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%\journal{Journal of Magnetic Resonance}

\begin{document}

\begin{frontmatter}

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\title{Solving PDEs using the finite element method with the Matlab PDE Toolbox}

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\section{Solving PDE with Neumann boundary conditions}

Please write Matlab code to solve for the unknown function $\omega$ that is the solution the PDE below that has zero forcing term, zero initial condition, and nonhomogeneous (non-zero) Neumann boundary conditions
\begin{equation}
\label{eq:wProblem}
\begin{split}
\frac{\partial}{\partial t} \omega(\bx,t) 
-\nabla\left(\Dintr \nabla \omega(\bx,t)\right)&=0,  \quad \quad \quad \quad \quad \quad \quad \quad \bx \in \Omega,\\
\Dintr \nabla \omega(\bx,t) \cdot \bnu(\bx) &= \Dintr F(t) \bug\cdot\bnu(\bx), \quad \bx \in \partial \Omega, \\
\omega(\bx,0) &= 0,  \quad \quad \quad \quad \quad \quad \quad \quad \bx \in \Omega,
\end{split}
\end{equation}
$\bnu$ being the outward normal to the domain $\Omega$.  The vector $\bug$ is a vector in two dimensions and has unit norm.
You need to solve this in the time interval 
\ben 
t\in[0,TE], \quad TE = \delta+\Delta,
\een
where $\delta$ and $\Delta$ are given constants.  
The time-dependent function $F(t)$ in the Neumann boundary condition is defined in the following way:
\begin{equation}
F(t) \equiv \int_0^{t} f(s)ds.
\end{equation}
where $f(t)$ is related to $\delta$ and $\Delta$ by:
\begin{equation}
\label{eq:f(t)}
f(t) =
\begin{cases}
1 & 0< t \leq  \delta, \\
0 & \delta < t \leq  \Delta, \\
-1 & \, \Delta < t \leq \Delta + \delta, \\
0 & \text{elsewhere},
\end{cases} \, 
\end{equation}
In particular,
\begin{equation}
\label{eq:F(t)}
F(t) =
\begin{cases}
t & 0 < t \leq  \delta, \\
\delta & 0 +\delta < t \leq  \Delta, \\
\Delta+\delta-t & \,0 + \Delta < t \leq \Delta + \delta, \\
0 & \text{elsewhere}.
\end{cases} \, 
\end{equation}

\begin{todo}
When you construct the Neumann boundary conditions, please generate the FE matrices without time dependence by writing the function \begin{verbatim}NeumannBC_notime.m \end{verbatim}
\end{todo}
\begin{ans}
Please include your Matlab files below using the command:
\begin{verbatim}
\lstinputlisting[style=Matlab-editor]{../project_files/NeumannBC_notime.m}
\end{verbatim}
\end{ans}
\begin{todo}
Add the time dependence to the ODE solver by writing the function
\begin{verbatim}
odefun_NeumannBC.m 
\end{verbatim}
\end{todo}
\begin{ans}
Please include your Matlab files below using the command:
\begin{verbatim}
\lstinputlisting[style=Matlab-editor]{../project_files/odefun_NeumannBC.m}
\end{verbatim}
\end{ans}

\section{Solving PDE with forcing term}

Writing $\omega$, which solves the problem \eqref{eq:wProblem}, as the sum
\begin{equation}
\omega(\bx,t) = \widetilde{\omega}(\bx,t) + F(t)\; \bx \cdot \bug, \bx \in \Omega, \quad t\in[0,TE]
\end{equation}
where $\widetilde{\omega}(\bx,t)$ satisfied the following diffusion equation with a forcing term and
homogeneous boundary condition:
\begin{align}
\frac{\partial}{\partial t} \widetilde{\omega}(\bx,t)
 -\nabla\left(\Dintr \nabla \widetilde{\omega}(\bx,t)\right)&= -f(t)\bx\cdot\bug, &\quad \bx \in \Omega, \, t\in[0,TE],\\
\Dintr \nabla \widetilde{\omega}(\bx,t) \cdot \bnu(\bx) &= 0, &\quad \bx \in \Gamma, \, t\in[0,TE],\\
\widetilde{\omega}(\bx,t) &= 0,&\quad \bx \in \Omega,
\end{align}
\begin{todo}
Please compute $\omega$ from $\widetilde{\omega}$ after solving the above PDE with a forcing term.
\end{todo}
\begin{ans}
Please include your Matlab files below.
\end{ans}



