# To find the Null space of a matrix which contains trig functions. the main objective is to find the green function of the differential equation

(* I am trying to find the null space of a matrix I used NUllspace command but I am getting the empty matrix and I also tried gauss elimination. All I am doing is I am finding out a green function a fourth order differential equation. I Used GreenFunction command in Mathematica I have given 4 boundary condition still I could not able to get the Green function solution *)

        W1 = C2*Sin[\[Beta]*x] + C4*Sinh[\[Beta]*x]
W2 = B2*Sin[\[Beta]*(1 - x)] + B4*Sinh[\[Beta]*(1 - x)]
(*Compatability condition*)
ccd1 = (W1 /. {x -> z}) - (W2 /. {x -> z});
ccs1 = ((D[W1, {x}]) /. {x -> z}) - ((D[W2, {x}]) /. {x -> z})
ccm1 = ((D[W1, {x, 2}]) /. {x -> z}) - ((D[W2, {x, 2}]) /. {x -> z})
ccsh1 = ((D[W1, {x, 3}]) /. {x -> z}) - ((D[W2, {x, 3}]) /. {x -> z})
R = Normal@
CoefficientArrays[{ccd1, ccs1, ccm1, ccsh1}, {C2, C4, B2, B4}][]
S = R // MatrixForm
NN1 = [NullSpace[R]]
(*Greenfunction *)
GreenFunction[{(u''''[x] - \[Beta]^4*u[x]), u == 0, u'' == 0,
u == 0, u'' == 0}, u[x], {x, 0, 1}, y]

• It's not surprising that NullSpace returns an empty matrix since there's no reason that any of the eigenvalues would be identically zero for all beta and z. – aardvark2012 Sep 10 '17 at 11:13

As pointed out by aardvark2012, your matrix is invertible and thus its null space is empty. To see for which parameter values it is not invertible, you can do Simplify[Det[S] == 0] to get the simple answer β Sin[β] Sinh[β] == 0. So, for example:

NullSpace[S /. β -> π]


gives

{{Csc[π z] Sin[π (1 - z)], 0, 1, 0}}


and

NullSpace[S /. β -> 8 I π]


gives

{{-((Csch[8 π z] (Cos[8 π (1 - z)] +Cot[8 π z] Sin[8 π (1 - z)]) Sinh[8 π (1 - z)])/(Cosh[8 π (1 - z)] + Coth[8 π z] Sinh[8 π (1 - z)])),
Csc[8 π z] Sin[8 π (1 - z)],
-((Cos[8 π (1 - z)] Sinh[8 π z] +
Cot[8 π z] Sin[8 π (1 - z)] Sinh[8 π z])/(Cosh[8 π z] Sinh[8 π (1-z)] + Cosh[8 π (1 - z)] Sinh[8 π z])),
1}}