Solve second order ode with Green's functions

[the king]

-u''(z)+α²u(z)=f(z), u(0)=g(z), u(z)=0 as z→∞

-u''(z)+α²u'(z)=f(z), u(0)=g(z), u(z)=0 as z→∞

I am interested to solve these two boundary problems using Green's functions. It is noticed that z is complex variable. Can someone help me to do this?

 

[Marioeden]

Well, first things first, find your green's function for the operator given.

In other words, solve -G''(z;z') + [a^2]G(z;z') = diracdelta(z - z')

Then simply integrate the product of G(z;z') with f(z') w.r.t z' and you're done :)

 

[Mute]

Well, first things first, find your green's function for the operator given.

In other words, solve -G''(z;z') + [a^2]G(z;z') = diracdelta(z - z')

Then simply integrate the product of G(z;z') with g(z') w.r.t z' and you're done :)

g(z) is the initial condition; integrating G(z;z') against f(z') should give the solution in this case, although...

-u''(z)+α²u(z)=f(z), u(0)=g(z), u(z)=0 as z→∞

-u''(z)+α²u'(z)=f(z), u(0)=g(z), u(z)=0 as z→∞

I am interested to solve these two boundary problems using Green's functions. It is noticed that z is complex variable. Can someone help me to do this?

Is z the only variable in this DE? How can the initial condition u(0) = g(z) depend on z?