Partial Differential Equations Tutorial Sheet, #12
Learning Targets
- Manipulate solutions to Partial Differential Equations
- Use separation of variables solution to solve a PDE
- Derive particular solutions that solve initial and boundary conditions
Additional Resources
Tutorials
- Step-by-step Question : Takes you through an example step by step the way we learn it - this question alone made me understand PDEs.
- Alternate PDE Method : This may not make much sense until you have your head around it - A slightly different method compared to what we’re taught but the beauty is they all work!
Problem sheet
Skill Building Questions
Problem 1.
Which of the following are solutions to Laplace’s equation, $\nabla^2 f(\mathbf{x}) = 0$? For those that are not, can you come up with a differential equation involving the Laplacian, $\nabla^2$, that $f(\mathbf{x})$ is a solution of?
(a) $e^{-y} \sin x $
$\Rightarrow{}\quad$ The Laplacian operator is defined as:
$\Rightarrow{}\nabla^2= \frac{\partial^2 } {\partial x^2}+ \frac{\partial^2} {\partial y^2} + \frac{\partial^2} {\partial z^2}$
$\Rightarrow{}\nabla^2 f(\mathbf{x}) = \frac{\partial^2 }{\partial x^2}{e^{-y} \sin x} + \frac{\partial^2}{\partial y^2}{e^{-y} \sin x}$
$\Rightarrow{} = \frac{\partial }{\partial x}{e^{-y} \cos x} - \frac{\partial}{\partial y}{e^{-y} \sin x}$
$\Rightarrow{} = {e^{-y} \sin x - e^{-y} \sin x} = 0$
NB This is an example for this kind problem in case you get confused by these short answers :)
(b) $\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$
(c) $f(x + i y)$
(d) $4 \arctan(y/x) + \log(x^2 +y^2)$
(e) $\frac{x+y}{z}$
Problem 2.
Show that the function $u\left(x,y\right)=\ln(1+xy^2)$ satisfies the partial differential equation:
\[2 \frac{\partial{} ^2u}{ \partial{} x^2} + y^3 \frac{ \partial{} ^2u }{ \partial{} y \partial{} x } = 0\]$$ \frac{ {\partial{}}^2u}{\partial{}y\partial{}x}=\frac{\partial{}}{\partial{}y}\left(\frac{\partial{}u}{\partial{}x}\right)=\frac{2y\left(1+xy^2\right)-y^2(2xy)}{ {(1+xy^2)}^2}=\frac{2y+2xy^3-2xy^3}{ { {(1+xy}^2)}^2}=\frac{2y}{ { {(1+xy}^2)}^2} $$
$$ 2\frac{ {\partial{}}^2u}{\partial{}x^2}+y^3\frac{ {\partial{}}^2u}{\partial{}y\partial{}x}=-\frac{2y^4}{ { {(1+xy}^2)}^2}+\frac{ {2y}^4}{ { {(1+xy}^2)}^2}=0 $$
Separation of Variables 0 – 1D Wave Equation
Go through the process of solving the general soultion to the 1D Wave equation, As we did in the lecture notes.
\[\frac{\partial^2 f(x, t)}{\partial t^2} = c^2 \frac{\partial^2 f(x, t)}{\partial x^2}\]- Assume a solution that is seperable, e.g., $f(x, t) = X(x)T(t)$.
- Plug this into the PDE, and let the derivatives work on like terms.
- Divide the whole thing by $X(x)T(t)$ and collect like terms.
- Set these terms to be constant, to define ODEs and a dispersion relation.
- Solve the ODEs and combine the solutions together.
Once you have done this, you can return the question later and try to solve it using different trial solutions (Sines and cosines, exponentials, hyperbolic functions) and different choice of constants (powers, sign). See how they each behave and how the choices interact with each other.
Separation of Variables 1 – Choice of constants
In the Better Constants section of the notes, We looked at the 1D wave equation,
\[\frac{\partial^2 f(x, t)}{\partial t^2} = c^2 \frac{\partial^2 f(x, t)}{\partial x^2}\]and chose ODE constants, $-\omega^2$ and $-k^2$, to solve the PDE.
You were asked to see what happened if instead we used positive squared coefficients, i.e.,
\[\begin{align*} T^{\prime\prime}(t) &= +\gamma^2 T(t) \\ X^{\prime\prime}(x) &= +\kappa^2 X(x) \end{align*}\]Show, by using a separable solution $f(x, t) = X(x)T(t)$ on the PDE, that we can return the ODEs above, and that their relationship is,
\[\gamma^2 = c^2 \kappa^2\]Then show by solving the ODEs, that exponentially growing and decaying solutions can be produced. Write the general solution to the PDE in this form.
