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How to solve a higher order partial differential equation with boundaries

Posted 10 years ago

I was trying to solve the equation of base excited cantilever. But I couldn't get that. Please help. My input is:

M = EI*D[y[x, t], {x, 2}];
V = EI*D[y[x, t], {x, 3}];
th = EI*D[y[x, t], x];
M1 = M /. x -> 0;
M2 = M /. x -> L;
V1 = V /. x -> 0;
V2 = V /. x -> L;
th1 = th /. x -> 0;
th2 = th /. x -> L;
y1 = y[x, t] /. x -> 0;
y2 = y[x, t] /. x -> L;
s = DSolve[{PA*D[y[x, t], {t, 2}] + EI*D[y[x, t], {x, 4}] + 
     CI*D[y[x, t], {x, 4}, t] == 0, y1 == 0, th1 == 0, M2 == 0, 
   V2 == 0}, y[x, t], {x, t}]

But I couldn't get output. The output is:

DSolve[{CI y^(4,1)(x,t)+EI y^(4,0)(x,t)+PA y^(0,2)(x,t)==0,y(0,t)==0,EI y^(1,0)(0,t)==0,EI y^(2,0)(L,t)==0,EI y^(3,0)(L,t)==0},y(x,t),{x,t}]
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POSTED BY: ihsan faseel
4 Replies

Looks difficult, even setting all constants equal to 1 one gets

In[8]:= DSolve[{D[y[x, t], {t, 2}] + D[y[x, t], {x, 4}] + D[y[x, t], {x, 4}, t] == 0,
  (y[x, t] /. x -> 0) == 0, (D[y[x, t], x] /. x -> 0) == 0, (D[y[x, t], {x, 2}] /. x -> L) == 0, 
  (D[y[x, t], {x, 3}] /. x -> L) == 0}, y[x, t], {x, t}]
Out[8]= (* actually the input *)

so give L a value and try NDSolve:

In[9]:= With[{L = 5},
 NDSolve[{D[y[x, t], {t, 2}] + D[y[x, t], {x, 4}] + D[y[x, t], {x, 4}, t] == 0,
   (y[x, t] /. x -> 0) == 0, (D[y[x, t], x] /. x -> 0) == 0, (D[y[x, t], {x, 2}] /. x -> L) == 0, 
   (D[y[x, t], {x, 3}] /. x -> L) == 0}, y[x, t], {x, t}]
 ]

During evaluation of In[9]:= NDSolve::underdet: There are more dependent variables, {y[x,t],(y^(0,1))[x,t],(y^(0,2))[x,t]}, than equations, so the system is underdetermined. >>
Out[9]= (* actually the input *)

first job you should work on is to get a numerical solution with unit constants. Seemingly the problem is incorrect formulated.

POSTED BY: Udo Krause
Posted 10 years ago

sir, actually "E","I" ,"P","A","C" are constants. like : E-Youngs modulus. I-moment of inertia. A-area. C-damping coefficient.

So what should i do sir.Should i define all these with there values.Actually i want to get a solution containing these constants as itself.

POSTED BY: ihsan faseel
Posted 10 years ago

"EI" is not defined, as "PA" and "CI" are not defined either.

POSTED BY: Simon Tyran

It appears that there are more unknowns than equations.

POSTED BY: S M Blinder
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