System of Coupled Higher Order Differential Equations

[saheedimran]

Hi all,
I am a PhD student in Structural Mechanics and I need some help/hints in solving a system of coupled higher order differential equations. The equations and the boundary conditions are specified in the attached zip file. Any useful information or suggestions on how to proceed with the solution will be highly appreciated.

Thanks.

 

[stapel]

The equations and the boundary conditions are specified in the attached zip file.

If you can't be bothered to type out your question, it's unlikely that anybody is going to be bothered to type out a response. In addition, nobody having any Internet-security savvy at all is going to touch a ZIP file of unknown origin. Sorry.

 

[saheedimran]

System of Coupled Higher Order Differential Equation

Sorry, my bad.



I have the following equations with the aim of finding the following terms:

σ1, σ2, σ3, σ4, W1, W2, W3, U1, U2, U3

N₁'-σ₁=0

N₂'+σ₁-σ₂=0

N₃'+σ₂=0

M₁''+c₁ σ₁'-σ₃=0

M₂''+c₂σ₁'+c₂σ₂'+σ₃-σ₄=0

M₃''+c₃σ₂'+σ₄=0

v₁=v₂+K₁σ₁

v₂=v₃+K₂σ₂

w1=w₂-K₃σ₃

w₂=w₃-K₄σ₄

c_k (k=1,2,3) and K_i (i =1,2,3,4) are constant coeffcients.

N₁'=D₁U₁''-E₁σ₁''+F₁σ₃'

N₂'=D₂U₂''+E₂(σ₁''-σ₂'')+F₂(σ₃'+σ₄')

N₃'=D₃U₃''+E₃σ₂''+F₃σ₄'

M₁''=-A₁W₁''''+B₁σ₁'''-T₁σ₃''

M₂''=-A₂W₂''''+B₂(σ₁'''+σ₂''')+T₂(σ₃''-σ₄'')

M₃''=-A₃W₃''''+B₃σ₂''+T₃σ₄''


Additional equations
φ₁+W₁'=R₁(Q₁-m₁σ₁)

φ₂+W₂'=R₂[Q₂-m₂(σ₁+σ₂)]

φ₃+W₃'=R₃(Q₃-m₃σ₂)

Q₁'-σ₃=0

Q₂'+σ₃-σ₄=0

Q₃'+σ₄=0

A_k, B_k, D_k, E_k, F_k, T_k, R_k, m_k (k=1,2,3) are constant coefficients.
All the primed letters denote differentiation with respect to x.

Boundary conditions

At x=0

U_k=0, φ_k=0, Q_k=0 (k=1,2,3)

x=L

M_k=0, N_k=0, Q_k=0 (k=1,2,3)