\begin{todo}
The inner product of the functions $f$ and $g$ (for projection and norm operation) is defined by
\be{}
<f,g> \equiv \int_{\Omega} f(\bx) g(\bx) d\bx.
\ee
If you have the values of $f$ and $g$ at the nodes, how do you compute the above integral?  Remember $f(\bx)  \approx \sum_1^{N_p} f_j \psi_j(\bx) $, where $\psi_j(\bx)$ is the $\bbP_1$ finite elements basis function on the node $j$
and $f_j$ is the value of $f$ at node $j$.  And $g(\bx) \approx \sum_1^{N_p} g_i \psi_i(\bx) $.  You want to compute
\ben
\int_{\Omega}  \left( \sum_1^{N_p} g_i \psi_i(\bx) \right) \left(  \sum_1^{N_p} f_j \psi_j(\bx) \right) d\bx.
\een
\end{todo}
\begin{ans}
Please answer with words and equations.  Do not include Matlab code.
\end{ans}

\begin{todo}
Write a function
\begin{verbatim}
seqprofile_expintegral.m
\end{verbatim}
that integrates 
\be{eq:expintegral}
\int_0^t e^{-\lambda_n(t-s)}f(s) ds
\ee
analytically, for the $f(t)$ defined in Eq. \ref{eq:f(t)}.  Break into different cases for $t \in [0,\delta], [\delta,\Delta],[\Delta,\Delta+\delta]$.
The function should take as inputs $t$ and $\lambda$ :
\begin{verbatim}
function value = seqprofile_expintegral(time,lambda)
  global BDELTA SDELTA
\end{verbatim}
Be sure to treat the case $\lambda = 0$.  
\end{todo}
\begin{ans}
Please give the formulas for the integral in Eq. \ref{eq:expintegral} here.  Do not include Matlab code.
\end{ans}

\section{Solving PDE with eigenfunctions}
The function $\widetilde{\omega}(\bx,t)$ also can be expanded in the basis of Laplace eigenfunctions.
Let $\phi_n(\bx)$ and $\lambda_n$ be the $L^2$-normalized eigenfunctions and eigenvalues associated to the Laplace operator 
with homogeneous Neumann boundary conditions:
\begin{alignat}{3}
\label{eq:eigen1}
-\nabla \Dintr \left(\nabla \phi_n(\bx)\right) &= \lambda_n \phi_n(\bx),&\quad \bx &\in& \Omega,\\
\Dintr \nabla \phi_n(\bx) \cdot \bnu(\bx) &= 0, &\quad \bx &\in& \Gamma.
\label{eq:eigen2}
\end{alignat}
We can write $\widetilde{\omega}(\bx,t)$ in the basis of the eigenfunctions as
\begin{equation}
\label{eq:weigen}
\widetilde{\omega}(\bx,t) = 
\sum_{n=1}^\infty (-a_n)\phi_n(\bx)\int_0^t e^{-\lambda_n(t-s)}f(s)ds,
\end{equation}
where the coefficients are
\begin{align}
a_n  
= \int_{\Omega} \bx\cdot\bug \phi_n(\bx) d\bx.
\end{align}

To calculate $\phi_n(\bx)$ and $\lambda_n$  in Matlab, you need to use the Matlab function pdeeig, with the command:
\begin{verbatim}
[V,L]=pdeeig(heatmodel,C_COEFF,0,1,[-inf,EigLim]);
\end{verbatim}
$V$ contains the  $\phi_n$ and $L$ contains $\lambda_n$, for $n = 1,\cdots n_{eig}$.  How many eigenfunctions get computed depends on the interval on which you are looking for eigenvalues.  As a first try, the range I gave is $\lambda_n \in [-\infty,10C_{COEFF}]$.

