How to use solve correctly with this partial differential equation?

Questions
1

Hi,
Could you please help me to find problems in the following attachment?
I want to find the coefficients (a1, b1, c1…).First, I wrote transformed governing equations (PDE to ODE), and then I imposed boundary and initial conditions by aid of solve command.
Two Problems are involved:

  1. Is it correct to use Integral form in my equations?, For example in [Sigma]zz1, there is double integral.
    2)Is it correct to use z-> -h in the boundary conditions?
    Thanks.
2

It seems to me that your double integrals do not parse correctly. Repeatedly click inside your formulas and see how the terms group together. You had better enter the double integrals again using the unambiguous ascii syntax

Integrate[
 Integrate[function, {\[Xi], 0, Infinity}], {s, (-I) Infinity, 
  I*Infinity}]

and avoid the traditional integral notation when typing the input, because it can be interpreted the wrong way.