Separation of Variables 2 – 2D Diffusion Equation
In the notes, we highlighted the 2D Diffusion Equation as an example of using separation of variables.
\[\frac{\partial u(x, y, t)}{\partial t} = \alpha \nabla^2 u(x, y, t)\]Find the general solution, $u(x, y, t)$, to this PDE.
Exam Style Question
Problem 3.
Vibrations on a guitar string can be modelled by the wave equation,
$\frac{\partial ^{2}u}{\partial t^{2}}-\frac{\tau\partial ^{2}u}{\rho \partial x^{2}}=0$
where $u(x,t)$ is the displacement of the string from its equilibrium position, $\tau$ is its tension, and $\rho$ is the linear density of the string. This is a clone of the 19-20 progress test.
(a) Use separation of variables to produce linear ODEs for the separated parts.
State any relationship between constants you define.
$X\left ( x \right )T{}''\left ( t \right )-\frac{\tau }{\rho }X{}''\left ( x \right )T\left ( t \right )=0$
$\Rightarrow{}$ And therefore $\frac{T{}''\left ( t \right )}{T\left ( t \right )}-\frac{\tau }{\rho }\frac{X{}''\left ( x \right )}{X\left ( x \right )}=0$
$\Rightarrow{}$Extract ODEs, $\frac{T{}''\left ( t \right )}{T\left ( t \right )}=-\omega ^{2},\frac{X{}''\left ( x \right )}{X\left ( x \right )}=-k^{2}$
$\Rightarrow{}$Then,$\boxed{T{}''\left ( t \right )=-\omega ^{2}T\left ( t \right ),X{}''\left ( x \right )=-k^{2}X\left ( x \right )}$, with $\boxed{\omega ^{2}=\frac{\tau }{\rho }k^{2}}$
(b) Solve the ODEs in terms of sinusoidal solutions.
$\boxed{X(x) = \cos k x, \sin k x}$
Accept complex exponentials, accept equivalent correct forms, accept sum with coefficients, A cos + B sin etc.
(c) The string has a length L, and is fixed at both ends, such that $𝑢(x = 0, 𝑡) = 𝑢(x = L, 𝑡) = 0$. How does this constrain your solutions? Write a general solution of the PDE, $ 𝑢(x, 𝑡)$, that is subject to these constraints.
Fixed at $x=L$ constrains to $k=\frac{n\pi }{L}$.
$\boxed{u\left ( x,t \right )=\left ( A\cos\omega t+B\sin\omega t \right )\sin\frac{n\pi x}{L}}$
(d) Write an expression for the fundamental (lowest) frequency allowed by the string.
Fundamental frequency, set $n=1$.
$\boxed{\omega =\sqrt{\frac{\tau }{\rho }}\frac{\pi }{L}}$
(e) If a guitar has a neck length of 0.65 m, and a string has linear density 5 g/m, what tension does the string need to be to sound an A note at a frequency $\omega = 2\pi × 110$ Hz?
$\Rightarrow \tau =\frac{L^{2}\omega ^{2}}{\pi ^{2}}\rho $
Insert given values, $\Rightarrow{}\tau =\frac{\left ( 0.65m \right )^{2}\left ( 2\pi \times 110Hz \right )^{2}}{\pi ^{2}}0.005\frac{kg}{m}$
$\boxed{\tau =102N}$
Extension Questions
Problem 4.
Prove for the 1D diffusion equation, $\frac{ \partial f(x,t)}{ \partial t} = \alpha \frac{ \partial^2 f(x,t)}{\partial x^2}$, that the total area under the curve $f(x,t)$ does not change over time. (You may assume that $\frac{ \partial f(x,t)}{ \partial x} \rightarrow 0$ for $x \rightarrow \pm\infty$).
Problem 5.
Solve the 3D wave equation by separation of variables to show, $u(\mathbf{x}, t) = \exp(i\mathbf{k}\cdot\mathbf{x} - i \omega t)$, is a solution. With $\mathbf{k} = (k_x, k_y, k_z)^T$ What is the resulting relationship between $\mathbf{k}$ and $\omega$?
