Laplace Transform Method for a PDE with Neumann conditions

see attached

Laplace Transform Derivation of Source Solution

see attachment

source problem of heat equation

Derive the source solution by performing integral transforms of the equation: (see attachment)

PDE with Time-Dependent Domain

Consider the diffusion equation attached on the time-dependent domain where a is a constant. We wish to solve the initial and boundary value problem... (SEE ATTACHMENT)

Helmholtz Equation

a)Solve the Helmholtz equation when u is a function of r only in 2-D. b)Solve the Helmholtz equation when u is a function of r only in 3-D. (see attachment for full question)

LaPlace Eqn with Neumann BC on a circle

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Fundamental Solution of N-Dimensional Laplace equation

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Dipoles and Laplacian

Consider a dipole of strength D, oriented along the x-axis and located at the point x =E/2... Take the limit as E->0,ED = fixed and show that the limiting potential is given by... (See attachment for full question)

given that y=x is one solution, find a second linearly independent solution

(x^2 - x + 1)y" - (x^2 + x)y' + (x + 1)y = 0

Fourier Series of Function f; Laplace's Equation; Heat Equation

Please assist me with the attached Fourier Series problems. Thanks!