Third-order nonlinear ODE with boundary condition

[rosecat]

I'm trying to solve a third-order nonlinear ordinary differential equation. I couldn't get the answer even using Mathematica.

The equation is:

u'''(t) + u/2 u''(t) = 0

with conditions u(0)=0, u'(0)=0, u(10)=1.

I need to get both analytic solution and numerical solution. For the numerical solution, I'm thinking the Newton iterative algorithm. But for the analytic solution, I really have no idea.

 

[jackmell]

I'm trying to solve a third-order nonlinear ordinary differential equation. I couldn't get the answer even using Mathematica.

The equation is:

u'''(t) + u/2 u''(t) = 0

with conditions u(0)=0, u'(0)=0, u(10)=1.

I need to get both analytic solution and numerical solution. For the numerical solution, I'm thinking the Newton iterative algorithm. But for the analytic solution, I really have no idea.

I mean if you're using Mathematica, just use NDSolve:

myguesses = {0.1, 0.15, 0.2}; 
sols = (First[NDSolve[{Derivative[3][u][t] + (u[t]/2)*Derivative[2][u][t] == 0, u[0] == 0, Derivative[1][u][0] == 0, 
        u[10] == 1}, u, t, Method -> {"Shooting", "StartingInitialConditions" -> {u[0] == 0, Derivative[1][u][0] == 0, 
           Derivative[2][u][0] == #1}}]] & ) /@ myguesses; 
Plot[Evaluate[u[t] /. sols], {t, 0, 10}, PlotStyle -> {Black, Blue, Green}]

tweek it as you see fit. As far as an analytic solution, in an act of utter desperation, I would resort to power series.

 

[rosecat]

I mean if you're using Mathematica, just use NDSolve:

myguesses = {0.1, 0.15, 0.2}; 
sols = (First[NDSolve[{Derivative[3][u][t] + (u[t]/2)*Derivative[2][u][t] == 0, u[0] == 0, Derivative[1][u][0] == 0, 
        u[10] == 1}, u, t, Method -> {"Shooting", "StartingInitialConditions" -> {u[0] == 0, Derivative[1][u][0] == 0, 
           Derivative[2][u][0] == #1}}]] & ) /@ myguesses; 
Plot[Evaluate[u[t] /. sols], {t, 0, 10}, PlotStyle -> {Black, Blue, Green}]

tweek it as you see fit. As far as an analytic solution, in an act of utter desperation, I would resort to power series.

Thank you jackmell!

For the numerical solution, I have to write out the algorithm and programming in MATLAB. No library could be used.

For the analytic solution, I tried DSolve but It didn't work. I am trying the power series.

Again, thanks a lot!