- We did not cover Chapter 7 in detail. Here I present implementation of the theory from Chapter 7 in Mathematica.
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This item is based on Section 7.3.
- In this Mathematica notebook I illustrate vibrations of a rectangular membrane. Three animations that follow below are produced using this notebook. One needs to be patient with this notebook. Manipulations with complicated 3D functions do not perform satisfactory. Therefore, I exported Mathematica images into the animated gifs which show more realistic vibrations.
-
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; in fact the membrane has been hit from below by a baton.
Place the cursor over the image to start vibrations.
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Above 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.
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This item is based on Sections 7.7 and 7.8. We study a circular drum of radius 1.
- Here is the Mathematica notebook which I used to produce the animations below. Here is a pdf file of this Mathematica notebook so that you can view it without access to Mathematica.
- In the above notebook we derived the formulas for normal modes of vibrations of a circular drum with the radius 1 and the constant $c = 1$. Below I show vibrations of 30 normal modes with $m=0,1,2,3,4,5$ and $n=1,2,3,4,5$.
- 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 $n=5$.
-
The initial velocity of the drum below is given by the following 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.
-
Here we study a circular drum with a hole in the middle. It is a drum of radius 3 and with a hole of radius 1.
- Below I show vibrations of 30 normal modes with $m=0,1,2,3,4,5$ and $n=1,2,3,4,5$ of the drum with radius 3 and with a hole of radius 1.
- 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 $n=5$.
-
The initial velocity of the drum below is given by the following polynomial equation:
\[
g(r,\theta,0) = (r-1)(3-r) \exp\bigl(-20(r-2)^2 \bigr) \, \sin\bigl((\pi/2)\exp\bigl(-30(\theta-\pi)^2 \bigr) \bigr)
\]
Place the cursor over the image to start vibrations.
-
Here is a short summary of topics that could appear on the exam:
- Know geometric meaning of the method of characteristics and how to apply it to solve quasi-linear first order PDEs,
- Know the physical principles used in derivation and be able to derive one-dimensional diffusion and heat equation. Understand the physical meaning of the boundary conditions.
- Know how to find the equilibrium temperature distribution of non-homogeneous one-dimensional heat equation with uniform and non-uniform thermal properties, Section 1.4.
- Know how to apply the method of separation of variables to solve one-dimensional heat equation with Dirichlet, Neumann and periodic boundary conditions, Sections 2.1, 2.2, 2.3 and 2.4. Know the relation of these problems to Fourier series.
- Know how to apply the method of separation of variables to solve Laplace's equation in rectangular and circular regions. Section 2.5 and problems assigned there.
- Fourier series: You should know the statements of the convergence theorems. You should know how to calculate a Fourier series of simple functions, like those that we did in class. You should know conditions under which a Fourier series can be differentiated term-by-term and calculations related to this process.
- You should know the Method of eigenfunction expansion, p. 118, problems 3.4.11, 3.4.12.
- You should be able to derive the vibrating string equation and understand the meaning of related boundary conditions.
- Understand the problem solved in Section 5.8 and the corresponding problem solved in class. Based on the discussion in Section 5.8 you should be able to determine graphically eigenvalues of simple Sturm-Liouville problems like in Problems 5.8.3 (hint: part (a) should be done by solving the differential equation), 5.8.4, 5.8.8 (hint: (a) should be done by solving the differential equation), 5.8.9.
- Know separation of variables for two-dimensional heat and wave equation, Sections 7.2 and 7.3. Exercises: 7.3.1 (a), (b), (c), 7.3.3, 7.3.4, 7.3.5, 7.3.6 (b).
-
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 eigenfunctions corresponding to the negative eigenvalue.
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.
- These animations were created in this Mathematica notebook. Recall that I save Mathematica notebooks with all output deleted. To recreate the output, go to the Menu item Evaluation and choose Evaluate Notebook. The shortcut is Alt+v o. The notebook should evaluate in under one minute. Here is a pdf file of this notebook. You can peek at the code without having Mathematica.
-
An interesting situation arises when a part of a string is rigid (dipped in super-glue, for example).
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.
-
Here is a guide for Chapter 5:
- Read Section 5.1 and Section 5.2.
- Read Section 5.3. Pay special attention to Boundary conditions and 5.3.2: Regular Sturm-Liouville problems, in particular Statements of theorems (skip Rayleigh quotient). Read 5.3.3.
- Skip Section 5.4.
