ◽  
Nonlinear Ordinary Differential Equation ◽  
Experimental Results ◽  
Limit Case ◽  
Reasonable Approximation ◽  
Constituent Material ◽  
Actual Solution ◽  
Wrinkled Membrane

The axisymmetric inflation problem for a wrinkled membrane is solved by means of a simple nonlinear ordinary differential equation. The solution is illustrated in full details. Both the free and constrained cases are addressed, in the limit case where the membrane is fully wrinkled. In the constrained inflation problem, no slippage is allowed between the membrane and the constraining surfaces. It is shown that an actual membrane can in no way reach the fully wrinkled configuration during free inflation, regardless of the membrane's initial configuration and constituent material. The fully wrinkled solution is compared to some finite element results obtained by means of an expressly developed iterative–incremental procedure. When the values of the inflating pressure and length of the meridian lie within a suitable applicability range, the fully wrinkled solution may represent a reasonable approximation of the actual solution. A comparison with some numerical and experimental results available in the literature is illustrated.

scholarly journals Approximate solution of nonlinear ordinary differential equation using ZZ decomposition method

2020 ◽  
Vol 4 (1) ◽  
pp. 448-455
Author(s):  
Mulugeta Andualem ◽  
◽  
Atinafu Asfaw ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Approximate Solution ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Nonlinear Ordinary Differential Equation ◽  
Initial Value ◽  
Transform Method ◽  
Adomian Decomposition

Nonlinear initial value problems are somewhat difficult to solve analytically as well as numerically related to linear initial value problems as their variety of natures. Because of this, so many scientists still searching for new methods to solve such nonlinear initial value problems. However there are many methods to solve it. In this article we have discussed about the approximate solution of nonlinear first order ordinary differential equation using ZZ decomposition method. This method is a combination of the natural transform method and Adomian decomposition method.

Dynamic Models for Magnetorheological Dampers and its Feedback Linearization

2009 ◽  
Vol 79-82 ◽  
pp. 1205-1208 ◽  
Author(s):  
Cheng Zhang ◽  
Lin Xiang Wang
Keyword(s):  
Differential Equation ◽  
Phase Transition ◽  
Ordinary Differential Equation ◽  
Feedback Linearization ◽  
Dynamic Models ◽  
Nonlinear Ordinary Differential Equation ◽  
Transition Theory ◽  
Magnetorheological Dampers ◽  
Differential Model ◽  
Mr Dampers

In the current paper, the hysteretic dynamics of magnetorheological dampers is modeled by a differential model. The differential model is constructed on the basis of a phenomenological phase transition theory. The model is expressed as a second order nonlinear ordinary differential equation with bifurcations embedded in. Due to the differential nature of the model, the hysteretic dynamics of the MR dampers can be linearized and controlled by introducing a feedback linearization strategy.

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