Homework for Math 46

Homework assignments will generally be due at the class time on Wednesday of the week following the week when the homework was assigned, unless announced otherwise. Late homework will not be accepted unless it is excused in advance for serious reasons. Unexcused late and missing papers count zero.

 

Lectures

Sections in Text

Homework assignment

Day 1, Monday March 29

Read Sections 1.1.2 and 1.1.3

Exercise 1 on page 7; Attention there is a typo, it should be P instead of T in the problem formulation.

Exercise 4 page 8 (In this problem choose π1 from the computation on page 6 and choose π2 that does not involve the distance r.)

Exercise 3 page 17 (First take the speed v to the left to get that some expression equals zero.)

Exercise 6 page 17;

due  Wednesday April 7 in written form

Sketches of solution can be found here

Day 2, Wednesday March 31

Read Section 1.1.3 and Section 1.2

Exercise 3 on page 30 and Exercise 12 on page 33.

(Hint: in exercise 12 use ℓ rather than L for the initial condition, so that there are no mixtures between the initial conditions and dimensions. When rescaling the problem choose xc=ℓ and do not attempt to solve the equation.)

due  Wednesday April 7 in written form

Sketches of solution can be found here

Day 3, Thursday April 1

x-hour instead of the lecture on

Friday April 16

Read Section 1.2

No additional homework

Day 4, Friday April 2

Read Section 1.3.1

Exercise 1.c on page 40 (will be counted as one exercise when grading). Note that the book formulation has a typo, you should change y to u in this problem. You do not have to compute the integral of t2exp(t2/2)

 

Exercise 1.d and 1.e on page 40 (together are to be counted as one exercise when grading). In 1.e if the integrals are hard do not compute them and leave them as the part of the answer.

 

Exercise 1.f and 1.j on page 40 (together are to be counted as one exercise when grading). In these problems you do not have to compute the integrals of exp(s2) and of 1/(s2+c) for constants c of various signs.

 

Exercise 1.n on page 40 will be counted as one exercise when grading.

 

due  Wednesday April 7 in written form

Sketches of solution can be found here

Day 5, Monday April 5

Read Section 4.1.1

Exercise 1, page 214. The following part will be graded as one problem: Show that the functions fm(x)=cos(mπx/ ℓ) form an orthogonal family in L2[0, ℓ]. Find the explicit formula for the Fourier coefficients of f with respect to this family. Hint: use the integral formula ∫cos(ax)cos(bx)dx=(sin(a-b)x)/(2(a-b))+ (sin(a+b)x)/(2(a+b)) when a2≠ b2

 

Exercise 1, page 214. The following part will be graded as one problem. Find the Fourier cosine series of f(x)=1-x on [0,1].

 

Exercise 3, page 214

due  Wednesday April 14 in written form

Sketches of solution can be found here

Day 6, Wednesday April 7

Read Section 4.1.1 and Section 4.1.2

Show that L2 is a vector space. (Hint you may want to use the Cauchy Schwarz inequality.)

 

Simplified version of Exercise 5, page 215.

Find P0(x), P1(x), P2(x), P3(x). Approximate

ex  as c0*P0(x)+c1*P1(x)+c2*P2(x). Do not compute errors and do not sketch anything.

 

Exercise 1, page 219. Do not sketch anything. Hint: your life will be much easier if you will use various facts about integration of odd and even functions.

due  Wednesday April 14 in written form

Sketches of solution can be found here

Day 7, Thursday April 8

x-hour

Read Section 4.1.2

No homework

Day 8, Friday April 9

End of the second week of schedule adjustment

Read Section 4.2

Exercise 3, page 225

Exercise 4, page 225

Exercise 7, page 225. Use energy argument to show that the eigenvalues have to be nonnegative. Do not find eigenvalues or eigenfunctions in Exercise 7.

due  Wednesday April 14 in written form

Sketches of solution can be found here

Day 9, Monday April 12

Final day for electing use of the Non-Recording option

Section 4.3.1

Exercise 1, page 243

Exercise 2 parts a and b, page 243 (these two parts combined will be counted as one exercise when grading)

Exercise 3, page 244. The answer should be given in terms of whether the vector b does belong or does not belong to some explicitly given vector subspace.

due  Wednesday April 21 in written form

Sketches of solution can be found here

Day 10, Wednesday April 14

Section 4.3.3

Exercise 4.b page 244. Hint rewrite K(u(x)) as an integral from 0 to x of some expression plus the integral from x to 1 of some expression. Obtain the differential equation by differentiating the equation K(u(x))=λu(x) twice using the Leibnitz formula from Exercise 1. Get the conditions on u(0) and u’(1).