- We covered Section 5.5 in detail. Pay special attention to the integral form of Lagrange's identity (book calls it Green's formula). My approach to this part will differ from the approach in the book. Notice that I will give a simple way of answering 5.5.1. Suggested problems: 5.5.1 and 5.5.4.
- Read Appendix to 5.5. This was covered in Math 304. It is good to read it for comparison.
- Skip Section 5.6 and Section 5.7.
- We will cover Section 5.8 in detail. Recommended problems: 5.8.1, 5.8.2, 5.8.3 in (a) ignore the comment in parenthesis and ignore question (c), 5.8.4, 5.8.5, 5.8.6, 5.8.8 solve (a) directly, 5.8.9, 5.8.11.
- Skip Section 5.9 and Section 5.10.
- I wrote a webpage on Symmetry of Sturm-Liouville eigenvalue problems
- Below is a list of problems for Chapter 4.
- 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.
- Let $L \gt 0$ and let $f:[-L,L]\to \mathbb R$ be a continuous piecewise smooth function. If \[ f(x) = a_0 + \sum_{k =1}^{+\infty} \biggl( a_k \cos\Bigl(\!\tfrac{k \pi}{L} x\!\Bigr) + b_k \sin\Bigl(\!\tfrac{k \pi}{L} x\!\Bigr) \biggr), \qquad \forall x \in (-L,L), \] then \[ a_0 = \frac{1}{2L} \int_{-L}^L f(\xi) d\xi \] and for all $k \in {\mathbb N}$, \[ a_k = \frac{1}{L}\int_{-L}^L f(\xi) \cos\Bigl(\!\tfrac{k \pi}{L} \xi\!\Bigr) d\xi, \quad b_k = \frac{1}{L}\int_{-L}^L f(\xi) \sin\Bigl(\!\tfrac{k \pi}{L} \xi\!\Bigr) d\xi. \] Similarly if $f:[0,L]\to \mathbb R$ is a continuous piecewise smooth function and if \[ f(x) = a_0 + \sum_{k =1}^{+\infty} a_k \cos\Bigl(\!\tfrac{k \pi}{L} x\!\Bigr) , \qquad \forall x \in [0,L], \] then \[ a_0 = \frac{1}{L} \int_{0}^L f(\xi) d\xi \] and for all $k \in {\mathbb N}$, \[ a_k = \frac{2}{L}\int_{0}^L f(\xi) \cos\Bigl(\!\tfrac{k \pi}{L} \xi\!\Bigr) d\xi. \] Similarly, if $f:[0,L]\to \mathbb R$ is a continuous piecewise smooth function and if \[ f(x) = \sum_{k =1}^{+\infty} b_k \sin\Bigl(\!\tfrac{k \pi}{L} x\!\Bigr), \qquad \forall x \in (0,L), \] then for all $k \in {\mathbb N}$, \[ b_k = \frac{2}{L}\int_{0}^L f(\xi) \sin\Bigl(\!\tfrac{k \pi}{L} \xi\!\Bigr) d\xi. \]
- The reasoning in the previous item applies even if $f$ is a function of $x$ and $t$. In that case the coefficients $a_k$ and $b_k$ are functions of $t$. This is used in the subsection Method of eigenfunction expension (Section 3.4) to solve some nonhomogeneous equations.
- 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.
- I created this Mathematica file in class today.
-
We first prove a Differentiation term-by-term Theorem for Fourier series.
