Solution of the Blasius and Sakiadis equation by generalized iterative differential quadrature method. (English) Zbl 1231.65119
Summary: The Blasius and Sakiadis equation was solved earlier with different numerical methods. In this study, it is solved by using the generalized iterative differential quadrature method (GIDQM). And more than one condition are imposed at the same point without using any higher-order polynomial or \(\delta \)-point approximation in GIDQM although it is one of the most important drawbacks in the differential quadrature method. The procedure is started with an initial guess value and true results are obtained by iterations. More grid points are used. Hence, the solution of the Blasius equation is calculated precisely and shows good agreements when compared with other works.
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |
Keywords:
Blasius equation; Sakiadis flow; generalized iterative differential quadrature method; boundary value problem; numerical examples; boundary layer flowReferences:
| [1] | Bellman, Differential quadrature and long term integration, Journal of Mathematical Analysis and Applications 34 pp 235– (1971) · Zbl 0236.65020 · doi:10.1016/0022-247X(71)90110-7 |
| [2] | Bellman, Differential quadrature: a technique for the rapid solution of non-linear partial differential equations, Journal of Computational Physics 10 pp 40– (1972) · Zbl 0247.65061 · doi:10.1016/0021-9991(72)90089-7 |
| [3] | Bellomo, Nonlinear models and problems in applied sciences from differential quadrature to generalized collocation methods, Mathematical and Computer Modelling 26 pp 13– (1997) · Zbl 0898.65074 · doi:10.1016/S0895-7177(97)00142-8 |
| [4] | Jang, Application of differential quadrature to static analysis of structural components, International Journal for Numerical Methods in Engineering 28 pp 561– (1989) · Zbl 0669.73064 · doi:10.1002/nme.1620280306 |
| [5] | Malik, Implementing multiple boundary conditions in the DQ solution of higher-order PDE’s application to free vibration of plates, International Journal for Numerical Methods in Engineering 39 pp 1237– (1996) · Zbl 0865.73079 · doi:10.1002/(SICI)1097-0207(19960415)39:7<1237::AID-NME904>3.0.CO;2-2 |
| [6] | Wu, A differential quadrature as a numerical method to solve differential equations, Computational Mechanics 24 pp 197– (1999) · Zbl 0976.74557 · doi:10.1007/s004660050452 |
| [7] | Liu, Numerical solution for differential equations of Duffing-type non-linearity using the generalized differential quadrature rule, Journal of Sound and Vibration 237 (5) pp 805– (2000) · Zbl 1237.65073 · doi:10.1006/jsvi.2000.3050 |
| [8] | Liu, Application of generalized differential quadrature rule in Blasius and Onsager equations, International Journal for Numerical Methods in Engineering 52 pp 1013– (2001) · Zbl 0996.76072 · doi:10.1002/nme.251 |
| [9] | Girgin, Combining differential quadrature method with simulation technique to solve nonlinear differential Equations, International Journal for Numerical Methods in Engineering 75 pp 722– (2008) · Zbl 1195.70005 · doi:10.1002/nme.2282 |
| [10] | Girgin, Combining modified integral quadrature method with simulation technique to solve nonlinear initial and boundary value problems, International Journal of Nonlinear Sciences and Numerical Simulation 10 (4) pp 475– (2009) · doi:10.1515/IJNSNS.2009.10.4.475 |
| [11] | Quan, New insights in solving distributed system equations by the quadrature method-I: analysis, Computer and Chemical Engineering 13 pp 779– (1989) · doi:10.1016/0098-1354(89)85051-3 |
| [12] | Shu, Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids 15 pp 791– (1992) · Zbl 0762.76085 · doi:10.1002/fld.1650150704 |
| [13] | Howarth, On the solution of the laminar boundary layer equations, Proceeding of the Royal Society of London A164 pp 547– (1938) · JFM 64.1452.01 · doi:10.1098/rspa.1938.0037 |
| [14] | Arikoglu, Inner-outer matching solution of Blasius equation by DTM, Aircraft Engineering and Aerospace Technology: An International Journal 77 (4) pp 298– (2005) · doi:10.1108/00022660510606367 |
| [15] | Yu, The solution of the Blasius equation by the differential transform method, Mathematical Computer Modelling 28 pp 101– (1998) · Zbl 1076.34501 · doi:10.1016/S0895-7177(98)00085-5 |
| [16] | Cortell, Numerical solutions of the classical Blasius flat-plate problem, Applied Mathematics and Computation 170 pp 706– (2005) · Zbl 1077.76023 · doi:10.1016/j.amc.2004.12.037 |
| [17] | Abbasbandy, A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method, Chaos, Solitons and Fractals 31 pp 257– (2007) · doi:10.1016/j.chaos.2005.10.071 |
| [18] | Ahmad, An approximate analytic solution of the Blasius problem, Communications in Nonlinear Science and Numerical Simulation 14 (4) pp 1021– (2008) · Zbl 1221.76135 · doi:10.1016/j.cnsns.2007.12.010 |
| [19] | Hashim, Comments on ’A new algorithm for solving classical Blasius equation’ by L. Wang, Applied Mathematics and Computation 176 pp 700– (2006) · Zbl 1331.65103 · doi:10.1016/j.amc.2005.10.016 |
| [20] | Fang, A note on the extended Blasius equation, Applied Mathematics Letters 19 pp 613– (2006) · Zbl 1126.34301 · doi:10.1016/j.aml.2005.08.010 |
| [21] | Andersson, Sakiadis flow with variable fluid properties revisited, International Journal of Engineering Science 45 pp 554– (2007) · Zbl 1213.76062 · doi:10.1016/j.ijengsci.2007.04.012 |
| [22] | Sakiadis, Boundary-layer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface, AIChE Journal 7 pp 221– (1961) · doi:10.1002/aic.690070211 |
| [23] | Tsou, Flow and heat transfer in the boundary layer on a continuous moving surface, International Journal of Heat and Mass Transfer 10 pp 219– (1967) · doi:10.1016/0017-9310(67)90100-7 |
| [24] | Takhar, Boundary layer flow due to a moving plate: variable fluid properties, Acta Mechanica 90 pp 37– (1991) · Zbl 0753.76149 · doi:10.1007/BF01177397 |
| [25] | Pop, The effect of variable viscosity on the flow and heat transfer to a continuous moving flat plate, International Journal of Engineering Science 30 pp 1– (1992) · doi:10.1016/0020-7225(92)90115-W |
| [26] | Pantokratoras, Further results on the variable viscosity on flow and heat transfer to a continuous moving flat plate, International Journal of Engineering Science 42 pp 1891– (2004) · Zbl 1211.76043 · doi:10.1016/j.ijengsci.2004.07.005 |
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