- I profusely apologize for not linking to this notebook earlier. My full intention was to help you with the Mathematica part of the assignment more. I wrote a notebook with some solutions from Section 5.8 but I forgot to link to that notebook earlier. Here it is now. I even made a pdf printout of this notebook to help you even more. Again, my apologies.
- Here is the Mathematica notebook which I used today. The drum animation that posted yesterday was produced using this notebook.
- The above notebook is based on Section 7.7. Suggested exercises for this section are 7.7.1, 7.7.3, 7.7.5, 7.7.6 (b),
- In the above notebook we derived the formulas for normal modes of vibrations of a circular drum or radius 1 and the constant $c = 1$. Below I show vibrations of 30 normal modes with $m=0,1,2,3,4,5$ and $n=1,2,3,4,5$. I will post the formulas for individual modes later.
- In the modes below we have $m=0,1,2,3,4,5$ and $n=1$.
- In the modes below we have $m=0,1,2,3,4,5$ and $n=2$.
- In the modes below we have $m=0,1,2,3,4,5$ and $n=3$.
- In the modes below we have $m=0,1,2,3,4,5$ and $n=4$.
- In the modes below we have $m=0,1,2,3,4,5$ and $n=5$.
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The initial velocity of the drum below is given by the following polynomial equation:
\[
g(r,\theta,0) = 2(1-r) \exp\bigl(-100(r-.5)^2 \bigr) \, \sin\bigl((\pi/2)\exp\bigl(-50(\theta-\pi)^2 \bigr) \bigr)
\]
Place the cursor over the image to start vibrations.
- Yesterday we covered Section 7.3. Do the exercises: 7.3.1 (a), (b), (c), 7.3.3, 7.3.4, 7.3.5, 7.3.6 (b).
- Today we started Section 7.7.
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Yesterday I posted membranes whose initial shapes or velocities were not symmetric. I thought that would result in interesting unpredictable vibrations. However, vibrations of membranes with symmetric initial shape are also interesting. The initial shape of the membrane below is given by the following polynomial equation:
\[
u(x,y,0) = \bigl(16 x y (1 - x) (1 - y)\bigr)^{10}, \quad 0\leq x \leq 1, \quad 0\leq y \leq 1.
\]
Place the cursor over the image to start vibrations.
- In this Mathematica notebook I illustrate vibrations of a rectangular membrane. Three animations that follow below are produced using this notebook.
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The initial shape of the membrane below is given by the following polynomial equation:
\[
u(x,y,0) = 10 x y (1 - x) (1 - y)^2, \quad 0\leq x \leq 1, \quad 0\leq y \leq 1.
\]
Place the cursor over the image to start vibrations.
-
The initial shape of the membrane below is given by a slightly different polynomial equation:
\[
u(x,y,0) = 10 x y (1 - x)^2 (1 - y)^2, \quad 0\leq x \leq 1, \quad 0\leq y \leq 1.
\]
Place the cursor over the image to start vibrations.
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The initial velocity is given to the membrane below the membrane being hit from below by a baton.
Place the cursor over the image to start vibrations.
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Here is an animation of a solution of the Problem 4 on the assignment. I hope that this animation clarifies the statement of the problem. To solve the problem you should state an ODE eigenvalue problem. However, in this case you have to be somewhat creative. For example, instead of just functions you can consider pairs of functions, that is vector valued functions. After you setup the problem you need to find the eigenvalues. It is interesting that the problem has infinitely many (but not all) eigenvalues that can be calculated explicitly. At first I missed those and did not get my approximation right. Also, you will have to "invent" the orthogonality relation among eigenfunctions. It is a sort of natural. When choosing your fundamental system of solutions you have many choices. For example functions $\cos(\mu (x-5)), \sin(\mu (x-5))$ might be a good choice for a part of a solution.
Place the cursor over the image to start vibrations.
Notice that the red part of the string is rigid, while the blue part is governed by the vibrating string equation.
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This is my interpretation of, what the book calls in Section 5.8, the "physical boundary conditions of the third kind".