 

Exercise 13 part a, page 245. Hint: this is a integral equation with a separable kernel.

 

Exercise 16, page 245. When the book says two terms in its power series expansion, it means the Taylor polynomial of power two of the function of many variables.

due  Wednesday April 21 in written form

Sketches of solution can be found here

Friday April 16,

No class, instead we had an x-hour on April 1

 

 

Day 11, Monday April 19

Section 4.3.4 and start 4.3.2

Exercise 13 part b page 245

Exercise 15, page 245

Exercise 18, page 246

Due Wednesday April 28 in written form

Problems of this type may appear on the first Midterm. Volterra equations will not be included into the Midterm material.

Sketches of solution can be found here

Day 12, Wednesday April 21

Section 4.3.2

Exercise 6, page 244

Exercise 8, page 244

Exercise 10, page 245

Due Wednesday April 28 in written form

Sketches of solution can be found here

Thursday April 22

First Midterm Exam

6-8 PM in Carpenter 013

 

 

Day 13, Friday April 23

Section 4.4.1

Exercise 3, page 257

Exercise 4, page 257

In these problems you may want first to multiply the equations by a function to make them into an Sturm-Liouville problem.

Exercise 5, page 257.

In these problems write the expression for Greens function g(x,ζ) AND write the solution of the differential equation as an integral involving the Green’s function g(x,ζ)

Due Wednesday April 28 in written form

Sketches of solution can be found here

Day 14, Monday April 26

Section 4.4.1 and 4.4.3

Exercise 2, page 257 If there is no Green’s function find for which functions f(x) the equation can be solved. Find the appropriate function  G(x,ζ) described in Theorem 4.23 and give an answer in terms of this function.

 

Exercise 7, page 258

Due Wednesday May 5 in written form

Sketches of solution can be found here

Day 15, Wednesday April 28

Section 4.4.3 and Section 4.5.1

Exercise 2 on page 267

Exercise 3 on page 267

Due Wednesday May 5 in written form

Sketches of solution can be found here

Day 16, Thursday April 29

x-hour

Section 4.5.1

No homework

Day 17, Friday April 30

Section 4.5.2 and Section 4.5.3

Exercise 1 page 267,

Exercise 5, page 268

Exercise 8, page 268

Exercise 9, page 268

Exercise 11, page 268

In these problems unless the interval is explicitly stated you can assume that it is (-a, a) for a finite positive a.

Due Wednesday May 5 in written form

Sketches of solution can be found here

Day 18, Monday May 3

Section 4.5.3 and Section 6.1.1

Exercise 4, page 267

Exercise 6, page 268

Exercise 3, page 345

Exercise 2 part a, page 345

Exercise 2 part b, page 345 (these two parts of exercise 2 will be counted as different problems when graded)

Hint: in exercise 2 it is useful to use the following trick fix one of the variables say t=τ and denote

u(x, τ) by a function g τ(x). This would allow you to convert a PDE to an ODE. Be careful the constants you get when solving this ODE are in fact functions depending on the fixed t=τ.

Due Wednesday May 12 in written form

Sketches of solution can be found here

Day 19, Wednesday May 5

Section 6.1.1 and Section 6.2.1

Exercise 2, part e on page 345

Exercise 5, page 346

Exercise 7, page 346

Due Wednesday May 12 in written form

Sketches of solution can be found here

Day 20, Friday May 7

Final day for dropping a fourth course without a grade notation of "W" 

Section 6.2.2 and Section 6.2.5

Exercise 2  on page 365

Exercise 3 on page 365 (note that the domain Ω in this problem is a domain in Rn)

Due Wednesday May 12 in written form

Sketches of solution can be found here

Day 21, Monday May 10

Section 6.2.2 and Section 6.2.3

Page 352: Learn and understand the proofs of statements (iii), (iv), (v), and the proof of how to get  the vector form of the divergence theorem from the other one

Exercise 10 on page 366 (do not do the physical interpretation)

Exercise 13 on page 367

Due Wednesday May 19 in written form

Problems and proofs of this type may appear on the second Midterm.

