solve a system of ode by bvp4c

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Day Rosli
Day Rosli on 13 Jan 2016
Commented: Taylor Nichols on 24 Jan 2019
Hi, I am trying so solve a system of boundary value problem. I have tried looking into the examples given yet I still have problem in writing the code and running the program.
Briefly, I have 3 equations to solve. I have convert them to first order system in order to put it in matlab. Where, I let W=y1, W'=y2, U=y3, U'=y4, U''=y5, V=y6, V'=y7, V''=y8. I have the boundary 0f x from [0 infinity], and I choose infinity=20. I also need to plot graphs of x-axis:x (from 0 to 20) for y-axis:U, V and W respectively.
I know there are lots of mistakes here hence please correct me if Im wrong. I have attached the code here.
Thank you :))
  3 Comments
Day Rosli
Day Rosli on 13 Jan 2016
So this is the equation and boundary conditions:
2*U + x*(1-n/n+1)*U' + W' = 0
U^2 - (V+1)^2 + [W+(1-n/n+1)*x*U]*U' - {(U'^2+V'^2)^((n-1)/2)*U''+U'*[(n-1)*(U'U''+V'V'')*(U'^2+V'^2)^((n-3)/2)]}=0
2*U*(V+1) + [W+(1-n/n+1)*x*U]*V' - {(U'^2+V'^2)^((n-1)/2)*V''+V'*[(n-1)*(U'U''+V'V'')*(U'^2+V'^2)^((n-3)/2)]}=0
bc:
U(0)=V(0)=W(0)=0, U(infinity)=0, V(infinity)=-1.
So I transformed them into 1st order by letting
W=y1, W'=y2, U=y3, U'=y4, U''=y5, V=y6, V'=y7, V''=y8.
Thus, I have this equation.
y(2) = -2*y(3)-x*(1-n/n-1)*y(4);
y(4) = (-y(3)^2+(y(6)+1)^2+((y(4)^2+y(7)^2)^((n-1)/2)*y(5)+y(4)*((n-1)*(y(4)*y(5)+y(7)*y(8))*(y(4)^2+y(7)^2)^((n-3)/2))))/(y(1)+(1-n/n+1)*x*y(3))
y(7) = (-2*y(3)*(y(6)+1)+((y(4)^2+y(7)^2)^((n-1)/2)*y(8)+y(7)*((n-1)*(y(4)*y(5)+y(7)*y(8))*(y(4)^2+y(7)^2)^((n-3)/2))))/(y(1)+(1-n/n+1)*x*y(3))
bc: f(1)(0)=f(3)(0)=f(6)(0)=0, f(3)(infinity)=0, f(6)(infinity)=-1.
Taylor Nichols
Taylor Nichols on 24 Jan 2019
how did you arange the boundary conditions in your code if you don't mind me asking?
I've only done a ODE with one dependent variable in which I arranged my conditions this way
This is a 4th order equation btw.
bc = [yl(1) - hi;
yl(2);
yr(1) - ho;
yr(2)];

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Accepted Answer

Torsten
Torsten on 15 Jan 2016
Solve your equations for W', U'' and V''.
Let
y(1) = W, y(2) = U, Y(3) = U', Y(4) = V, Y(5) = V'.
Your system then reads
Y(1)' = F0(Y(1),Y(2),Y(3),Y(4),Y(5))
Y(2)' = Y(3)
Y(3)' = F1(Y(1),Y(2),Y(3),Y(4),Y(5))
Y(4)' = Y(5)
Y(5)' = F2(Y(1),Y(2),Y(3),Y(4),Y(5))
Now use one of the ODE solvers in the usual manner.
Best wishes
Torsten.
  3 Comments
Torsten
Torsten on 15 Jan 2016
Edited: Torsten on 15 Jan 2016
Order equations (2) and (3) such that they look like
a1*U'' + b1*V'' = c1
a2*U'' + b2*V'' = c2
with a1, b1, c1, a2, b2, c2 expressions only depending on U, U', V, V', W and W'.
Then solve this (linear) system of equations from above for U'' and V'':
U'' = (c1*a2-c2*a1)/(b1*a2-b2*a1)
V'' = -(c1*b2-c2*b1)/(b1*a2-b2*a1)
Best wishes
Torsten.
P.S.: It seems you forget parentheses around expressions, e.g. (1-n/n+1) should be (1-n/(n+1)), I guess.
Day Rosli
Day Rosli on 16 Jan 2016
Thank you again, for your help. Unfortunately, I am not very clear of some of the explanation above (Sorry, my bad). Okay, from what I understand so far, I have to write the equations so that it will look like this, right?
a1*U'' + b1*V'' = c1
a2*U'' + b2*V'' = c2
Okay, so this is what I got from rearranging equations for Y(3)' and Y(5)'
[((U'^2+V'2)^((n-1)/2)+U'(n-1)(U'^2+V'^2)^((n-3)/2)U']U''+[U'(n-1)(U'^2+V'^2)^((n-3)/2)V']V''=U^2-(V+1)^2+(W+x(1-n/(n+1))U)U'
[V'(n-1)(U'^2+V'^2)^((n-3)/2)U']U''+[((U'^2+V'2)^((n-1)/2)+V'(n-1)(U'^2+V'^2)^((n-3)/2)V']V''=-2U(V+1)+(W+x(1-n/(n+1))U)V'
in which
a1=((U'^2+V'2)^((n-1)/2)+U'(n-1)(U'^2+V'^2)^((n-3)/2)U'
b1=U'(n-1)(U'^2+V'^2)^((n-3)/2)V'
c1=U^2-(V+1)^2+(W+x(1-n/(n+1))U)U'
a2=V'(n-1)(U'^2+V'^2)^((n-3)/2)U'
b2=((U'^2+V'2)^((n-1)/2)+V'(n-1)(U'^2+V'^2)^((n-3)/2)V'
c2=-2U(V+1)+(W+x(1-n/(n+1))U)V'
so, what do you mean by
'Then solve this (linear) system of equations from above for U'' and V'':
U'' = (c1*a2-c2*a1)/(b1*a2-b2*a1)
V'' = -(c1*b2-c2*b1)/(b1*a2-b2*a1)' ?
Kindly need your help, thank you in advance.

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More Answers (1)

Torsten
Torsten on 18 Jan 2016
I mean that the function F1 from above is given by
F1(W,U,U',V,V') = (c1*a2-c2*a1)/(b1*a2-b2*a1)
and the function F2 from above by
F2(W,U,U',V,V') = -(c1*b2-c2*b1)/(b1*a2-b2*a1)
Now you can define your array "dydx" in function "ex11ode" by
dydx = [-2Y(2)+x(1-n/n+1)Y(3)
Y(3)
F1(Y(1),Y(2),Y(3),Y(4),Y(5))
Y(5)
F2(Y(1),Y(2),Y(3),Y(4),Y(5))]
Best wishes
Torsten.
  9 Comments
Torsten
Torsten on 25 Jan 2016
Something like
c=(sol.y(4,:).^2+sol.y(5,:).^2).^((1-n)/(n+1));
plot(sol.x,c);
after the call to bvp4c should work.
Best wishes
Torsten.
Day Rosli
Day Rosli on 25 Jan 2016
Thank you Torsten! It works well.

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