# What is wrong in my code for shooting method to solve 4 ODEs?

Posted 8 months ago
409 Views
|
0 Replies
|
0 Total Likes
|
 The set of ODEs that I need to solve $g=f''\frac{g^{2}+\lambda\gamma^{2}}{g^{2}+\gamma^{2}}$ $g'=\frac{1}{3}f'^{2}-\frac{2}{3}ff''+Mnf'$ $(1+Rd)\theta''+\frac{2}{3}Prf\theta'+N_{b}\theta'\phi'+N_{t}\theta'^{2}=0$ $\phi''+\frac{2}{3}Lef\phi'+\frac{N_{t}}{N_{b}}\theta''=0$And the boundary conditions are: $f=0,\; f'=1,\; \theta=1,\; \phi=1\; at\; \eta=0,$ $f'\rightarrow0,\; \theta\rightarrow0,\; \phi\rightarrow0\; as\; \eta\rightarrow\infty$I need to employ shooting method i.e guess the missing initial condition till the boundary conditions are satisfied.And obtain the solutions and graphs of $f'(\eta), g(\eta), \theta(\eta), \phi(\eta)$I have used NDSolve to carry out the numerical analysis, the following is the code: ODEs[\[CapitalOmega]1_, \[CapitalOmega]2_, \[CapitalOmega]3_, \ \[Lambda]_, \[Gamma]_, Mn_, Rd_, Lew_, Nb_, Nt_, Pr_] := {f''[\[Eta]] == g[\[Eta]]*(g[\[Eta]]^2 + \[Gamma]^2)/(g[\[Eta]]^2 + \[Lambda]*\ \[Gamma]^2), g'[\[Eta]] == (1/3)*(f'[\[Eta]])^2 - (2/3)*f[\[Eta]]*f''[\[Eta]] + Mn*f'[\[Eta]], \[Theta]''[\[Eta]] == -(1/1 + Rd)*(2/3)*Pr* f[\[Eta]]*\[CapitalTheta]'[\[Eta]] - (Nb/1 + Rd)*\[CapitalTheta]'[\[Eta]]*\[Phi]'[\[Eta]] - (Nt/1 + Rd)*(\[CapitalTheta]'[\[Eta]])^2, \[Phi]''[\[Eta]] == -(2/3)*Lew* f[\[Eta]]*\[Phi]'[\[Eta]] - (Nt/Nb)*\[CapitalTheta]''[\[Eta]], f[0] == 0, f'[0] == 1, \[Theta][0] == 1, \[Phi][0] == 1, g[0] == \[CapitalOmega]1, \[CapitalTheta]'[ 0] == \[CapitalOmega]2, \[Phi]'[0] = \[CapitalOmega]3} Soln[\[CapitalOmega]1_, \[CapitalOmega]2_, \[CapitalOmega]3_, \ \[Lambda]_, \[Gamma]_, Mn_, Rd_, Lew_, Nb_, Nt_, Pr_] := NDSolve[ODEs[\[CapitalOmega]1, \[CapitalOmega]2, \[CapitalOmega]3, \ \[Lambda], \[Gamma], Mn, Rd, Lew, Nb, Nt, Pr], {f, g, \[CapitalTheta], \[Phi]}, {\[Eta], 0, 10}] EndCondition[\[CapitalOmega]1_?NumericQ, \[CapitalOmega]2_? NumericQ, \[CapitalOmega]3_?NumericQ, \[Lambda]_? NumericQ, \[Gamma]_?NumericQ, Mn_?NumericQ, Rd_?NumericQ, Lew_?NumericQ, Nb_?NumericQ, Nt_?NumericQ, Pr_?NumericQ] := {(First[f[\[Eta]] /. Soln[\[CapitalOmega]1, \[CapitalOmega]2, \[CapitalOmega]3, \ \[Lambda], \[Gamma], Mn, Rd, Lew, Nb, Nt, Pr]] /. \[Eta] -> 10), (First[ f'[\[Eta]] /. Soln[\[CapitalOmega]1, \[CapitalOmega]2, \[CapitalOmega]3, \ \[Lambda], \[Gamma], Mn, Rd, Lew, Nb, Nt, Pr]] /. \[Eta] -> 10), (First[g[\[Eta]] /. Soln[\[CapitalOmega]1, \[CapitalOmega]2, \[CapitalOmega]3, \ \[Lambda], \[Gamma], Mn, Rd, Lew, Nb, Nt, Pr]] /. \[Eta] -> 10), (First[\[CapitalTheta][\[Eta]] /. Soln[\[CapitalOmega]1, \[CapitalOmega]2, \[CapitalOmega]3, \ \[Lambda], \[Gamma], Mn, Rd, Lew, Nb, Nt, Pr]] /. \[Eta] -> 10), (First[\[Phi][\[Eta]] /. Soln[\[CapitalOmega]1, \[CapitalOmega]2, \[CapitalOmega]3, \ \[Lambda], \[Gamma], Mn, Rd, Lew, Nb, Nt, Pr]] /. \[Eta] -> 10)} EndCondition[0, 0, 0, 0.5, 1, 1, 1, 1, 1, 1, 5] But it is unable to give a result so that I can proceed further and use 'FindRoot' to modify the initial guessI have attached a pdf, from where I have taken the code, it gives an example similar to mine. Attachments: