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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}]  Attachments:
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Posted 10 years ago
 It appears that there are more unknowns than equations.
Posted 10 years ago
 "EI" is not defined, as "PA" and "CI" are not defined either.
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 10 years ago
 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.
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