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GreenFunction[{u''''[x] - b^4*u[x], u''[0] == 0, u'''[0] == 0, 
  u''[1] == 0, u'''[1] == 0}, u[x], {x, 0, 1}, y]

Actually, i am trying to find the green function of a linear fourth order differential equation using Mathematica. But seems like I am not getting any results by using Mathematica predefined function "GreenFunction "

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Up to Wiki, the Green function $G(x,y)$ of the linear differential operator L = u''''[x] - b^4*u[x] can be treated as any solution of the ODE

DSolve[u''''[x] - b^4*u[x] == DiracDelta[y - x], u[x], x]

{{u[x] -> E^(-b x) C[2] + E^(b x) C[4] + C1 Cos[b x] + C[3] Sin[b x] - (1/( 4 b^3))E^(-b x - b y) (E^(2 b x) HeavisideTheta[ y - x HeavisideTheta[1 - x] - HeavisideTheta[-1 + x]] HeavisideTheta[-y + HeavisideTheta[1 - x] + x HeavisideTheta[-1 + x]] - E^(2 b y) HeavisideTheta[ y - x HeavisideTheta[1 - x] - HeavisideTheta[-1 + x]] HeavisideTheta[-y + HeavisideTheta[1 - x] + x HeavisideTheta[-1 + x]] - 2 E^(2 b x) HeavisideTheta[-1 + x] HeavisideTheta[ y - x HeavisideTheta[1 - x] - HeavisideTheta[-1 + x]] HeavisideTheta[-y + HeavisideTheta[1 - x] + x HeavisideTheta[-1 + x]] + 2 E^(2 b y) HeavisideTheta[-1 + x] HeavisideTheta[ y - x HeavisideTheta[1 - x] - HeavisideTheta[-1 + x]] HeavisideTheta[-y + HeavisideTheta[1 - x] + x HeavisideTheta[-1 + x]] + 2 E^(b x + b y) Cos[b y] HeavisideTheta[x - y] Sin[b x] - 2 E^(b x + b y) Cos[b x] HeavisideTheta[x - y] Sin[b y])}}

However, Mathematica uses a slight modification of the above definition:

DSolve[u''''[x] - b^4*u[x] == DiracDelta[x - y], u[x], x]

{{u[x] -> E^(-b x) C[2] + E^(b x) C[4] + C1 Cos[b x] + C[3] Sin[b x] - (1/( 4 b^3))E^(-b x - b y) HeavisideTheta[ x - y] (-E^(2 b x) + E^(2 b y) + 2 E^(b x + b y) Cos[b y] Sin[b x] - 2 E^(b x + b y) Cos[b x] Sin[b y])}}

I am not a specialist in ODEs so I am sorry if I am mistaken.

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  • $\begingroup$ What is that C[2], C[3] is. They are integration constant? $\endgroup$ – Vijay Kumar S Jan 5 '18 at 8:15
  • $\begingroup$ @Vijay Kumar S: The notations C[1], C[2], C[3], C[4] stand for arbitrary constants. $\endgroup$ – user64494 Jan 7 '18 at 13:46
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GreenFunction[] gives no result because your boundary conditions u''''[1]==0 are of the same order as the ode is!!

With correct bc you can calculate greens function using DSolveValue

green = Simplify@DSolveValue[{u''''[x] - b^4*u[x] == DiracDelta[x - y] ,u''[0] == 1,u'''[0] == 0, u''[1] == 0, u''' [1] == 0}, u , {x, 0, 1}]
(* green[x] *)  

Don't know why GreenFunction[] doesn't work...

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  • $\begingroup$ sorry it is third order boundary condition , but still it don't work $\endgroup$ – Vijay Kumar S Jan 5 '18 at 7:59
  • $\begingroup$ All your bc ==0 and you expect a non vanishing solution? $\endgroup$ – Ulrich Neumann Jan 5 '18 at 8:12
  • $\begingroup$ yes, is it not possible? I guess it is possible $\endgroup$ – Vijay Kumar S Jan 5 '18 at 8:19
  • $\begingroup$ You're right, sorry my answer was to fast! $\endgroup$ – Ulrich Neumann Jan 5 '18 at 8:23

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