Place the cursor over the image to start vibrations.
Notice that the red part of the string is rigid, while the blue part is governed by the vibrating string equation.
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This is my interpretation of, what the book calls in Section 5.8, the "non-physical boundary conditions of the third kind". However, I am not sure that the adjective non-physical is appropriate here. As you can see the string breaks here.
Place the cursor over the image to start vibrations.
In the above animation one end of a blue string is attached to an end a rigid red bar which is free to move up and down. The other end of the bar is fixed. Mathematically the red bar establishes a relation between the position of the end of the spring and the slope of the string at that end.
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The setting shown in this item is the same as in the previous animation. This string does not break since the initial shape of the string is orthogonal to the negative eigenfunction.
Place the cursor over the image to start vibrations.
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The setting shown below is also termed "non-physical" in the book. However, in this case there are no negative eigenvalues. Consequently, for an arbitrary initial shape of the string the string keeps oscillating without breaking.
Place the cursor over the image to start vibrations.
- Today in class I handed out the assignment which is due on Friday, November 30, 2012.
- Section 4.2: 4.2.1, 4.2.2.
- Section 4.3: 4.3.1, 4.3.2.
- Section 4.4: 4.4.1, 4.4.3, 4.4.4, 4.4.6, 4.4,7, 4.4.8.
- This is the Mathematica notebook that I used on Friday. It has a Manipulate command which explores natural modes of vibration.
- Section 5.8: 5.8.1, 5.8.2, 5.8.3, 5.8.4, 5.8.5, 5.8.6, 5.8.8, 5.8.9, 5.8.12.
- Section 3.4: 3.4.5, 3.4.6, 3.4.9, 3.4.11, 3.4.12, 3.4.13.
- In this Mathematica notebook I solved some exercises from Chapter 3 in Mathematica. Here is the pdf printout of the same notebook. This pdf file is not suitable for printing.
- Section 3.3: the assigned problems are 3.3.1 (do few), 3.3.2 (a), (d), 3.3.3, 3.3.4, 3.3.5 (a), 3.3.6 (a), (b), (c) (there is a mistake in (c): the values 0 and 1 of the function should be swapped), 3.3.7, 3.3.8, 3.3.9, 3.3.10, 3.3.12, 3.3.13, 3.3.14, 3.3.15, 3.3.17 (it is more interesting to deduce the same formula using Fourier series), 3.3.18.
- In this Mathematica notebook I solved some exercises from Chapter 3 in Mathematica. You can use this notebook to see the code that I used and to check your calculations. Here is the pdf printout of the same notebook. This pdf file is not suitable for printing.
- As I demonstrated in class you should be able to find basic Fourier series "by hand". In this Mathematica notebook I did some basic series in Mathematica, just to check calculations. Here is the pdf printout of the same notebook.
- In this Mathematica notebook I solved some exercises from Chapter 3 in Mathematica, just to check calculations. Here is the pdf printout of the same notebook.
- Section 3.2: the assigned problems are 3.2.1, 3.2.2, 3.2.3, 3.2.4.
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Recall the figure from yesterday in which I illustrated a periodic extension of a function defined on an interval.
In the figure below the function $f$ is the restriction of the function $x \mapsto x$ (in blue) to the interval $(1,3]$. The red function is the periodic extension.
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There are several drawbacks with the plot in the figure above. First, the blue plot is NOT the function $f$. The function $f$ is the restriction of the blue plot to the interval $(1,3]$. Second, the values of the red function, the extension, at the jumps are not clear. I show the function $f$ (in blue) and its periodic extension (in red) with the complete information in two separate plots below.
