I have a linear problem with double derivate of space and time, which has Dirichlet boundary condition in (1)2 and Neumann's boundary condition in (1)3:δ2uδt2−c2Δu=f in Ωu=0 on δΩ or∇u⋅n=0 on δΩwith conditionsu(0,x)=u0(x) in Ωδuδt(0,x)=v0(x) in ΩWhich is the weak form of the problem?My attempt:∫(δ2uδt2−c2Δu)⋅vdx=∫fv⋅dx,for ∀v∈Vn. Handling two different parts separately∫Ωδ2uδt2vdV+c2[∫Ω∇u⋅∇vdV−∫δΩ∇u⋅n^⋅vdA]=∫fv⋅dx

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I have a linear problem with double derivate of space and time, which has Dirichlet boundary condition in (1)2 and Neumann&#039;s boundary condition in (1)3:δ2uδt2−c2Δu=f  in  Ωu=0  on  δΩ  or∇u⋅n=0  on  δΩwith conditionsu(0,x)=u0(x)  in  Ωδuδt(0,x)=v0(x)  in  ΩWhich is the weak form of the problem?My attempt:∫(δ2uδt2−c2Δu)⋅vdx=∫fv⋅dx,for ∀v∈Vn. Handling two different parts separately∫Ωδ2uδt2vdV+c2[∫Ω∇u⋅∇vdV−∫δΩ∇u⋅n^⋅vdA]=∫fv⋅dx

sergiotheguyqgs · Accepted Answer

I finally got the result which should be right one. By integrating and multiplying Equation (1) with v ∫vu¨dx+c2vΔu=∫fvdx using Green I we get ∫vu¨dx−∫∂c2v∂u∂ndS+∫∇c2v⋅∇udx=∫fvdx. So the weak form is (v,u¨)+c2(∇v,∇u)−c2(v,∂u∂n)∂Ω=(f,v)

I have a linear problem with double derivate of space and time, which has Dirichlet boundary conditi

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