PDE solution by 3 methods

Please use: 1.) LaPlace Transform and 2.) Fourier Transforms methods and 3.) our old friend separation of variables with eigenvalues expansion to solve each problem. It is not necessary to evaluate an inverse transform. Where convenient, show any solution as a convolution of two functions and indicate how these functions are determined. See attached file for full problem description.

PDE by 3 methods

PDE by 3 methods. See attached file for full problem description.

Laplace Equation in a disk

I am looking for a solution of Laplace Equation in a unit disk. And I need to compare it to the answer given by Poisson integral formula. See attached file for full problem description.

PDE: Harmonic Function

Suppose U is a positive harmonic function which is defined everywhere in the plane. Show that U must be a constant function. This is a question regarding Poisson Integral Formula in PDE. I need to find it using the Harnak's inequalities.

PDE; Harmonic in the disk and Poisson Integral Formula

Is value of U at the origin a constant? Please prove it with detail explanation. See attached file for full problem description.