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This brings us to a new concept: the Fourier periodic extension of a piecewise continuous function. It is not difficult to see that a periodic extension of a piecewise continuous function (pwc) is itself piecewise continuous. The Fourier periodic extension of a pwc function is a modification of its periodic extension. The modification is done at the jumps: the value of the Fourier periodic extension at a jump $y$ of $\tilde{f}$ is
\[
\frac{1}{2} \bigl(\tilde{f}(y+)+\tilde{f}(y-)\bigr)
\]
The concept of the Fourier periodic extension is not explicitly defined in our textbook. In fact most textbooks do not define this concept, although it is implicitly present in each. Since there is no standard notation for the Fourier periodic extension, I will denote it by $\tilde{f}_{\!\!\rm Fourier}$. Here is the formal definition:
- Let $a$ and $b$ be real numbers such that $a \lt b$ and let $f:[a,b] \to \mathbb R$ be a piecewise continuous function. Let $\tilde{f}:{\mathbb R} \to {\mathbb R}$ be the periodic extension of $f$. The Fourier periodic extension of $f$ is the following function \[ \tilde{f}_{\!\!\rm Fourier}(x) = \begin{cases} \tilde{f}(x) & \text{if $\tilde{f}$ is continuous at $x$} \\[10pt] \tfrac{1}{2}\!\! \bigl(\tilde{f}(x+)+\tilde{f}(x-)\bigr) & \text{if $\tilde{f}$ is not continuous at $x$} \end{cases}, \qquad x\in\mathbb R. \]
- You can find several examples of piecewise smooth functions, their periodic extensions and their Fourier periodic extensions on this webpage.
-
Unfortunately the book does not have formal definitions of the concepts that we are studying in Chapter 3. Therefore, I give few formal definitions here.
-
Let $a$ and $b$ be real numbers such that $a \lt b$. A function $f:[a,b] \to \mathbb R$ is said to be piecewise continuous on $[a,b]$ if the following conditions are satisfied:
- there exists a finite set $\{x_1,\ldots,x_n\} \subset (a,b)$ such that $x_1 \lt \cdots \lt x_n$ and $f$ is continuous on each interval \[ (a,x_1), \quad (x_k,x_{k+1}), \ k=1,\ldots,n-1, \quad (x_n,b); \]
- all the following one-sided limits exist \[ \lim_{x\downarrow a} f(x), \quad \lim_{x\uparrow x_k} f(x), \quad \lim_{x\downarrow x_k} f(x), \ k=1,\ldots,n, \quad \lim_{x\uparrow b} f(x). \]
-
Let $a$ and $b$ be real numbers such that $a \lt b$. A function $f:[a,b] \to \mathbb R$ is said to be piecewise smooth on $[a,b]$ if the following conditions are satisfied:
- there exists a finite set $\{x_1,\ldots,x_n\} \subset (a,b)$ such that $x_1 \lt \cdots \lt x_n$ and $f$ is continuous and it has a continuous derivative $f'$ on each interval \[ (a,x_1), \quad (x_k,x_{k+1}), \ k=1,\ldots,n-1, \quad (x_n,b); \]
- all the following one-sided limits exist \[ \lim_{x\downarrow a} f(x), \quad \lim_{x\uparrow x_k} f(x), \quad \lim_{x\downarrow x_k} f(x), \ k=1,\ldots,n, \quad \lim_{x\uparrow b} f(x); \]
- all the following one-sided limits exist \[ \lim_{x\downarrow a} f'(x), \quad \lim_{x\uparrow x_k} f'(x), \quad \lim_{x\downarrow x_k} f'(x), \ k=1,\cdots,n, \quad \lim_{x\uparrow b} f'(x). \]
- Let $a$ and $b$ be real numbers such that $a \lt b$ and let $f:(a,b] \to \mathbb R$ be a function. The function $\tilde{f}: {\mathbb R} \to {\mathbb R}$ defined by \[ \tilde{f}(x) = f\left(\!x- \bigl(\bigl\lceil\!\tfrac{x-a}{b-a}\! \bigr\rceil -1 \bigr)(b-a)\!\right), \quad x \in {\mathbb R}. \] is called the periodic extension of $f$.
