A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

2012 ◽  
Vol 67 (12) ◽  
pp. 665-673 ◽  
Author(s):  
Kourosh Parand ◽  
Mehran Nikarya ◽  
Jamal Amani Rad ◽  
Fatemeh Baharifard
Keyword(s):  
Viscous Flow ◽  
Flat Plate ◽  
Numerical Algorithm ◽  
Bessel Functions ◽  
Nonlinear Ordinary Differential Equation ◽  
Finite Domain ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Third Order ◽  
Domain Truncation

In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on ℝ and is convergent for any x ∊ℝ. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.

scholarly journals Solution of the Falkner–Skan Equation Using the Chebyshev Series in Matrix Form

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Abdelrady Okasha Elnady ◽  
M. Fayek Abd Rabbo ◽  
Hani M. Negm
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Numerical Method ◽  
Matrix Representation ◽  
Point Of View ◽  
Finite Domain ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Infinite Domain ◽  
Computational Point

A numerical method for the solution of the Falkner–Skan equation, which is a nonlinear differential equation, is presented. The method has been derived by truncating the semi-infinite domain of the problem to a finite domain and then expanding the required approximate solution as the elements of the Chebyshev series. Using matrix representation of a function and their derivatives, the problem is reduced to a system of algebraic equations in a simple way. From the computational point of view, the results are in excellent agreement with those presented in published works.

Transient Thermal Stresses in a Flat Plate With a Circular Opening

1995 ◽  
Author(s):  
Stephen C. Lipp ◽  
Paul D. Herrington ◽  
Edwin P. Russo
Keyword(s):  
Boundary Conditions ◽  
Flat Plate ◽  
Thermal Stresses ◽  
Integral Transform ◽  
Welding Process ◽  
Outer Radius ◽  
Finite Domain ◽  
Circular Opening ◽  
Infinite Domain ◽  
Thermal Boundary Conditions

Abstract Solutions for stress distributions in a flat plate with a circular opening which is undergoing radial and temporal changes in temperature are presented. A typical example would be a hot pipe penetrating a wall or a welding process. The problem is formulated as a partial differential equation on an infinite-domain problem with inhomogeneous boundary conditions. As such, the solution of the problem is obtained through integral-transform methods involving integrals difficult to numerically evaluate. The problem may also be formulated as a finite-domain problem with eigenvalue solution. The results in this case depend on the number of eigenvalues calculated which depends on the time value chosen. Three types of thermal boundary conditions are considered: (1) convection on the boundary, (2) insulated boundary, and (3) sudden heating or cooling, that is, thermal shock. Some sample results comparing the integral-transform and eigenvalue methods are presented. The results indicate that temperature values computed by both methods for an aluminum sample with a 1.5 in (0.0381 m) hole at time 10 s are identical to four decimal places. The finite domain outer radius for the eigenvalue solution was taken to be 18 in (0.4572 m). However, there is a discrepancy when comparing radial and tangential stress. This difference is the stress caused by the constant temperature from the finite boundary to infinity when imposed on the finite-domain problem.

Solutions of the Blasius and MHD Falkner-Skan boundary-layer equations by modified rational Bernoulli functions

2017 ◽  
Vol 27 (8) ◽  
pp. 1687-1705 ◽  
Author(s):  
Velinda Calvert ◽  
Mohsen Razzaghi
Keyword(s):  
Boundary Layer ◽  
Rational Functions ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Nonlinear Boundary Value Problems ◽  
Infinite Domain ◽  
Third Order ◽  
Content Type ◽  
Boundary Layer Equations ◽  
Bernoulli Functions

Purpose This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third-order nonlinear boundary value problems on the semi-infinite domain. Design/methodology/approach The approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations. Findings The method is computationally very attractive and gives very accurate results. Originality/value Many problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations.

Laminar Viscous Flow past a Semi-Infinite Flat Plate

1970 ◽  
Vol 28 (3) ◽  
pp. 776-779 ◽  
Author(s):  
Akira Yoshizawa
Keyword(s):  
Viscous Flow ◽  
Flat Plate ◽  
Flow Past

Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model

2009 ◽  
Vol 20 (2) ◽  
pp. 187-214 ◽  
Author(s):  
WAN CHEN ◽  
MICHAEL J. WARD
Keyword(s):  
Hopf Bifurcation ◽  
Stefan Problem ◽  
Differential Algebraic Equations ◽  
Finite Domain ◽  
Quasi Equilibrium ◽  
Algebraic Equations ◽  
Source Terms ◽  
One Dimensional ◽  
The One ◽  
Oscillatory Instabilities

The dynamics and oscillatory instabilities of multi-spike solutions to the one-dimensional Gray-Scott reaction–diffusion system on a finite domain are studied in a particular parameter regime. In this parameter regime, a formal singular perturbation method is used to derive a novel ODE–PDE Stefan problem, which determines the dynamics of a collection of spikes for a multi-spike pattern. This Stefan problem has moving Dirac source terms concentrated at the spike locations. For a certain subrange of the parameters, this Stefan problem is quasi-steady and an explicit set of differential-algebraic equations characterizing the spike dynamics is derived. By analysing a nonlocal eigenvalue problem, it is found that this multi-spike quasi-equilibrium solution can undergo a Hopf bifurcation leading to oscillations in the spike amplitudes on an O(1) time scale. In another subrange of the parameters, the spike motion is not quasi-steady and the full Stefan problem is solved numerically by using an appropriate discretization of the Dirac source terms. These numerical computations, together with a linearization of the Stefan problem, show that the spike layers can undergo a drift instability arising from a Hopf bifurcation. This instability leads to a time-dependent oscillatory behaviour in the spike locations.

On Hypersonic Viscous Flow Over an Insulated Flat Plate With Surface Mass Transfer

1962 ◽  
Vol 29 (9) ◽  
pp. 1024-1028 ◽  
Author(s):  
C. L. TIEN
Keyword(s):  
Mass Transfer ◽  
Viscous Flow ◽  
Flat Plate ◽  
Surface Mass ◽  
Surface Mass Transfer

Solution of third-order Emden–Fowler-type equations using wavelet methods

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Arshad Khan ◽  
Mo Faheem ◽  
Akmal Raza
Keyword(s):  
Nuclear Reactions ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Third Order ◽  
Content Type ◽  
Resolution Level ◽  
The Stability ◽  
Comparison Of The Results ◽  
Wavelet Methods ◽  
Nonlinear Nature

Purpose The numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs. Design/methodology/approach Because of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods. Findings This paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased. Originality/value This paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.

On the Integral of the Product of Three Bessel Functions over an Infinite Domain

2012 ◽  
Vol 14 ◽  
Author(s):  
Sunil Auluck
Keyword(s):  
Bessel Functions ◽  
Infinite Domain
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