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MATH 110 functions
创建日期：2024-05-21 16:37:15
所属分类：数学mathematics
标签： MATH 110
functions

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MATH 110 PRACTICE FINAL

Please justify all your steps!
1. (a) Calculate the cosine series of the function x on the interval [0, 1]. You should do
this using integration by parts.
(b) Let An be the coefficients of the cosine series of x, as in (a). Calculate
A20/4 +
∞∑
n=1
A2n.
Hint: You should be able to do this even if you could not solve (a).
(c) What is the value of the cosine series of (a) for x = −1/2? Again, you should be
able to answer this question even if you could not solve (a).
2. (a) Find the solution of the heat equation wt = wxx with initial and boundary condi-
tions
w(0, t) = w(pi, t) = 0, t > 0,
w(x, 0) = sin(2x) + sin(3x) 0 < x < pi.
(b) Find the solution of the steady state problem vxx = 0 with boundary conditions
v(0) = 0 and v(pi) = 1 (remember that for steady state temperature we have vt = 0,
i.e. we can consider v as a function of x only).
(c) Find the solution of the heat equation ut = uxx with initial and boundary condi-
tions
u(0, t) = 0 u(pi, t) = 1, t > 0,
u(x, 0) = sin(2x) + sin(3x) + x/pi, 0 < x < pi.
3. Let D be the disk x2 + y2 ≤ a2. We have shown in class that the eigenfunctions v of
−∆ satisfying −∆v = λv are given by
v(r, θ) = Jm(
√
λr)(Am cosmθ +Bm sinmθ), m = 0, 1, 2, ...
(a) What are the eigenfunctions v(r, θ) = f(r) which do not depend on θ and which satisfy
the boundary condition f ′(a) = 0? What are their eigenvalues? (It is OK to describe
the eigenvalues as the zeros of one or several functions; it may not be possible to give
a more explicit description).
(b) Let λn be the eigenvalues of (a) with eigenfunctions vn. One can show that the
function f(r) = r2 can be written as
r2 =
∞∑
n=1
Anvn.
How does one calculate the coefficients An? It is OK to write down formulas for An
involving integrals without calculating them.
More problems on the back side!
4. (a) Consider the two explicit functions:
φ(x) = 2 sin(pix)− 4 sin(2pix) + 3 sin(4pix)− 10 sin(5pix) + 5 sin(6pix),
ψ(x) = − sin(pix) + 6 sin(2pix)− 2 sin(5pix) + 3 sin(6pix)− sin(7pix).
Please compute the integral:
〈φ, ψ〉 =
∫ 1
0
φ(x)ψ(x) dx.
Hint : You may freely quote the values of certain integrals we have covered. The
answer can be given quickly, but you need to explain clearly what you are doing. Do
not use a calculator to get a numerical answer.)
(b) Let Jm be the m-th Bessel function of the first kind, and let zm,n be its n-th zero.
Calculate ∫ a
0
J5(z5,3r/a)J5(z5,7r/a) r dr.
Hint : Consider the theorem in Lecture 25 and its application to Bessel functions.
5. Consider the Laplace equation ∆u = 0 on the square [0, pi] × [0, pi]. We assume the
boundary conditions
ux(0, y) = 0 = ux(pi, y),
u(x, 0) = 0 and u(x, pi) = 3 cos 2x+ 2 cos 5x.
(a) Calculate separated solutions u(x, y) = X(x)Y (y) subject to the given homoge-
neous boundary conditions.
(b) Calculate the solution of the Laplace equation subject to all boundary conditions.
6. Consider the Laplace equation ∆u = 0 on the upper semidisk D of radius 1, given
in polar coordinates by r ≤ 1 and 0 ≤ θ ≤ pi, subject to the boundary conditions
u(r, 0) = 0 = u(r, pi).
(a) Write down the ordinary differential equations for R(r) and for Θ(θ), with bound-
ary conditions, for separated solutions u(r, θ) = R(r)Θ(θ).
(b) Calculate the separated solutions u(r, θ) = R(r)Θ(θ) in (a).
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