Differentiation term-by-term Theorem. Let $L \gt 0$ and let $f:[-L,L] \to \mathbb R$ be a function. Assume that
- $f$ is continuous on the closed interval $[-L,L],$
- $f$ is piecewise smooth function on $[-L,L],$
- $f'$ is piecewise smooth function on $[-L,L],$
To toggle the proof of the Differentiation term-by-term Theorem clickSince $f'$ is piecewise smooth its Fourier series converges pointwise to the Fourier periodic extension of $f'.$ Denote the coefficients of this Fourier series by $A_0,$ $A_k$ and $B_k.$ Let us calculate $A_0.$ Since $f$ is continuous and $f'$ is piecewise continuous the Fundamental Theorem of Calculus applies and it yields \[ A_0 = \frac{1}{2L} \int_{-L}^L f'(x) dx = \frac{1}{2L} \bigl( f(L) - f(-L) \bigr). \] Let us calculate $A_k$. Since $f(x)\cos\bigl(k\pi x/L\bigr)$ is continuous and $f'$ is piecewise continuous the Integration by Parts Theorem applies and it yields \begin{align*} A_k & = \frac{1}{L} \int_{-L}^L f'(x) \cos\left(\frac{k \pi}{L} x\right) dx \\ & = \frac{1}{L} \left( f(L) \cos\bigl(k\pi\bigr) - f(-L) \cos\bigl(-k\pi\bigr) + \frac{k \pi}{L} \int_{-L}^L f(x) \sin\left(\frac{k \pi}{L} x\right) dx \right) \\ & = \frac{1}{L} \left( (-1)^k \bigl( f(L) - f(-L) \bigr) + \frac{k \pi}{L} \int_{-L}^L f(x) \sin\left(\frac{k \pi}{L} x\right) dx \right) \\ & = \frac{1}{L} \left( (-1)^k \bigl( f(L) - f(-L) \bigr) + k \pi \, b_k \right) \\ \end{align*} Let us calculate $B_k$. Since $f(x)\sin\bigl(k\pi x/L\bigr)$ is continuous and $f'$ is piecewise continuous the Integration by Parts Theorem applies and it yields \begin{align*} B_k & = \frac{1}{L} \int_{-L}^L f'(x) \sin\left(\frac{k \pi}{L} x\right) dx \\ & = \frac{1}{L} \left( f(L) \sin\bigl(k\pi\bigr) - f(-L) \sin\bigl(-k\pi\bigr) - \frac{k \pi}{L} \int_{-L}^L f(x) \cos\left(\frac{k \pi}{L} x\right) dx \right) \\ & = \frac{1}{L} \left( - k \pi \, a_k \right) \end{align*} -
The book proves the following Differentiation term-by-term Theorem for Fourier sine series.
Differentiation term-by-term Theorem. Let $L \gt 0$ and let $f:[0,L] \to \mathbb R$ be a function. Assume that
- $f$ is continuous on the closed interval $[0,L],$
- $f$ is piecewise smooth function on $[0,L],$
- $f'$ is piecewise smooth function on $[0,L].$
- Applying the above theorems one can calculate the coefficients of the Fourier series for functions whose derivatives and integrals replicate itself. For example for function $e^x$ or $\cosh(x),$ or an even extension of $\sin(x)$ and similar.
-
First I want to emphasize the concept of Fourier periodic extension of a function.
Recall a function from yesterday: the restriction of the function $x \to x$ to the interval $[1,4).$ A detailed graph of this function is below.
In the figure below the function $f$ is the restriction of the function $x \mapsto x$ (in blue) to the interval $[1,4).$
-
It is clear that the above function has a periodic extension with period $3.$ Below I show the given function in blue and its periodic extension in red. Since the periodic extension has jumps, I emphasize its values at the jumps.
-
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 is itself piecewise continuous. The Fourier periodic extension of a piecewise continuous function function is a modification of the periodic extension. The modification is done at the jumps: the value of the Fourier periodic extension at a jump $x$ of $\tilde{f}$ is
\[
\frac{1}{2} \bigl(\tilde{f}(x+)+\tilde{f}(x-)\bigr)
\]
The concept of the Fourier periodic extension is not explicitly defined in our textbook. In fact most textbooks on this topic 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 \[ \tilde{f}_{\!\!\rm Fourier}(x) = \begin{cases} \tilde{f}(x) & \text{if $\tilde{f}$ is continuous at $x$} \\[10pt] \tfrac{1}{2}\!\! \bigl(\tilde{f}(x+)+\tilde{f}(x-)\bigr) & \text{if $\tilde{f}$ is not continuous at $x$} \end{cases}, \qquad x\in\mathbb R. \]
-
One more example. The given function is in blue, its periodic extension in red and its Fourier periodic extension in green.
- You can find several examples of piecewise smooth functions, their periodic extensions and their Fourier periodic extensions on this webpage.
-
Here I present two important convergence theorems for Fourier series.
- Let $L \gt 0$ and let $f:[-L, L] \to \mathbb R$ be a piecewise continuous function. The series \[ a_0 + \sum_{k=1}^{+\infty} \biggl( a_k \cos\Bigl(\!\tfrac{k \pi}{L} x\!\Bigr) + b_k \sin\Bigl(\!\tfrac{k \pi}{L} x\!\Bigr) \biggr) \] where \[ a_0 = \frac{1}{2L} \int_{-L}^L f(\xi) d\xi \] and, for $k \in {\mathbb N}$, \[ a_k = \frac{1}{L}\int_{-L}^L f(\xi) \cos\Bigl(\!\tfrac{k \pi}{L} \xi\!\Bigr) d\xi, \quad b_k = \frac{1}{L}\int_{-L}^L f(\xi) \sin\Bigl(\!\tfrac{k \pi}{L} \xi\!\Bigr) d\xi, \] is called the Fourier series of $f$.
