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.

scholarly journals A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Mohamed R. Ali
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Power Series ◽  
Fractional Order ◽  
Nonlinear Ordinary Differential Equation ◽  
Lie Symmetry ◽  
Series Method ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Bo Equation

We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order, we complete the solutions by utilizing the power series method (PSM).

scholarly journals On finite series solutions of conformable time-fractional Cahn-Allen equation

2020 ◽  
Vol 9 (1) ◽  
pp. 194-200 ◽  
Author(s):  
Asim Zafar ◽  
Hadi Rezazadeh ◽  
Khalid K. Ali
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Exact Solutions ◽  
Fractional Order ◽  
Symbolic Computation ◽  
Nonlinear Ordinary Differential Equation ◽  
Hyperbolic Function ◽  
Series Solutions ◽  
Finite Series ◽  
New Exact Solutions

AbstractThe aim of this article is to derive new exact solutions of conformable time-fractional Cahn-Allen equation. We have achieved this aim by hyperbolic function and expa function methods with the aid of symbolic computation using Mathematica. This idea seems to be very easy to employ with reliable results. The time fractional Cahn-Allen equation is reduced to respective nonlinear ordinary differential equation of fractional order. Also, we have depicted graphically the constructed solutions.

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.

Steady flow in locally exponential tubes

1982 ◽  
Vol 383 (1784) ◽  
pp. 231-245 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Reynolds Number ◽  
Viscous Flow ◽  
Steady Flow ◽  
Stream Function ◽  
Nonlinear Ordinary Differential Equation ◽  
Wide Applicability ◽  
First Approximation ◽  
Axial Variation

Daniels & Eagles (1979) developed a theory for steady, axisymmetric, viscous flow in tubes of a slowly varying radius. This flow, for tubes of exponentially varying radius, was governed to a first approximation by a nonlinear ordinary differential equation when the Reynolds number was high and the scale of the axial variation in radius was proportional to the Reynolds number. In the present work this theory is modified and extended so that the Daniels & Eagles solutions may be applied to a wide variety of ‘locally exponential’ tubes. Numerical calculations are made of several terms in the series for the Stokes stream function, and a number of examples are presented, which indicate that the theory has a wide applicability.

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.

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