\[\frac{n^2}{c^2}\frac{\partial^2 u(\mathbf{x}, t)}{\partial t^2} - \nabla^2 u(\mathbf{x}, t) = 0\]$\Rightarrow{}$ Start with a separation of variables solution, let $u(\mathbf{x}, t) = X(x)Y(y)Z(z)T(t)$. then, \begin{align*}\frac{n^2}{c^2}{T''(t)}{X(x)Y(y)Z(z)} -X''(x)Y(y)Z(z)T(t) - Y''(y)X(x)Z(z)T(t)) - Z''(z)X(x)Y(y)T(t) = 0 \end{align*} \begin{align*} \frac{n^2}{c^2}\frac{T''(t)}{T(t)} - \frac{X''(x)}{X(x)} - \frac{Y''(y)}{Y(y)} - \frac{Z''(z)}{Z(z)} = 0 \end{align*} $\Rightarrow{}$ There is some freedom of choice as to how you define the constants here. In order to match to the test solution, $u(\mathbf{x}, t) = \exp(i\mathbf{k}\cdot\mathbf{x} - i \omega t)$, the most straightforward choice is, \begin{align*} \frac{T''(t)}{T(t)} = -\omega^2 \quad \frac{X''(x)}{X(x)} = -k_x^2 \quad \frac{Y''(y)}{Y(y)} = -k_y^2 \quad \frac{Z''(z)}{Z(z)} = -k_z^2 \space . \end{align*} $\Rightarrow{}$ But here we'll solve as if we assumed arbitrary constants instead, \begin{align*} \frac{T''(t)}{T(t)} = a \quad \frac{X''(x)}{X(x)} = b \quad \frac{Y''(y)}{Y(y)} = c \quad \frac{Z''(z)}{Z(z)} = d \space . \end{align*} $\Rightarrow{}$ With $\frac{n^2}{c^2} a - b - c - d = 0$. Solving the ODEs, we get, \begin{align*} T(t) &= A_+ e^{+ \sqrt{a} t} + A_- e^{- \sqrt{a} t} \newline X(x) &= B_+ e^{+ \sqrt{b} x} + B_- e^{- \sqrt{b} t} \end{align*} $\Rightarrow{}$ And equivalently for $Y(y)$ and $Z(z)$. To match with our test solution, $u(\mathbf{x}, t) = \exp(i\mathbf{k}\cdot\mathbf{x} - i \omega t)$, We'll keep the negative exponential from the $T(t)$ function, and the positive exponentials from the spatial functions. This allow us to build the solution, $u(\mathbf{x}, t) \propto \exp(\sqrt{b} x + \sqrt{c} y + \sqrt{d} z - \sqrt{a} t)$, Matching coefficients requires,} $$ \sqrt{a} = i \omega \quad \sqrt{b} = i k_x \quad \sqrt{c} = i k_y \quad \sqrt{d} = i k_z \quad . $$ $\Rightarrow{}$ Which gives, $u(\mathbf{x}, t) \propto \exp(i k_x x + i k_y y + i k_z z - i \omega t)$, or $\exp(i\mathbf{k}\cdot\mathbf{x} - i \omega t)$. and a dispersion relations,} $$ \boxed{ \frac{n^2}{c^2} \omega^2 = k_x^2 + k_y^2 + k_z^2 } $$
Problem 6.
Show that a Gaussian function, $f(x,t) = \frac{1}{\sigma(t)\sqrt{2\pi}} \exp\left( -\frac{x^2}{2\sigma(t)^2} \right)$ , solves the 1D diffusion equation, on the condition that $\sigma(t) = \sqrt{\sigma(0)^2 + 2 \alpha t}$.
Problem 7.
For an initial concentration, $f(x,0) = \frac{1}{\sigma_0}\exp\left( -\frac{(x-x_0)^2}{2 \sigma_0^2} \right)$, governed by the diffusion equation, calculate the concentration at $x=0\,\mu\mathrm{m}$ and $t=10\,\mathrm{s}$, for $x_0 = 5\,\mu\mathrm{m}$, $\sigma_0 = 1\,\mu\mathrm{m}$, and $\alpha = 2\,\mu\mathrm{m}^2\mathrm{s}^{-1}$. (Use the solution given in Q6 to help)
Answers
For Printing
Revision Questions
The questions included are optional, but here if you want some extra practice.
- Lboro Questions : Not exactly the same as our method but very close - you should be able to work though these and get to either the final answer or the ‘step before’ the answer.
- Tutorial/Questions : Sort of a tutorial as well, walks you through the steps but also provides examples along the way that ‘build up’. Look particularly at sections 9-4 to 8, again might not be exactly the same at every step but it’s easy to figure which stages are relevant.
- Other Questions : Gives them in a different context, not necessarily focused on our method.
- More Other Questions : Many use different methods, but try to get the separation of variables down. If you wanted, try go onto linking the summation.
- Laplace PDEs : Good context questions, though the end form may not be what we were taught.