- For $n \in {\mathbb N}$, the $n$th partial sum of the Fourier series of $f$ is \[ S_n^f(x) = a_0 + \sum_{k=1}^{n} \biggl( a_k \cos\Bigl(\!\tfrac{k \pi}{L} x\!\Bigr) + b_k \sin\Bigl(\!\tfrac{k \pi}{L} x\!\Bigr) \biggr) \]
- Pointwise Convergence Theorem. If $f$ is piecewise smooth on $[-L,L]$, then for every $x \in {\mathbb R}$ we have \[ \lim_{n \to +\infty} S_n^f(x) = \tilde{f}_{\!\!\rm Fourier}(x). \] Loosely speaking, the Fourier series of $f$ converges pointwise to the Fourier periodic extension of $f$.
- Notice that if the periodic extension of $f$ is a continuous function, then the Fourier periodic extension of $f$ coincides with the periodic extension of $f$. In other words, if $\tilde{f}$ is a continuous function, then $\tilde{f}_{\!\!\rm Fourier} = \tilde{f}$.
- Uniform Convergence Theorem. If $f$ is piecewise smooth and the periodic extension of $f$ is continuous, then the sequence of functions $\bigl\{S_n^f\bigr\}_{n=1}^{+\infty}$ converges uniformly on ${\mathbb R}$ to $\tilde{f}$. This means that for every $\varepsilon > 0$ there exists $N_\epsilon$ such that for all $n \gt N_\varepsilon$ and for all $x \in {\mathbb R}$ we have \[ \bigl|S_n^f(x) - \tilde{f}(x) \bigr| < \varepsilon. \]
- Section 3.2: the assigned problems are 3.2.1, 3.2.2, 3.2.3, 3.2.4.
- Our next topic is 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.
-
The book does not have formal definitions of the concepts that are studied 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 \[ \tilde{f}(x) = f\left(\!x- \Bigl(\left\lceil\!\tfrac{x-b}{b-a}\! \right\rceil \Bigr)(b-a)\!\right), \quad x \in {\mathbb R}. \] is called the periodic extension of $f$.
- Let $a$ and $b$ be real numbers such that $a \lt b$ and let $f:[a,b) \to \mathbb R$ be a function. The function $\tilde{f}: {\mathbb R} \to {\mathbb R}$ defined by \[ \tilde{f}(x) = f\left(\!x- \left\lfloor\!\tfrac{x-a}{b-a}\! \right\rfloor (b-a)\!\right), \quad x \in {\mathbb R}. \] is called the periodic extension of $f.$
- The reasoning for the above definition of the periodic extension is as follows. For every $x\in\mathbb R$ there exists unique $k \in \mathbb Z$ such that \[ a + k (b-a) \leq x \lt a + (k+1)(b-a). \] The last expression is equivalent to \[ k \leq \frac{x - a}{b-a} \lt k+1. \] Therefore \[ k = \left\lfloor \frac{x - a}{b-a} \right\rfloor. \] Consequently \[ x - \left\lfloor\!\tfrac{x-a}{b-a} \right\rfloor(b-a) \in [a,b). \]
-
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:
-
I do understand that the last two definitions might look somewhat weird. The only reason for that is that the ceiling function and the floor function are 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 and floor 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 $[1,4)$. 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.
In the figure below the function $f$ is the restriction of the function $x \mapsto \cos(x)$ (in blue) to the interval $[0,\pi)$. The red function is the periodic extension.
- A Mathematica implementation of the solution of Laplace's equation in a disk is in this Mathematica notebook.
- It is almost as if snown has known that I have written up quite detailed html notes for today's lecture: Solution of Laplace's equation in a rectangle.
- A Mathematica implementation of the solution is in this Mathematica notebook. As usual, before saving a notebook I delete all output. To recreate the output, click the menu item Evaluation, then Evaluate Notebook.
- A Mathematica implementation of the solution of the heat equation with three standard boundary conditions is in this Mathematica notebook. As usual, before saving a notebook I delete all output. To recreate the output, click the menu item Evaluation, then Evaluate Notebook.
- Section 2.5 The assigned Exercises 2.5.1 (b), 2.5.3 (a), (b), 2.5.5 (a), (d), 2.5.6, 2.5.7, 2.5.8, 2.5.9
-
We are working on Chapter 2. Here is the summary of what we did:
- Section 2.2. The assigned Exercises 2.2.3, 2.2.4.
- Section 2.3. The assigned 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.
