1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x...
Thank you. 5. Find the solution u(x, y) of Laplace's equation in the rectangle 0<<a, 0<y<b that satisfies the boundary conditions u(0, y = 0, u(a, y) = 0,uy 3,0) = 0, 2,b) = g(1), where J2 0<x<a/2 g(x) = - 0<a/2 <<a
Problem 4. (25 points) Find the solution to the 2-dimensional Laplace's equation OLY + = 0 inside the square 0<x<1 0 <y <1 subject to the boundary conditions V(x,0) = 0 = V(x, 1) V(0,y) = 0 V(1,y) = 2 sin (31 y)
2. Solve for the bounded solution of Laplace's equation v2T=0 in the UHP: [2] < 0, y > 0 with the following boundary conditions given on y = 0: T(x,0) = {A on x < l1, B on li < x < l2,C on x > la} A, B, C are real constants.
We can expect the solution u(x,y) to be in the form X(x)Y(y). or I believe that these are the correct forms of X(x) and Y(y). 2. Laplace's equation Consider Laplace's equation on the rectangle with 0 < x < L and 0 < < H: PDE BC BC BC u(x,0) 0, u(z, H) = g(z). (10) where a mixture of Dirichlet and Neumann boundary conditions is specified, and only one of the sides has a boundary condition that is nonhomogeneous...
Solve heat equation in a rectangle du = k ( ou + dou), 0<x<t, 0<y< 1, t> 0 u(x, 0, 1) = 0, uy(x,1,1) = 0, with boundary conditions u(O, y,t) = 0, u(r, y, t) = 0, and initial condition u(x, y,0) = (y – į v?) sin(2x).
(Laplace's equation in polar coordinates) (a) Find the solution to Laplace's equation on a disk with boundary condition u(1,0) = 5 + sin(40). (You do not need to derive the general solution to the polar Laplace's equation.) (b) Verify that the solution to (a) satisfies the mean value property. (Hint: compare the average value of u(r, 0) on the boundary r = 1 to the value of u(r,) at r=0.) (c) Find the minimum and maximum of the solution to...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
(a) Find the solution to Laplace's equation on a disk with boundary condition u(1,0) = 5 + sin(40). (You do not need to derive the general solution to the polar Laplace's equation.) (b) Verify that the solution to (a) satisfies the mean value property. (Hint: compare the average value of u(r, ) on the boundary r=1 to the value of u(r,() at r = 0.) (c) Find the minimum and maximum of the solution to (a) and verify they occur...
Solve the following Laplace's equation in the semi-annular plate (on the right) Urr + - + = 0, a<r<b, O<O<T, u(a,0) = (1-0), u(b, ) = 0, ur,0) = 0, ur.) = 0. 4= (1-1)
(a) Find the solution u(x, y) of Laplace's equation in the semi-infinite strip 0<x<a, y>0, that satisfies the boundary conditions u(0, y)-0 u(a, y)-0, y > 0, and the additional condition that u(x, y) -0 as yoo, etnyla sin nTX where Cn X where Cn- NTX) where Cn = u(x, y) - -Ttny/a sin(where Cn u(x, y) n=1 u(x, y) - (b) Find the solution if f(x) = x(a-x) V(x)- (c) Let a9. Find the smallest value of yo for...