اريد مساعدة ضرووووري في حل مسائل في طرق نظرية دالة جرين

[أ. دلال]

السلام عليكم ورحمة الله
ارجوو منكم المساعدة في حل هذه المسائل ياليت يكون بالتفصيل

. This problem concerns the Green’s function for the diffusion equation in an arbitrary number of spatial
dimensions d.
(a) Use an eigenfunction expansion to show that the Green’s function is
http://arb-up.com/a4b745namzb7
where (t) is the Heaviside step function.
(b) Find the Green’s function by taking a Fourier transform of the diusion equation with respect to
the spatial variables. You should, of course, get the same answer.

[أ. دلال]

This problem concerns the Green’s function for the Helmholtz equation in two dimensions, with
outgoing-wave boundary conditions.
(a) Use an eigenfunction expansion to show that the Green’s function is in terms of a Hankel function.
[You will need to evaluate some integrals. Look them up in a table of integrals or in Mathematica.]
(b) Derive the Green’s function by rst solving the Helmholtz equation away from the source point
and then normalizing [as we did in class for the Laplace equation Green’s function].