The Iterative Transformation Method

Abstract In a transformation method, the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. Therefore, a transformation method, like a shooting method, is an initial value method. The main difference between a transformation and a shooting method is that the former is conceived and derive its formulation from the scaling invariance theory. This paper is concerned with the application of the iterative transformation method to several problems in the boundary layer theory. The iterative method is an extension of the Topfer’s non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. This iterative method provides a simple numerical test for the existence and uniqueness of solutions. Here we show how the method can be applied to problems with a homogeneous boundary conditions at infinity and in particular we solve the Sakiadis problem of boundary layer theory. Moreover, we show how to couple our method with Newton’s root-finder. The obtained numerical results compare well with those available in the literature. The main aim here is that any method developed for the Blasius, or the Sakiadis, problem might be extended to more challenging or interesting problems. In this context, the iterative transformation method has been recently applied to compute the normal and reverse flow solutions of Stewartson for the Falkner–Skan model (Fazio, 2013) .

[1]  W. Gautschi.  Numerical analysis: an introduction , 1997 .

[2]  L. Howarth,et al.  On the solution of the laminar boundary layer equations. , 1938, Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences.

[3]  N. S. Asaithambi,et al.  A numerical method for the solution of the Falkner-Skan equation , 1997 .

[4]  H. B. Keller,et al.  Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation , 1971 .

[5]  L. A. Rubel.  An estimate of the error due to the truncated boundary in the numerical solution of the Blasius equation , 1955 .

[6]  Riccardo Fazio,et al.  The falkneer-skan equation: Numerical solutions within group invariance theory , 1994 .

[7]  Riccardo Fazio.  The Iterative Transformation Method for the Sakiadis Problem , 2014, 1410.2043.

[8]  Riccardo Fazio.  A moving boundary hyperbolic problem for a stress impact in a bar of rate-type material , 1992 .

[9]  I. Tani.  On the Solution of the Laminar Boundary Layer Equations , 1949 .

[10]  W. A. Coppel.  On a differential equation of boundary-layer theory , 1960, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[11]  Riccardo Fazio,et al.  A Similarity Approach to the Numerical Solution of Free Boundary Problems , 1998, SIAM Rev..

[12]  R. J. Goldstein,et al.  Flow and heat transfer in the boundary layer on a continuous moving surface , 1967 .

[13]  F. Allan,et al.  On the analytic solutions of the nonhomogeneous Blasius problem , 2005 .

[14]  A. Acrivos,et al.  A moving-wall boundary layer with reverse flow , 1976, Journal of Fluid Mechanics.

[15]  J. Miller.  Numerical Analysis , 1966, Nature.

[16]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[17]  M. B. Friedman,et al.  An analytical solution of a nonlinear, singular boundary value problem in the theory of viscous fluids , 1968 .

[18]  L. Peletier,et al.  Reverse flow solutions of the Falkner-Skan equation for ? > 1 , 1972 .

[19]  J. Spurk.  Boundary Layer Theory , 2019, Fluid Mechanics.

[20]  I. Boyd,et al.  Blasius boundary layer solution with slip flow conditions , 2002 .

[21]  L. Quartapelle,et al.  Galerkin-Laguerre Spectral Solution of Self-Similar Boundary Layer Problems , 2012 .

[22]  A. Acrivos,et al.  A method for integrating the boundary-layer equations through a region of reverse flow , 1972, Journal of Fluid Mechanics.

[23]  Saeid Abbasbandy,et al.  A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method , 2007 .

[24]  Ram N. Mohapatra,et al.  On solutions of some singular, non-Linear differential equations arising in boundary layer theory , 1991 .

[25]  M. Gad-el-Hak.  The Fluid Mechanics of Microdevices—The Freeman Scholar Lecture , 1999 .

[26]  Abdul-Majid Wazwaz,et al.  The variational iteration method for solving two forms of Blasius equation on a half-infinite domain , 2007, Appl. Math. Comput..

[27]  B.-L. Kuo,et al.  Application of the differential transformation method to the solutions of Falkner-Skan wedge flow , 2003 .

