Similar Equations to the Blasius Equation

[xingxian]

Homework Statement

Hi guys. I am actually not looking for solutions. I am now working on a numerator integrator for the Blasius Equation:

y''' +y''*y =0

where the boundary condition is y(0)=y'(0)=y''(0)-1=0

I have got the numerical integrator up and running, and obtain a solution which tends to a linear function as x tends to infinity.

However, I was told by my lecturer that I should test my integrator on another problem similar to Blasius, and we must know the exact analytic solution to that question to test how the error behaves with my integrator.

So may I know if you know any good model problem for this purpose?

I have found a few but non of them are really very suitable.

f'''+(1-f'^2)=0

or

y'''+2y''y-3y'^2=0

Don't think these are good. Any suggestion?

Really appreciate it!

 

[jackmell]

Homework Statement

Hi guys. I am actually not looking for solutions. I am now working on a numerator integrator for the Blasius Equation:

y''' +y''*y =0

where the boundary condition is y(0)=y'(0)=y''(0)-1=0

I have got the numerical integrator up and running, and obtain a solution which tends to a linear function as x tends to infinity.

However, I was told by my lecturer that I should test my integrator on another problem similar to Blasius, and we must know the exact analytic solution to that question to test how the error behaves with my integrator.

So may I know if you know any good model problem for this purpose?

I have found a few but non of them are really very suitable.

f'''+(1-f'^2)=0

or

y'''+2y''y-3y'^2=0

Don't think these are good. Any suggestion?

Really appreciate it!

Fire up Mathematica and start running the differential equation solver first starting with your equation which Mathematica can's solve, then start modifying it until Mathematica comes up with a solution:

DSolve[y'''[t]+y''[t] y[t]==0,y,t]

no

DSolve[y'''[t]+y'[t]^2==0,y,t]

no

DSolve[y'''[t]+y[t] y'[t]==0, y,t]

. . . close enough for me. Also, looks like you wrote it as an IVP when I think it's a boundary-value problem and I'm not sure, but I don't think DSolve can solve these. Check. BVPs are usually solved numerically in Mathematica.

 

[xingxian]

Hi Jackmell!

Thanks so much for your reply!

Actually we were given and IVP. but then I realized that it can be a bvp but that will only affect the solution curve by a constant factor, so didnt bother too mcuh.

I know actually I can just fire up many program to do dsolve. (I am using Maple by the way). But the thing is that my lecturer specifically asked us to confirm the correctness of the ODe integrator whcih we wrote on model problems where the analytic answer is known. and the analytic model problems haev to be clsoe to the blasius equation to demonstrate the suitability of the algorithms.

f'''+(1-f'^2)=0

or

y'''+2y''y-3y'^2=0

the reason why I chose these two are because they are both 3rd order nonlinear ODE.

Really appreciate it if you or anyone has any idea on what model problems to choose. thanks a lot!

 

[Dick]

How about y=3/(x+1)? It solves y'''+y*y''=0, though you'll have to change the boundary conditions, of course.