- Section 2.4. The assigned Exercises 2.4.1 (a), (b), (c), 2.4.2, 2.4.3, 2.4.4, 2.4.6, 2.4.7.
- Today I demonstrated how to use Mathematica v8 to implement and illustrate the solution of the heat equation. Here is the notebook that I used in class. The file is called 20171031.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 "20171031.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.
- We are working on Section 2.3. The assigned 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.
-
We are working on
Chapter 2. The exercises for the first few sections are as follows:
- Section 2.2. The assigned Exercises 2.2.3, 2.2.4.
- Section 2.3. The assigned 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.
- Here is a derivation of the formula for the Laplacian in polar coordinates. Please study the examples presented after the derivation.
- Several homework problems assigned yesterday deal with polar and spherical coordinates.
- The formula for the Laplacian in spherical coordinates is more complicated: \[ \nabla^2 w = {1 \over r^2} {\frac{\partial}{\partial r}} \left(r^2 {\frac{\partial w}{\partial r}} \right) + {\frac{1}{r^2 \sin \theta}} {\frac{\partial}{\partial \theta}} \left( (\sin \theta) {\frac{\partial w}{\partial \theta}} \right) + {1 \over r^2 (\sin \theta)^2} {\partial^2 w \over \partial \varphi^2}. \] Here $w$ is a function of $r \in [0,+\infty)$, $\theta \in [0, 2\pi)$ and $\varphi \in [0,\pi]$.
- The formula for the Laplacian in cylindrical coordinates is just a variation on the formula for the Laplacian in polar coordinates: \[ \nabla^2 w = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial w}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 w}{\partial \theta^2} + \frac{\partial^2 w}{\partial z^2}. \] Here $w$ is a function of $r \in [0,+\infty)$, $\theta \in [0, 2\pi)$ and $z \in \mathbb R$.
- 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 (d) (e) (f), 1.4.2, 1.4.3, 1.4.4, 1.4.5, 1.4.6, 1.4.7.
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In addition to the above exercises do Exercises 1.4.12 and 1.4.13 from the 4th edition. For those who do not have this edition I post these to exercises below.
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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 a link to the Chapter that we are covering now.
- Today we started discussing 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.
- I derived the diffusion equation today. Here is a summary of that derivation. The 3rd edition derives only the heat equation in Section 1.2. The 4th and the 5th edition derive the diffusion equation at the end of Section 1.2.
- I also explained the differences which occur in 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 and 5th 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 I gave the geometric interpretation of the partial differential equation \begin{equation*} A(x,y,u) \, u_x(x,y) + B(x,y,u) \, u_y(x,y) = C(x,y,u) \end{equation*} subject to the initial condition \begin{equation*} u(x,0) = f(x), \quad x \in \mathbb R. \end{equation*} Briefly: the given PDE can be rewritten as \begin{equation*} \bigl\langle A(x,y,u), B(x,y,u), C(x,y,u) \bigr\rangle \bigl\langle u_x(x,y) , u_y(x,y) , -1 \bigr\rangle = 0. \end{equation*} Here $\bigl\langle a , b , c \bigr\rangle$ denotes the vector represented by the oriented line segment $\overrightarrow{OP}$ where $P = (a,b,c).$ Recall that the vector $\bigl\langle u_x(x_0,y_0) , u_y(x_0,y_0) , -1 \bigr\rangle$ is orthogonal to the surface $\bigl(x,y,u(x,y)\bigr)$ at the point $(x_0,y_0,u(x_0,y_0).$ Therefore the equality \begin{equation*} \bigl\langle A(x,y,u), B(x,y,u), C(x,y,u) \bigr\rangle \bigl\langle u_x(x,y) , u_y(x,y) , -1 \bigr\rangle = 0. \end{equation*} means that the vector field \begin{equation*} \bigl\langle A(x,y,z), B(x,y,z), C(x,y,z) \bigr\rangle \end{equation*} is tangent to the surface $\bigl(x,y,u(x,y) \bigr).$ We do not know how to find a surface tangent to a vector field, but in Math 331 we learned how to find a curve which is tangent to a given vector field and which starts at a given point. For more details on this point of view to the method of characteristics see my page Method of characteristics with few examples.
- Today I demonstrated how to use Mathematica v8 to implement and illustrate the Method of characteristics. Here is the notebook 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 you 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.
- Today I gave an introduction to the Method of characteristics. On Monday we will do the general case of this method. I wrote a webpage that covers this general method: Method of characteristics.
- The information sheet
- We start with 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.
- Until your books arrive here is the section we are covering now.
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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.