[28]  John P. Boyd,et al.  The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems , 2008, SIAM Rev..

[29]  Mehdi Dehghan,et al.  Modified rational Legendre approach to laminar viscous flow over a semi-infinite flat plate , 2008 .

[30]  L. Talbot,et al.  The Theory of Laminar Boundary Layers in Compressible Fluids , 1964 .

[31]  Melvin Epstein,et al.  On the Solution of the Laminar Boundary-Layer Equations , 1959 .

[32]  Asai Asaithambi,et al.  Solution of the Falkner-Skan equation by recursive evaluation of Taylor coefficients , 2005 .

[33]  B. C. Sakiadis.  Boundary‐layer behavior on continuous solid surfaces: I. Boundary‐layer equations for two‐dimensional and axisymmetric flow , 1961 .

[34]  Riccardo Fazio,et al.  A survey on free boundary identification of the truncated boundary in numerical BVPs on infinite intervals , 2002 .

[35]  T. Y. Na,et al.  Computational methods in engineering boundary value problems , 1979 .

[36]  Riccardo Fazio.  A numerical test for the existeence and uniqeness of solution of free boundery problems , 1997 .

[37]  Similarity solutions of a boundary layer problem over moving surfaces , 1997 .

[38]  Riccardo Fazio,et al.  On the moving boundary formulation for parabolic problems on unbounded domains , 2007, Int. J. Comput. Math..

[39]  S. Goldstein,et al.  Concerning some Solutions of the Boundary Layer Equations in Hydrodynamics , 1930, Mathematical Proceedings of the Cambridge Philosophical Society.

[40]  M. Y. Hussaini,et al.  On similarity solutions of a boundary layer problem with an upstream moving wall , 1987 .

[41]  Riccardo Fazio,et al.  The Blasius problem formulated as a free boundary value problem , 1992 .

[42]  B. C. Sakiadis.  Boundary‐layer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface , 1961 .

[43]  Asai Asaithambi,et al.  A second-order finite-difference method for the Falkner-Skan equation , 2004, Appl. Math. Comput..

[44]  Hermann Weyl,et al.  On the differential equations of the simplest boundary-layer problems , 1942 .

[45]  L. Fox.  The Numerical Solution of Two-Point Boundary Problems in Ordinary Differential Equations , 1957 .

[46]  John P. Boyd,et al.  The Blasius Function in the Complex Plane , 1999, Exp. Math..

[47]  A. Yakhot,et al.  New Approach to Solution of the Falkner-Skan Equation , 2001 .

[48]  L. Collatz.  The numerical treatment of differential equations , 1961 .

[49]  Asai Asaithambi,et al.  A finite-difference method for the Falkner-Skan equation , 1998, Appl. Math. Comput..

[50]  K. Stewartson,et al.  Further solutions of the Falkner-Skan equation , 1954, Mathematical Proceedings of the Cambridge Philosophical Society.

[51]  Riccardo Fazio,et al.  Numerical transformation methods: Blasius problem and its variants , 2009, Appl. Math. Comput..

[52]  Riccardo Fazio.  Blasius problem and Falkner–Skan model: Töpfer’s algorithm and its extension , 2012, 1212.5057.

[53]  D. J. Evans,et al.  Similarity and numerical analysis for free boundary value problems , 1990 .

[54]  Riccardo Fazio,et al.  A Novel Approach to the Numerical Solution of Boundary Value Problems on Infinite Intervals , 1996 .

[55]  Riccardo Fazio,et al.  The iterative transformation method: numerical solution of one-dimensional parabolic moving boundary problems , 2001 .

[56]  Arthur Veldman,et al.  On a Generalized Falkner-Skan Equation, , 1980 .

[57]  H. Blasius.  Grenzschichten in Flüssigkeiten mit kleiner Reibung , 1907 .

[58]  V. M. Falkner.  L.A method of numerical solution of differential equations , 1936 .

[59]  Jiawei Zhang,et al.  An iterative method for solving the Falkner-Skan equation , 2009, Appl. Math. Comput..