XU Ding, XIE Gong-nan. Application of the Fixed Point Method to Solve the Nonlinear Falkner-Skan Flow Equation[J]. Applied Mathematics and Mechanics, 2015, 36(1): 78-86. doi: 10.3879/j.issn.1000-0887.2015.01.007
 Citation: XU Ding, XIE Gong-nan. Application of the Fixed Point Method to Solve the Nonlinear Falkner-Skan Flow Equation[J]. Applied Mathematics and Mechanics, 2015, 36(1): 78-86.

# Application of the Fixed Point Method to Solve the Nonlinear Falkner-Skan Flow Equation

##### doi: 10.3879/j.issn.1000-0887.2015.01.007
Funds:  The National Natural Science Foundation of China(11102150)
• Received Date: 2014-07-16
• Publish Date: 2015-01-15
• The Falkner-Skan flow equation is a strongly nonlinear differential equation, which describes the flow around a wedge. In order to overcome the difficulties originated from the semi-infinite interval and asymptotic boundary condition in this flow problem, transformations were simultaneously conducted for both the independent variable and the correponding function to convert the problem to a 2-point boundary value one within a finite interval. The deduced new-form nonlinear differential equation was subsequently solved with the fixed point method (FPM). The present analytical results obtained with the FPM agree well with the previous referential numerical ones. The accuracy of the present solution is conveniently improved through iteration under the FPM framework, which shows that the FPM makes a promising tool for nonlinear differential equations.
•  [1] Wang C Y. Exact solutions of the unsteady Navier-Stokes equations[J].Applied Mechanics Reviews,1989,42(11S): S269-S282. [2] Wang C Y. Exact solutions of the steady-state Navier-Stokes equations[J].Annual Review of Fluid Mechanics,1991,23: 159-177. [3] Falkner V M, Skan S W. Some approximate solutions of the boundary layer equations[J].Philosophical Magazine,1931,12: 865-896. [4] White F M.Viscous Fluid Flow [M]. New York: McGraw-Hill, 1991. [5] Schlichting H, Gersten K.Boundary-Layer Theory [M]. Springer Verlag, 2000. [6] Blasius H. Grenzschichten in flüssigkeiten mit kleiner reibung[J].Z Math Phys,1908,56: 1-37. [7] Hiemenz K. Die grenzchicht an einem in den gleichformingen flussigkeitsstrom eingetauchten geraden kreiszylinder[J].Dinglers Polytech J,1911,326: 321-410. [8] Hartree D R. On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer[J].Mathematical Proceedings of the Cambridge Philosophical Society,1937,33(2): 223-239. [9] Asaithambi A. A finite-difference method for the Falkner-Skan equation[J].Applied Mathematics and Computation,1998,92(2/3): 135-141. [10] Fazio R. Blasius problem and Falkner-Skan model: T-pfer’s algorithm and its extension[J].Computers & Fluids,2013,〖STHZ〗73: 202-209. [11] Boyd J P. The Blasius function: computations before computers, the value of tricks, undergraduate projects, and open research problems[J].SIAM Review,2008,50(4): 791-804. [12] Boyd J P. The Blasius function in the complex plane[J].Experimental Mathematics,1999,8(4): 381-394. [13] LIAO Shi-jun. A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate[J].Journal of Fluid Mechanics,1999,385: 101-128. [14] Motsa S S, Sibanda P. An efficient numerical method for solving Falkner-Skan boundary layer flows[J].International Journal for Numerical Methods in Fluids,2012,69(2): 499-508. [15] Marinca V, Ene R D, Marinca B. Analytic approximate solution for Falkner-Skan equation[J].The Scientific World Journal,2014,2014: 617453. [16] Yun B I. New approximate analytical solutions of the Falkner-Skan equation[J].Journal of Applied Mathematics,2012,2012: 170802. [17] XU Ding, XU Jing-lei, XIE Gong-nan. Revisiting Blasius flow by fixed point method[J].Abstract and Applied Analysis,2014,2014: 953151. [18] Zeidler E.Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems[M]. Springer, 1986. [19] XU Ding, GUO Xin. Fixed point analytical method for nonlinear differential equations[J].Journal of Computational and Nonlinear Dynamics,2013,8(1): 011005. [20] XU Ding, GUO Xin. Application of fixed point method to obtain semi-analytical solution to Blasius flow and its variation[J].Applied Mathematics and Computation,2013,224: 791-802. [21] XU Ding, WANG Xian, XIE Gong-nan. Spectral fixed point method for nonlinear oscillation equation with periodic solution[J].Mathematical Problems in Engineering,2013,2013: 538716. [22] 郭欣, 王娴, 许丁, 谢公南. 混合层无粘稳定性分析的Legendre级数解[J]. 应用数学和力学, 2013,34(8): 782-794.(GUO Xin, WANG Xian, XU Ding, XIE Gong-nan. A Legendre series solution to Rayleigh stability equation of mixing layer[J].Applied Mathematics and Mechanics,2013,34(8): 782-794.(in Chinese)) [23] 赵学端, 廖其奠. 粘性流体力学[M]. 北京: 机械工业出版社,1992.(ZHAO Xue-duan, LIAO Qi-dian.Visocus Fluid Flow [M]. Beijing: China Machine Press, 1992.(in Chinese))

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## Article Metrics

Article views (946) PDF downloads(1125) Cited by()

/