Numerical Solution Of Non Linear Di Erential Equation By .
Jrnf Mathemaal otiVolume 4, Issue 1{A (2016), 93{100.nte IISSN: 2347-1557d its ApplicrnationAnaloucsInternational Journal of Mathematics And its Applicationsoni57atAvailable Online: http://ijmaa.in/ISSN: 2347-1s 5International Journal of Mathematics And its ApplicationsNumerical Solution of Non Linear Di erential Equationby Using Shooting TechniquesResearch ArticleDr.Summiya Parveen1 1 Department of Mathematics, College of Engineering Roorkee, India.Abstract:Many problems that occur in physics and engineering can be modelled by linear or nonlinear di erential equations. In thispaper we nd the solution of Blasius type equations which are nonlinear ordinary di erential equations on a semi-in niteinterval. The Blasius equation is a third-order non-linear ordinary di erential equation. The non-linear mathematicalmodel of the problem prohibits the use of the analytical methods. A numerical solution is the single approach for theseproblems. The two-point boundary problem was solved by a Runge-Kutta method and shooting method. Matlab functionsmake numerical solution of the mathematical models of the uid ow relatively simple and quick solutions are presentedfor Blasius equations with additional computations based on the numerical results obtained by the Matlab function.Numerical study on boundary layer equation due to stationary at plate, Matlab is the mathematical programming thatused to solve the boundary layer equation applied toolbox method. The numerical results show a good agreement withthe exact solution of Blasius equation and consistent with prior published result. The accuracy of the proposed methodis higher than other approximation analytical solutions; hence suggest that proposed method is e cient and practical.Keywords: Boundary layer, Blasius ow, Newton method, RungeKutta method, Shooting Technique.c JS Publication.1.Introduction1.1.Blasius EquationIf a uid ows past a solid, a uid layer is formed adjacent to the boundary of the solid. This layer is called a boundary layerand strong viscous e ects exist within this layer. Consider a uniform ow over a at surface, y 0, x 0, z Equations of the ow in the boundary layer are the continuity equation@u@v 0@x@y(1)@u@v@2u v @x@y@y 2(2)and the reduced Navier-Stokes equationuwhere u and v are respectively the components of the velocity vector and v represents the viscosity of the uid. Boundaryconditions areu(x; 0) 0x 0(3a)v(x; 0) 0x 0(3b)u(x; y) U as y (3c)E-mail: summiyaparveen82@gmail.com93
Numerical Solution of Non Linear Di erential Equation by Using Shooting Techniqueswhere U is the constant speed of the ow outside the boundary Layer. De ne a stream functionu @@; v @y@x(x; y) such that(4)then equation (1) is satis ed identically and equation (2) becomes@ @2@ @2@3 v 32@y @x@y@x @y@y(5)Blasius used a similarity transformation to reduce (1.5) to an ordinary di erential equation. A similarity transformation isbased on the symmetry analysis of a di erential equation [11? ]. When a symmetry property of a di erential equation isidenti ed it can be exploited to achieve a simpli cation. If it is an ordinary di erential equation then usually the order ofthe equation can be reduced. If it is a partial di erential equation then usually the dependent and independent variablescan be combined to achieve a reduction of order or a reduction of the partial di erential equation to an ordinary di erentialequation. In the case of (5) symmetry analysis leads to the following transformation [11]y a x (x; y) b x f ( )where a and b are constants and are chosen to makeand f ( ) dimensionless. They are taken asra b With this choice, UUis called the dimensionless similarity variable and f ( ) is called the dimensionless stream function. Nowf( )@Uy 01 f ( ) U @x2 x2x@0 Uf ( )@y@2a U f 00 ( ) @y 2xr2U U y 00U@ f 00 ( )3 f ( ) @x@y22xx2U 2 000@3 f ( ):@y 3xA substitution of the above derivatives in equation (5) reduces it tod3 f1d2 f f( ) 2 03d2d(6)Equation (6) is known as the Blasius equation. The boundary condition (3a) transforms tof 0 (0) 0(7a)f (0) 0(7b)while (3b) becomes94
