Problem 6
(a) Find the solution \(u(r, \theta)\) of Laplace's equation in the semicircular
region \(r
Problem 7
Find the solution of the heat conduction problem
$$
\begin{aligned} 100 u_{x x} &=u_{t}, & 0
Problem 7
Determine whether the given function is periodic. If so, find its fundamental period. $$ f(x)=\left\\{\begin{array}{ll}{0,} & {2 n-1 \leq x<2 n,} \\ {1,} & {2 n \leq x<2 n+1 ;}\end{array} \quad n=0, \pm 1, \pm 2, \dots\right. $$
Problem 7
In each of Problems 7 through 12 a function \(f\) is given on an interval of length \(L .\) In each case sketch the graphs of the even and odd extensions of \(f\) of period \(2 L .\) $$ f(x)=\left\\{\begin{array}{ll}{x,} & {0 \leq x<2} \\ {1,} & {2 \leq x<3}\end{array}\right. $$
Problem 7
Either solve the given boundary value problem or else show that it has no solution. \(y^{\prime \prime}+4 y=\cos x, \quad y(0)=0, \quad y(\pi)=0\)
Problem 7
Find the solution \(u(r, \theta)\) of Laplace's equation in the circular sector
\(0
Problem 7
Carry out the following steps. Let \(L=10\) and \(a=1\) in parts (b) through (d). (a) Find the displacement \(u(x, t)\) for the given \(g(x) .\) (b) Plot \(u(x, t)\) versus \(x\) for \(0 \leq x \leq 10\) and for several values of \(t\) between \(t=0\) and \(t=20 .\) (c) Plot \(u(x, t)\) versus \(t\) for \(0 \leq t \leq 20\) and for several values of \(x .\) (d) Construct an animation of the solution in time for at least one period. (e) Describe the motion of the string in a few sentences. \(g(x)=8 x(L-x)^{2} / L^{3}\)
Problem 7
assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let \(e_{n}(x)=f(x)-s_{n}(x)\). Find the least upper bound or the maximum value (if it exists) of \(\left|e_{n}(x)\right|\) for \(n=10,20\), and 40 . (c) If possible, find the smallest \(n\) for which \(\left|e_{x}(x)\right| \leq 0.01\) for all \(x .\) $$ f(x)=\left\\{\begin{array}{ll}{x,} & {-\pi \leq x<0,} \\ {0,} & {0 \leq x<\pi ;}\end{array} \quad f(x+2 \pi)=f(x)\right. $$
Problem 8
find the steady-state solution of the heat conduction equation \(\alpha^{2} u_{x x}=u_{t}\) that satisfies the given set of boundary conditions. $$ u(0, t)=T, \quad u_{x}(L, t)+u(L, t)=0 $$
Problem 8
assume that the given function is periodically extended outside the original interval. (a) Find the Fourier series for the given function. (b) Let \(e_{n}(x)=f(x)-s_{n}(x)\). Find the least upper bound or the maximum value (if it exists) of \(\left|e_{n}(x)\right|\) for \(n=10,20\), and 40 . (c) If possible, find the smallest \(n\) for which \(\left|e_{x}(x)\right| \leq 0.01\) for all \(x .\) $$ f(x)=\left\\{\begin{array}{lr}{x+1,} & {-1 \leq x<0,} \\ {1-x,} & {0 \leq x<1 ;}\end{array} \quad f(x+2)=f(x)\right. $$
