Question

Find u(x,y) harmonic in S with given boundary values: S = {(x,y): 1 < y < 3} , u(x,y) = 5 (if y=1) and = 7 (when y=3)

I have this problem to solve, and I'm not sure where to start. Any help would be appreciated. Thanks!

Answer #1

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first we make boundary condiiton homogeneous..

For the given function u(x, y) = cos(ax) sinh(3y),(a >
0);
(a) Find the value of a such that u(x, y) is harmonic.
(b) Find the harmonic conjugate of u(x, y) as v(x, y).
(c) Find the analytic function f(z) = u(x, y) + iv(x, y) in
terms of z.
(d) Find f ′′( π 4 − i) =?

Walter A . Strauss- Partial Differential Equations (2nd
Edition)
Chapter 7.1, Problem 10E
Let u(x，y) be the harmonic function in the unit disk with
boundary
values u(x， y) = x^2 on {x^2 + y^2 = 1}. Find the Rayleigh-Ritz
approximation of the
form, w0 + c1w1 = x^2 + c1(1 - x^2 - y^2).

a)Prove that the function
u(x, y) = x -y÷x+y
is harmonic and obtain a conjugate function v(x, y) such that
f(z) = u + iv is analytic.
b)Convert the integral
from 0 to 5 of (25-t²)^3/2 dt
into a Beta Function and evaluate the resulting function.
c)Solve the first order PDE
sin(x) sin(y)
∂u
∂x + cos(x) cos(y)
∂u
∂y = 0
such that u(x, y) = cos(2x), on x + y =
π
2

uxx = ut - u (0<x<1, t>0),
boundary conditions: u(1,t)=cost, u(0,t)= 0
initial conditions: u(x,0)= x
i) solve this problem by using the method of separation of
variables. (Please, share the solution step by step)
ii) graphically present two terms(binomial) solutions for
u(x,1).

Given the following utility function: U (X,Y) = 2X½ + Y and
given that U = 40
Part 1: Find Y1 for X = 4
Part 2: Find Y1 for X = 9
Part 3: Find Y1 for X = 16
Part 4: Find Y1 for X = 36
Part 5: Find Y1 for X = 49
Using graph paper construct the graph for indifference curve for
U = 40 Given : Py = 20, Px = 5 and I...

Let s = f(x; y; z) and x = x(u; v; w); y = y(u; v; w); z = z(u;
v; w). To calculate ∂s ∂u (u = 1, v = 2, w = 3), which of the
following pieces of information do you not need?
I. f(1, 2, 3) = 5
II. f(7, 8, 9) = 6
III. x(1, 2, 3) = 7
IV. y(1, 2, 3) = 8
V. z(1, 2, 3) = 9
VI. fx(1, 2, 3)...

Calculate the Y values corresponding to the X values given
below. Find the critical values for X for the given polynomial by
finding the X values among those given where the first derivative,
dy/dx = 0 and/or X values where the second derivative,
d2y/dx2 = 0. Be sure to indicate the sign (+
or -) of dy/dx and of d2y/dx2 tabled values.
Reference Power Point Lesson 13 as needed. Using the first and
second derivative tests with the information you...

Calculate the Y values corresponding to the X values given
below. Find the critical values for X for the given polynomial by
finding the X values among those given where the first derivative,
dy/dx = 0 and/or X values where the second derivative,
d2y/dx2 = 0. Be sure to
find the sign (+ or -) of dy/dx and of
d2y/dx2 at all X values. Reference Lesson 13
and the text Appendix A (pp 694 – 698), as needed. Using the...

Solve ut=uxx, 0 < x < 3, given the
following initial and boundary conditions:
- u(0,t) = u(3,t) = 1
- u(x,0) = 0
Please write clearly and explain your reasoning.

Given: The following boundary value problem:
y"+ lamda*y = 0;
0 < x < 2; y(0) =
0; y’(2) = 0
Find corresponding eigenvalues, (lamda)n
and normalized eigenfunctions yn
Expand the function f(x) = x, in terms of the eigen functions
obtained in (i)

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