Wave equation with mixed boundary conditions

Uxx means second derivative with respect to x Uyy means second derivative with respect to y Uxx + Uyy = 0, 0 < x < pi, 0 < y < 1 Ux(0,y) = 0 = U(pi,y), 0 < y < 1 U(x,0) = 1, U(x,1) = 0, 0 < x < pi Please show all work including how eigenvalues and eigenvectors are derived. Thank you

Elliptic Boundary Value Problem

Uxx means second derivative with respect to x Uyy means second derivative with respect to y Uxx + Uyy = 0, 0 < x < pi, 0 < y < pi U(x,0) = 0, U(x,pi) = 1, 0 < x < pi U(0,y) = 0, U(pi,y) = 1 0 < y < pi I know the problem has to be broken into 2 separate problems using U = V + W with zero conditions on 3 sides for each problem W and V. So please show the problem in this manner. Please show all work including the derivation of any eigenvalues or eigenvectors. THank you

Elliptic Boundary Value problem using Laplace and polar cordinates

(lap) means the Laplacian Vrr means the second derivative of V with respect to r V(theta theta) means the second derivative of V with respect to theta Solve: (lap)V(r,theta)= Vrr+(1/r)Vr+(1/r^2)V(theta theta)=0 0 < r < 1, -(pi) < theta < pi V(1,theta) = {1, -(pi/2) < theta < (pi/2) {0, elsewhere Please show all work including the derivation of any eigenvalues or eigenvectors. Thank you

It is an explanation for solving homogeneous linear partial differential equation. Find the General solutions of the equations r = a2t and (D + D’)z = sin x.

Linear Partial Differential Equation (I) Linear Homogeneous Partial Differential Equation with Constant Coefficients Problem 1: Find the General solution of the equation r = a2t. Problem 2: Find the General solution of the Equation (D + D’)z = sin x

It is an explanation for solving Non- Homogeneous Linear Partial Differential Equation with Constant Coefficients. Find the solution of the equation (D2 – D’2 + D – D’)z = e^(2x + 2y).

Linear Partial Differential Equation (II) Non- Homogeneous Linear Partial Differential Equation with Constant Coefficients Problem: Find the solution of the equation (D2 – D’2 + D – D’)z = e^(2x + 2y)