-
Let $a$ and $b$ be real numbers such that $a \lt b$. A function $f:[a,b] \to \mathbb R$ is said to be piecewise continuous on $[a,b]$ if the following conditions are satisfied:
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I do understand that the last definition might look somewhat weird. The only reason for that is that the ceiling function is almost completely absent from our curriculum. That is the fault of our curriculum. The book gives a descriptive definition in English of the concept of a periodic extension. The above formula involving the ceiling function is the only way that I was able to translate the definition from English into Mathish. The figures below illustrate the definition with some simple functions $f$. Here is the
Mathematica notebook which I used to produce these figures.
In the figure below the function $f$ is the restriction of the function $x \mapsto x$ (in blue) to the interval $(1,3]$. The red function is the periodic extension.
In the figure below the function $f$ is the restriction of the function $x \mapsto x^2-2$ (in blue) to the interval $(-2,2]$. The red function is the periodic extension.
- We finished Section 2.5 today. We skipped Subsection 2.5.3 and the last part of Subsection 2.5.4 Solvability condition.
- A Mathematica implementation of the solution of Laplace's equation in a disk is in this Mathematica notebook.
- The assigned Exercises for Section 2.5 are 2.5.1 (b), 2.5.3 (a), (b), 2.5.5 (a), (d), 2.5.6 (a), (b), 2.5.7 (a), (b), 2.5.8 (a), 2.5.14.
- I wrote a page with a solution of the Laplace equation in a rectangle. An implementation of this solution is in this Mathematica notebook.
- Your weekend homework is to read Section 2.4 The assigned Exercises for this section are 2.4.1 (a), (b), (c), 2.4.2, 2.4.3, 2.4.4, 2.4.6, 2.4.7.
- In addition to the initial conditions above doe 2.4.1 with the initial condition $\Bigl(\cos\bigl(\pi x/L\bigr)\Bigr)^6$. You can find the exact solution of this problem. The formula for the sixth power of cosine in terms of cosines of different frequencies is obtained in Exercise 17 in Section 4.7 of the third edition of Lay's linear algebra book which we use in Math 204 and 304. For my 304 I wrote a web-page about Chebyshev polynomials. At the end of this notebook there are formulas connecting powers of cosines to cosines of different frequencies that might clarify what is going on here. The whole web-page might be of interest to you since we will encounter Legendre polynomials later on in this class.
- Today in class I used Mathematica notebook with several solutions of the heat equation with Dirichlet boundary conditions.
- Today we discussed the separation of variables method Section 2.3. The assigned homework are Exercises 2.3.1 (a), (c), (d), (f), 2.3.2 (a), (b), (c), (d), (g), 2.3.3 (a), (b), (c), 2.3.4, 2.3.5, 2.3.6, 2.3.7, 2.3.8, 2.3.9, 2.3.11.
- Yesterday we did Section 2.2. The assigned homework are Exercises 2.2.3, 2.2.4.
- It is my impression that there are no differences in different editions in Section 2.3. Here is Chapter 2 of the 3rd edition if you want to compare.
- Here is a derivation of the formula for the Laplacian in polar coordinates.
- Today we discussed derivation of the heat equation in three variables from Section 1.5. The assigned homework are Exercises 1.5.1, 1.5.2, 1.5.5, 1.5.8 through 1.5.13. Some of these exercises relate to the Laplacian in polar and spherical coordinates that we will do tomorrow.
- Today we discussed the equilibrium temperature distribution. This is covered in Section 1.4. The assigned homework are Exercises 1.4.1, 1.4.2, 1.4.3, 1.4.4, 1.4.5, 1.4.6, 1.4.7.
-
In addition to the exercises above do Exercises 1.4.12 and 1.4.13 from the 4th edition.
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1.4.12: Suppose the concentration $u(x,t)$ of a chemical satisfies Fick's law and the initial concentration is given $u(x,0) = f(x)$. Consider a region $0 \leq x \leq L$ in which the flow is specified at both ends: $-k\frac{\partial u}{\partial x}(0,t) = \alpha$ and $-k\frac{\partial u}{\partial x}(L,t) = \beta$. Assume that $\alpha$ and $\beta$ are constants.
