- Here is the Mathematica notebook which I used to produce the animations below.
- The above notebook is based on Section 7.7.
- 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 $n=5$.
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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.
- Today 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). The animations below are produced using the principles from Section 7.3 implemented in Mathematica.
- 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.
-
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.
-
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.
-
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 orthogonality. 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.
-
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.
-
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.
- 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.
- Here is the Mathematica notebook that I created today in class. Here is a pdf file of this notebook. You can peek at the code without having Mathematica.
-
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
-
Here is a short summary of topics that could appear on the exam:
- 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 the meaning of related boundary conditions.
- Understand the problem solved in Section 5.8. 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 (a) should be done by solving the differential equation, 5.8.4, 5.8.8 (a) should be done by solving the differential equation, 5.8.9
- 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.
-
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.
- 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.
-
First I want to emphasize the cocept 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 \[ \tilde{f}(x) = f\left(\!x- \Bigl(\left\lceil\!\tfrac{x-a}{b-a}\! \right\rceil -1 \Bigr)(b-a)\!\right), \quad x \in {\mathbb R}. \] 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:
- Notice that this definition differs slightly from the definition given on Friday. The reason for the difference is that here we study function defined on $(a,b]$ and on Friday we studied a function defined on $[a,b).$
-
Today we reviewed two Mathematica notebooks in which I implemented the solution of Laplace's equation:
- Mathematica notebook with a solution of Laplace's equation in a rectangle.
- Mathematica notebook with a solution of Laplace's equation in a disk.
- 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 $f.$
-
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 $[1,4)$. 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.
-
We are done with 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.
- 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 skipped Subsections 2.5.3 and 2.5.4.
- A Mathematica implementation of the solution of Laplace's equation in a disk is in this Mathematica notebook.
- Today we covered 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
- Today we finished Section 2.4. The recommended exercises were listed on Friday.
-
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.
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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.
- 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 on Tuesday.
- 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.
-
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.
-
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.
-
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.
- 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.
- 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.
-
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.
- On Friday we discussed in detail Exercise 12.2.4. The PDE in this exercise is \[ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0. \] If we impose the initial condition $u(x,0) = F(x)$ for all $x \in \mathbb R$, then the solution of this problem is \[ u(x,t) = F(x-ct), \qquad x,t \in \mathbb R. \] Similarly, the solution of \[ \frac{\partial u}{\partial t} - c \frac{\partial u}{\partial x} = 0. \] subject to $u(x,0) = G(x)$ for all $x \in \mathbb R$, is \[ u(x,t) = G(x+ct), \qquad x,t \in \mathbb R. \] Our next topic are second order equations. It is interesting to point out that the one dimensional wave equation \[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \] can be solved by combining the above two solutions. Since \[ \frac{\partial^2 u}{\partial t^2} - c^2 \frac{\partial^2 u}{\partial x^2} = \left(\frac{\partial}{\partial t} - c \frac{\partial }{\partial x}\right) \left(\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x}\right) = \left(\frac{\partial}{\partial t} + c \frac{\partial }{\partial x}\right) \left(\frac{\partial u}{\partial t} - c \frac{\partial u}{\partial x}\right), \] Both functions $F(x-ct)$ and $G(x+ct)$ solve the one dimensional wave equation. Therefore, with $F$ and $G$ being arbitrary twice differentiable functions of one variable, the function \[ u(x,t) = F(x-ct) + G(x+ct) \] is a solution of the one-dimensional wave equation. More about this in Section 12.3.
- I used the above method in this notebook to solve the following problem \[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}, \qquad u(x,0) = \sin x, \quad \frac{\partial u}{\partial t}(x,0) = 0. \]
- We are done with the Method of characteristics. Do 2, 3, 4, 5, 6, 7, 8 in Exercises 12.2 and the problems in my summary of the Method of characteristics.
- Yesterday in class 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.
- I aslo wrote a summary for the Method of characteristics with few examples.
- 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.
- For those who are waiting for their books to 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.
