(40 marks) Find the solution of the two-dimensional Laplace equation
$$ u_{x x}+u_{y y}=0 \quad 0<x<1,0<y<1 $$
with the boundary conditions
$$ u(x, 1)=x, u(x, 0)=u(0, y)=0, u(1, y)=y $$
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25 points) Find the solution of the Laplace equation ur the domain 0-x-π and 0-y-T. The boundary condition at the left boundary is given by u(0, y)-sin(y/2). The boundary conditions at al other boundaries are zero. Express the solution as an infinite series = 0, over
Using Fourier transform, prove that a solution of the Laplace equation in the half plane: Urn+ Uyy=0,- << ,y>0, with the boundary conditions u(1,0) = f(t), - <I< u(x,y) +0,31 +0,+0, is given by r(2, y) == Love you > 0. Hint: 1. Take Fourier transform on the variable r, 2. Observe U(k, y) +0 as y → 00, 3. Use pt {e-Mliv = Vice in
3. Consider a Laplacian equation in a two-dimensional Cartesian coordinate as a2u + ax2 =0 in 0 < x <a, 0 <y <b ay2 Associate with the boundary conditions of 0ys b at x= 0, u = 0 0Sysb u = f(y) at x = a, 0 x a 0 = n at y 0, at y b, 0 x s a u = 0 (20 points) Find the solution of u(x, y) Reading assignment: Sect. 12.6 of the textbook...
(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.$$ \begin{array}{l} u_{t t}=c^{2} u_{x x}, \quad 0< x < \infty, t>0 \\ u(x, 0)=A e^{-\alpha x} \quad \text { and } \quad u_{t}(x, 0)=0, \quad 0< x < \infty \\ u(0, t)=A \cos \omega t, \quad t>0 \\ \lim _{x \rightarrow \infty} u(x, t)=0, \quad \lim _{x...
7. Consider the boundary value problem for the Laplace equation on the strip u (0, y) u (т, y) = 0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x, y) -ZYn (v)sinnx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y)-Yn (y) sin n. the Laplace equation and the boundary conditions. (i.e. find Yn. (3).) that satisfies...
7. Consider the boundary value problem for the Laplace equation on the strip u(0, y) u(n,y)=0, = a. Explain why it makes sense to look for a solution of the form b. Find all solutions of the form u(x,y) = Σ Yn (y) sin nx satisfying c. Among the solutions you found in part (b) find the unique solution u (x, y) = Σ Y, (y) sin na. the Laplace equation and the boundary conditions. (i.e. find Yn (y).) that...
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)
PROBLEM 1 IS SUPPOSED TO BE A WAVE EQUATION NOT HEAT EQUATION 1. Find the solution to the following boundary value initial value problem for the Heat Equation au 22u 22 = 22+ 2 0<x<1, c=1 <3 <1, C u(0,t) = 0 u(1,t) = 0 (L = 1) u(x,0) = f(x) = 3 sin(7x) + 2 sin (3x) (initial conditions) (2,0) = g(x) = sin(2x) 2. Find the solution to the following boundary value problem on the rectangle 0 <...
Consider the Laplace equation v²u(x, y, z) = 0 in 3D and find its most general solution for a parallelepiped defend by planar as (see the figure below) (x = 0,X = 2, y = 0, y = 2, (z=0,2 = 5, with the Dirichlet boundary conditions as u(0, y, z) = u(2, y, z) = 0 u(x, 0, 2) = u(x, 2, z) = 0 (u(x, y,0) = 0, u(x,y,5) = 17
Transform the following IBVP into a problem with homogeneous boundary conditions.$$ \begin{array}{l} u_{t t}=u_{x x}, \quad 0<x<1, t=>0 \\ a u(0, t)+b u_{x}(0, t)=f_{1}(t) \quad \text { and } \quad c u(1, t)+d u_{x}(1, t)=f_{2}(t), \quad t>0 \\ u(x, 0)=g(x), \quad u_{t}(x, 0)=h(x), \quad 0<x<1 \end{array} $$where \(a, b, c\) and \(d\) are constants.