Matrix Dirichlet problem with applications to hinged beam differential equations

In this study, we propose a construction of the Green kernel for the matrix Dirichlet problemY″−A2Y=FandY(0)=0=Y(1), where A∈Cn×nA∈Cn×n and F∈L1([0,1],Cn)F∈L1([0,1],Cn). We apply our result to the beam differential equationDy=y⁗−(α2+β2)y″+α2β2y=f′Dy=y⁗−(α2+β2)y″+α2β2y=f′ where α2≠β2α2≠β2, are defined on [0,1][0,1] with the Navier boundary conditionsy(0)=y(1)=y″(0)=y″(1)=0.y(0)=y(1)=y″(0)=y″(1)=0. In conclusion, we show that the pointwise maximum principle for the hinged beam equation holds if both α and β   are real or β¯=α=k(l+i), with |k|≤ϰ(|l|)2, where ϰ(l)∈(π,3π2) is the first positive root of the equation ltan⁡t=tanh⁡(lt)ltan⁡t=tanh⁡(lt).