Dr.Summiya Parveenand (1.3c) reduces tof 0( ) 1as(7c) Equation (6) together with the boundary conditions (7a), (7b) and (7c) is called the Blasius problem. Several methods havebeen used for numerical solution of Falkner-Skan equations. Meksyn [3] solved the Falkner-Skan equation through analyticalapproximations. Asaithambi [4, 5] usednite di erence method and piecewise linear functions for solving Falkner-Skanequation switch high accuracy. Recently Shi-JunLiao [6] applied the homotopy analysis to solve the Falkner-Skan equation.Khabibrakhmanov and D.Summers [7] used a spectral method with generalized Laguerre polynomials for solving the Blasiusequation ( 0). Moreover, the Blasius equation was solved by Rosales and Valencia [8] using Fourier series. Also, Veraand Valencia [9] solved the Falkner-Skan equation with heat transfer through an expansion in Fourier series. In this paper,write the Blasius equation as a rst order di erential system and obtain a numerical solution to the di erential using 4thorder Runge- Kutta method by using a guess1.2.and nd out the solution.Method of SolutionThe non-linear di erential equations (1) subject to the boundary conditions (2) constitute a two-point boundary valueproblem. In order to solve these equations numerically, we follow RungeKutta 4th order with shooting technique. In thismethod it is most important to choose the appropriate nite values oflarge value of . The solution process is repeated with another until two successive values of f 00 (0) di er only after a desired digit signifying the limit of the boundaryalong . The last value of is chosen as appropriate value of the limit for that particular set of parameters.The ordinary di erential equation (1) was rst converted into a set of three rst-order simultaneous equations. To solvethis system we require three initial conditions but we have only two initial conditions, f (0) and f 0 (0) on f ( ). The initialcondition f 00 (0) is not prescribed. However the values of f 0 ( ) is known at 0. Now we employ the numerical shootingtechnique based where this ending boundary condition is utilized to produce unknown initial conditions at 0 nally, theproblem has been solved numerically using Runge-Kutta 4th order.2.2.1.Description of the MethodReduction to a First Order SystemThe solve the falkner-skan equation numerically, the equation is reduced to a rst order system by introducing the threeauxiliary variables.f u1@f@2f u2 and u3 ;@@ 2So that we have the following system of three coupled ODEs:f1 ( ; u1 ; u2 ; u3 ) u01 u2f2 ( ; u1 ; u2 ; u3 ) u02 u2f3 ( ; u1 ; u2 ; u3 ) u03 u1 u3The rst order system can be written more compactly using vector notation.23 2f6 1 7 6 u267 6@f7 6 f1 ( ; u1 ; u2 ; u3 ): i:e 66 f 2 7 6 u3@45 4f3 u1 u337777595
Numerical Solution of Non Linear Di erential Equation by Using Shooting Techniquesit is important to note the ODE system is in normal form and then the boundary conditionu1 (0) 0u2 (0) 0u2 ( ) 1Where is the unknown free boundary used to truncate the semi-in nite interval to anite one. Which is to bedetermine as the part of the procedure in addition, an initial condition on the second derivatives is introduced to apply theShooting Method,@2f @ 2whereat 0;is the shooting angle. The shooting algorithm therefore consists of the following procedure:(1). Starting from a relatively large value ofzero toas the initial guess f 0 ( ) is evaluated by increasingthrough steps of h fromm.(2). If at some , f 0 ( ) 1, thenis decreased and f 0 ( ) evaluated until f 0 ( ) 1 for some . At this point, the asymptoticpro le is bracketed.(3). A newis then determined by Newton Method.(4). If f 0 ( ) does not cross unity from below asincreases from zero tom,below its correct value and Newton Method again determines the next(5). Finally, when the estimate forthen is checked for negativity. If negative,is.is approximately within an order of magnitude of the desired error, Runge-Kutta 4thorder method can be used to the initial value problems.2.2.Numerical SolutionA numerical solution of the Blasius problem usually uses the shooting method. In this method it is assumed thatf 00 (0) (8)and the problem is solved with di erent values of . Such values ofthat value ofwill lead to di erent values ofdfdas . We seekwhich will yield an f which satis eslim df 1:dFirst accurate numerical solution was obtained by Howarth [2]. More recently Asaithambi [4], and Cortell [10] have alsosolved the Blasius problem by the shooting method. In practice it is impossible to carry out calculations up to in nity.Hence an is arbitrarily xed and we demand thatdf 1 whend :We solve the Blasius equation by the shooting method. We start withare practically a constant 0.449287. With96 12 and 13 0:6. We nd the slope for 12 13 equal to 1.18352. This means the0:1 0:62 0:35 and so on. We present the resultsactual value should lie between 0.1 and 0.6. Therefore our next choice isin Table 1 and 2. 0:1 and nd that f 0 ( ) between