Sketches of solution can be found here

Day 22, Wednesday May 12

Sections 6.2.4, 6.3.1, and 6.3.2

Exercise 4, page 372

Exercise 5, page 372

Exercise 6, page 372

Due Wednesday May 19 in written form

Sketches of solution can be found here

Thursday May 13

Second Midterm Exam

6-8 PM in Carpenter 013

 

 

Friday May 14

No class. We shall have an x-hour on Thursday May 20

instead

 

 

Day 23, Monday May 17

Note that Tuesday May 18 is the final day to withdraw from a course

Section 6.3.2 and 6.4.1

In the following three problems Ω=[0,π]× [0,π]

 

Exercise 1: Solve the equation –Δu=1 for x in Ω; and u=0 for x in boundary of  Ω;

 

Exercise 2:  Solve the equation

–Δu=3sin(4x)sin(3y) for x in Ω; and u=0 for x in boundary of  Ω;

 

Exercise 3: Find the eigen values and the eigen functions corresponding to the homogeneous Neumann problem: –Δu=λu for x in Ω; and du/dn=0 for x in boundary of  Ω;

 

Due Wednesday May 26 in written form

 Sketches of solution can be found here

Day 24, Wednesday May 19

Section 6.4.1 and Section 6.4.2

Exercise 1 part a, page 381

Exercise 1 part b, page 382

These two parts will be counted as separate exercises when the homework will be graded.

Due Wednesday May 26 in written form

Sketches of solution can be found here

Day 25, Thursday May 20

x-hour instead of the class on Friday May 14

Section 6.4.2

No homework

Day 26, Friday May 21

Section 6.5.2

Exercise 5, page 396

Exercise 6, page 396

You may assume that the constant a in problem 5 is positive, since otherwise the integral in the definition of the Fourier transform will not converge.

Due Wednesday May 26 in written form

Sketches of solution can be found here

Day 27, Monday May 24

Section 6.5.2

Exercise 7, page 396

Hint: in this exercise you may want to use the completion of the square technique used in deriving the formula for the solution of the quadratic equation. You also may want to use Exercise 5 part b on page 396.

Exercise 11, page 397

Exercise 13, page 397

Due Wednesday June 2 in written form

Sketches of solution can be found here

Day 28, Wednesday May 26

Section 6.5.2

Exercise 8, page 397 (In this exercise you do not have to compute the inverse Fourier transform.)

Exercise 9, part b page 397

Exercise 9, part c page 397

The two parts of exercise 9 will be counted as separate exercises when graded.

To find the inverse Fourier transform in Exercise 9.c one has to introduce a convolution (u*v)(x) of two functions u,v:Rn→R, via the formula

(u*v)(x)=∫u(y)v(x-y)dy, where the integral is taken over Rn. Then one shows that

F(u*v)(ζ)=(F(u)(ζ))×(F(v)(ζ)),

similarly to Theorem 6.32

Due Wednesday June 2 in written form

Sketches of solution can be found here

Day 29, Friday May 28

Section 6.5.2 and sections 6.7.1, 6.7.2

Exercise1: Part A.

Find the second distributional derivative

of the  distributional Fourier transform of δ(x-x0).

Part B. Find the distributional Fourier transform of the second distributional derivative of δ(x-x0).

 

Exercise 2: Use the Laplace transform to solve the given initial value problem:

y(4)(x)-4y(x)=0 the initial conditions are: y(0)=1, y’(0)=0, y’’(0)=-2, y’’’(0)=0. You may want to use the method of partial fractions and table on page 384 to help you find the inverse Laplace transforms.

 

Exercise 1, page 415

 

Exercise 2, page 415

 

Due Wednesday June 2 in written form

Sketches of solution can be found here

Monday May 31

Memorial Day, No Class

 

 

Day 30, Wednesday June 2

Last day of classes

Section 6.7.2 and section 6.7.3

No homework

Friday June 4

Final Exam

3-6 PM in Kemeny 105