Trying to solve a system of 2 ordinary differential equations with boundary conditions

Hi there, I am trying to solve a system of ordinary differential equations with boundary conditions. These equations are part of my thesis and govern thick cylinders under internal pressure. I have checked my equations with the general theroy as suggested by Gabriel Lame for thick walled cylinders. The codes simplify as follows:

A,B,C and D are the functions of E and poisson's ratio.

the boundary conditions are t=1, y=0 and at t=R, y=-P. (As in the case of cylinder with internal pressure only)

I have tried to solve it in matlab but get an [ empty sym ] in return. I have limited experience with matlab so would appreciate any help possible.

My code:

syms E nu a11= (1/E); a12= (-nu/E); a33= (1/E); A= (a12)/(a11+a12); B= ((a33)-((2*a12^2)/(a11+a12))); C= (a11)/(a11^2-a12^2); D= (2*a12+a11)/(a11+a12);

syms x(t) y(t) eqns = [diff(x,t)==A*(x/t)+B*y, diff(y,t)==-C*(x/t^2)-D*y]; cond = [y(0) == 0, y(2)==1]; withSimplifications = dsolve(eqns, cond)

Answer 1

[Ketav Majumdar]

Correction this is my code so far

if true
  %syms E nu
a11= (1/E);
a12= (-nu/E);
a33= (1/E);
A= (a12)/(a11+a12);
B= ((a33)-((2*a12^2)/(a11+a12)));
C= (a11)/(a11^2-a12^2);
D= (2*a12+a11)/(a11+a12);

syms x(t) y(t) eqns = [diff(x,t)==A*(x/t)+B*y, diff(y,t)==-C*(x/t^2)-D*y]; cond = [y(0) == 0, y(2)==1]; withSimplifications = dsolve(eqns, cond)

end

Answer 2

[Torsten]

I doubt that your system can be solved analytically using dsolve.

Try a numerical solution with bvp4c.

If you start at t=0, problems will arise because t is in the denominator in both equations.

Furthermore, are you sure that you don't have to prescribe any boundary condition for x ?

Best wishes

Torsten.

[Ketav Majumdar]

well I dont have any boundary conditions for x as thats the radial displacement of the cylinder. Ill give numerical values a shot

Answer 3

[Ketav Majumdar]

So if managed to get something.but i get

 struct with fields:

    y: [1×1 sym]
    x: [1×1 sym]

This is my code so far:

 code:
 %Mechanical Properties of Material
E=200e9; 
nu=0.3;

%Constants A,B,C,D in the Equations
syms E nu
a11= (1/E);
a12= (-nu/E);
a33= (1/E);
A= (a12)/(a11+a12);
B= ((a33)-((2*a12^2)/(a11+a12)));
C= (a11)/(a11^2-a12^2);
D= (2*a12+a11)/(a11+a12);

 %Defining the System of 2ODES
syms x(t) y(t)
t=1;
eqns = [diff(x,t)==A*(x/t)+B*y, diff(y,t)==-C*(x/t^2)-D*y];
cond = [y(0) == 0, y(2)==1];
withSimplifications = dsolve(eqns, cond)
withoutSimplifications = dsolve(eqns, cond, 'IgnoreAnalyticConstraints', false)

end

how do I get the numerical values?

[Karan Gill]

Is this example in the documentation helpful? https://www.mathworks.com/help/symbolic/dsolve.html#bve5nw0-1_1

Answer 4

[Ketav Majumdar]

So i made changes and i have some solutions but i can't get a plot for y against t. I'm not sure how to get this solved with ode45. any suggestions?

 %Mechanical Properties of Material
E=200e9; 
nu=0.3;

 %Constants A,B,C,D in the Equations
a11= (1/E);
a12= (-nu/E);
a33= (1/E);
A= (a12)/(a11+a12);
B= ((a33)-((2*a12^2)/(a11+a12)));
C= (a11)/(a11^2-a12^2);
D= (2*a12+a11)/(a11+a12);

 %Defining the System of 2ODES
syms x(t) y(t)
t=1;
eqns = [diff(x,t)==A.*(x/t)+B*y, diff(y,t)==-C.*(x/t^2)-D*y];
cond = [y(0) == 0, y(2)==1];
[ySol(t) xSol(t)] = dsolve(eqns)