Dr.Summiya Parveenf 0 5620.9889370.3304680.9968080.3324218 1.000730.3314453 0.9987710.3319335 0.9997510.3321771 1.000240.3320553 0.9999960.3321162 1.000120.3320857 1.000060.3320705 1.000030.3320629 1.000010.3320591 1Table 1.Sequence of values ofconverging tof( )0.0 0f 0( )f 00 ( )00.3320590.2 0.00664105 0.0664081 0.3319850.4 0.026560.1327650.3314680.6 0.0597350.1989380.3300810.8 0.1061090.2647110.3273911.0 0.1655730.3297820.3230091.2 0.237950.3937780.316591.4 0.3229830.4562640.3078671.6 0.4203230.5167590.2966651.8 0.5295210.5747610.2829322.0 0.6500280.6297690.2667532.2 0.7811970.6813140.2483522.4 0.9222950.7289850.2280922.6 1.072510.7724590.2064552.8 1.230980.8115130.1840073.0 1.396820.8460480.1613593.2 1.56910.8760850.1391293.4 1.746960.9017650.1178763.6 1.929530.9233340.09808673.8 2.116040.9411220.08012584.0 2.305760.9555220.06423414.2 2.498050.9669610.05051934.4 2.692370.9758750.03897314.6 2.888260.9826870.02948294.8 3.085330.9877930.02187115.0 3.283290.9915460.01590685.2 3.481880.9942490.01134145.4 3.680940.9961590.007927865.6 3.880310.9974810.005431695.8 4.07990.9983790.003648356.0 4.279640.9989760.002401986.2 4.479480.9993660.001550226.4 4.679380.9996150.00098041997
Numerical Solution of Non Linear Di erential Equation by Using Shooting Techniquesf( )f 0( )f 00 ( )6.6 4.87932 0.999711 0.0006081916.8 5.07928 0.999867 0.0003695997.0 5.27926 0.999925 0.0002202077.2 5.47925 0.999959 0.0001284317.4 5.67924 0.999979 0.00007371767.6 5.87924 0.999990.00004134417.8 6.07924 0.999996 0.0000227099Table 2.8.0 6.27924 10.00001225038.2 6.47924 16.4618 10 68.4 6.67924 13.28575 10 6Numerical values of f ( ), f 0 ( ) and f 00 ( )Approximation solution Numerical solution0Table 3.3.000.4 0.02660.026560.8 0.10610.1061091.2 0.23790.237952.0 0.65000.6500282.8 1.23111.230983.0 1.3961.396824.0 2.30582.305765.0 3.28273.28328.0 6.27936.2792Comparisons the values of f ( ) for di erent authorsConclusionIn this study, we have considered the classical Blasius problem. This nonlinear di erential equation is successfully solved byemploying Runge-Kutta method with shooting method to obtain numerical solutions. It is found that present results arein good agreement compared to exact solutions by Blasius [1] for the values f ( ) and f 0 ( ) as well as the values of f 00 ( )in comparison with Howarth [2] as shown in Table 1-3 and Figure 1-3. The numerical results strongly display the e ciencyand accuracy of the proposed method in solving the nonlinear equation.98
Dr.Summiya ParveenReferences[1] H.Blasius, Z.U. Math. Phys., 56(1908), 137.[2] L.Howarth, On the solution of the laminar boundary layer equations, Proc. Roy. Soc. of London, 164(1938), 547-579.[3] D.Meksyn, New methods in laminar boundary-layer theory, Pergamon Press, (1961).[4] N.S.Asaithambi, A nu merical method for the solution of the Falkner-Skan equation, Appl. Math. Comput., 81(1997),259-264.[5] A.Asaithambi, A finite-difference method for the solution of the Falkner-Skan equation, Appl. Math. Comput., 92(1998),135-141.[6] Shi-Jun Liao, A uniformly valid analytic solution of two-dimensional viscous flow over a semi -infinite flat plate, J.Fluid Mech., 385(1999), 101-128.[7] I.K.Khabibrakhmanov and D.Summers, The use of generalized Laguerre polynomials in spectral methods for nonlineardifferential equations, Comput.Math. Appl., 36(2)(1998), 65-70.[8] M.Rosales and A.Valencia, A note on solution of blasius equation by fourier series, Adv. Appl. Fluid Mech., 6(1)(2009),33-38.[9] M.Rosales-Vera and A.Valencia, Solutions of Falkner-Skan equation with heat transfer by Fourier series, InternationalCommunications in Heat and Mass Transfer, 37(2010), 761-765.99
Numerical Solution of Non Linear Di erential Equation by Using Shooting Techniques[10] R.Cortell, Effects of heat source/sink, radiation and work doneby deformation on flow and heat transfer of a viscoelasticfluid over a stretching sheet, Comput.Math. Appl., 53(2007), 305-316.[11] T.Cebeci and H.Keller, Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equations,J. Comp. Physics, 7(1971), 289-300.100
write the Blasius equation as a rst order di erential system and obtain a numerical solution to the di erential using 4th order Runge- Kutta method by using a guess and nd out the solution. 1.2. Method of Solution The non-linear di erential equations (1) subject to the boundary conditions (2) constitute a two-point boundary value problem.
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