- Express the conservation law for the entire region.
- Determine the total amount of chemical in the region as a function of time (using the initial condition).
- Under which conditions is there an equilibrium chemical concentration and what is it?
- 1.4.13: Do Exercise 1.4.12 if $\alpha$ and $\beta$ are functions of time.
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1.4.12: Suppose the concentration $u(x,t)$ of a chemical satisfies Fick's law and the initial concentration is given $u(x,0) = f(x)$. Consider a region $0 \leq x \leq L$ in which the flow is specified at both ends: $-k\frac{\partial u}{\partial x}(0,t) = \alpha$ and $-k\frac{\partial u}{\partial x}(L,t) = \beta$. Assume that $\alpha$ and $\beta$ are constants.
- Here is the notebook that I used in class today.
- I have now posted a derivation of the one dimensional heat equation here. I should still add accompanying picture to make it clear what the quantities represent.
- Today we discussed different kinds of boundary conditions that can naturally arise in the setting of the heat equation. This is covered in Section 1.3. The assigned homework are Exercises 1.3.1, 1.3.2, 1.3.3. In Exercise 1.3.3 apply the conservation of heat energy law to the bath.
- It is my impression that there are no differences in different editions in Section 1.3. Here is Chapter 1 of the 3rd edition if you want to compare.
- I derived the diffusion equation today. Here are notes of that derivation. The 3rd edition derives only the heat equation in Section 1.2. The 4th edition derives the diffusion equation at the end of Section 1.2.
- Tomorrow I will finish the derivation of the heat equation.
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For homework do the following two exercises:
- Consider a rod of variable cross-sectional area $A(x), 0 \lt x \lt L$. Assume that all thermal quantities are constant across a section, the rod is made of a uniform material and that the rod is well insulated so that no heat energy can pass through the lateral surface. Assume no sources of heat energy in the rod. Derive the heat equation for this rod. (This is my version of Exercise 1.2.3 in the 4th ed.)
- Assume that we have an object whose mass is one unit of mass and whose specific heat depends on its temperature. Denote by $c(\tau)$ the specific heat of this object at temperature $\tau$. What is the heat energy of this object when it is at the temperature $T$?
- Exercises 1.2.3 and 1.2.4 (3rd ed.); in 4th ed. these are Exercises 1.2.8 and 1.2.9.
- Today in class I demonstrated how to use Mathematica v8 to implement and illustrate the Method of characteristics. Here is the notebooks that I used in class. The file is called Example_Medhod_of_Chara.nb. Right-click on the underlined word "Here"; in the pop-up menu that appears, your browser will offer you to save the file in your directory. Make sure that you save it with the exactly same name "Example_Medhod_of_Chara.nb". After saving the file you can open it with Mathematica v8. You need to find a campus computer with Mathematica v8 installed on it (for example BH 215).
- More information on how to use Mathematica you can find on my Mathematica page.
- If you have problems running files that I posted please let me know. If you spend some time learning how to use Mathematica you will enhance your understanding of math that we are studying.
- We also have Mathematica v5.2. Mathematica v5.2 is more widely available on campus but it will not run this file. The command structure of Mathematica v5.2 is very similar and I can produce a version that will run in v5.2 if there is enough interest. I decided to use v8 in this class since I will ask you to do assignments in Mathematica and it is easier to find information about how to use v8.
- The information sheet
- We will start with the review of Section 12.2.2 Method of Characteristics for First-Order PDEs. Suggested problems are 2, 3, 4, 5, 6, 7, 8 in Exercises 12.2.
- For those who are waiting for their books to arrive here is the section we are covering now.
- For those of you who got the book on loan please take an exceptionally good care of the book. For example you can make your own book cover as shown in this video. If the video starts with a political ad, please do not make you voting choices based on propaganda. And, by all means, register to vote and vote based on information you gather yourself not on propaganda.
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Our next topic is the heat (or diffusion) equation. The animation below is obtained by solving that equation.
Place the cursor over the image to see the diffusion of the dye.