Before you use  Eq. \ref{eq:weigen}, please first (numerically) normalize the eigenfunctions $\{\phi_n\}$ to have $L^2$-norm equal to 1, meaning:
\be{}
\int_{\Omega} |\phi_n(\bx)|^2 d\bx = 1.
\ee

\begin{todo}
Please solve Eqs \ref{eq:eigen1}-\ref{eq:eigen2} and compute $\omega$ using the formula in Eq. \ref{eq:weigen}.
\end{todo}
\begin{ans}
Please include your Matlab files below.
\end{ans}


\section{Miscellaneous questions}



\begin{todo}

Please compute the quantity $h(t)$ using both definitions given below:
\begin{align}
\label{eq:h1}
h(t) = \frac{1}{|\Omega|}\int_{\Omega} \bug\cdot\nabla\omega(\mathbf{x},t)d\mathbf{x}\\
= \frac{1}{|\Omega|}\int_{\Gamma} \omega(\by,t) \left(\bug\cdot\nu(\by)\right) ds_{\by}.
\label{eq:h2}
\end{align}
and compare with the computed results from using Eq. \ref{eq:h1} versus Eq. \ref{eq:h2}.
Put both curves on the same plot.
\end{todo}
\begin{ans}
Please
\begin{itemize}
\item  explain in words how you calculated the quantities in Eq. \ref{eq:h1} ($h_1(t)$) versus Eq. \ref{eq:h2} ($h_2(t)$).
\item  include the Matlab code.
\item Plot $| h_1(t)-h_2(t)|$.
\end{itemize}
\end{ans}

\begin{todo}
Using the exact eigenvalues given for a rectangle and for a disk, compare the computed eigenvalues with the exact eigenvalues. 
Put the exact eigenvalue and the numerical eigenvalues with 3 different $h_{max}$ on the same plot.  One Figure for rectangle, one Figure for disk.  
\end{todo}
\begin{ans}
Please include two Figures here.  In the caption of each Figure, include the geometry shape, and the size of the geometry.  
\end{ans}

\begin{todo}
For a rectangle with sides $l_x$ and $l_y$, for a disk with radius $R$, 
what should you use as 'EigLim' in the command
\begin{verbatim}
[V,L]=pdeeig(heatmodel,C_COEFF,0,1,[-inf,EigLim])
\end{verbatim}
to get approximately $P$ eigenvalues? 
In other words, what value do you use for 'EigLim' if someone gives you the number $P$ where 
$1\leq P \leq 25$?  
\end{todo}
\begin{ans}
Please 
\begin{itemize}
\item explain your answer in words
\item include Matlab code that gives 'EigLim' as a function of $P$ (and $l_x$ and $l_y$ or $R$)
\item Using your code, for P = 10, P = 20, for the rectangle and a disk (choose as you wish $l_x$ and $l_y$ or $R$, and include the values you chose), describe here how many eigenvalues you get from Matlab command
\begin{verbatim}
[V,L]=pdeeig(heatmodel,C_COEFF,0,1,[-inf,EigLim])
\end{verbatim}
\end{itemize}
\end{ans}
\begin{todo}
How can you compute 
$a_n  = \int_{\Omega} \bx\cdot\bug \phi_n(\bx) d\bx$ numerically?
How can you compute it using one of the finite element matrices?
\end{todo}
\begin{ans}
Include here at least one way of calculating $a_n$ using one of the finite element matrices.
\end{ans}
\begin{todo}
For the rectangle (choose your own $l_x$ and $l_y$) plot the eigenfunctions $\phi_n(\bx)$ corresponding to the 4 largest $|a_n|$, where 
$a_n  = \int_{\Omega} \bx\cdot\bug \phi_n(\bx) d\bx$ using the command 
\begin{verbatim}
pdeplot
\end{verbatim}
and include the value of $|a_n|$ in the Figure.  Make sure you find enough eigenfunctions so that the largest 4 $|a_n|$ are found.  
\end{todo}
\begin{ans}
Include the Figure with 4 subplots here.  Include the $EigLim$ you used.
\end{ans}
\begin{todo}
Compute $a_n$ analytically for a rectangle for $\bug = [0,1]$ and $\bug = [1,0]$. 
\end{todo}
\begin{ans}
Give the answer in equations please.  
\end{ans}
\begin{todo}
Fix a geometry and $\delta$ (SDELTA), increase $\Delta$ (BDELTA).  How does $h(t)$ change as $\Delta$ increases?  If you have $h(t)$ for $\delta$ and $\Delta$, how can you obtain $h_{new}(t)$ 
for $\delta_{new} = \delta$ and $\Delta_{new} = \Delta-\delta$?
\end{todo}
\begin{ans}
Please
\begin{itemize}
\item Plot $h(t)$ for several values of $\Delta$ in one plot.
\item Describe how to obtain $h_{new}(t)$ of $\delta_{new}$ and $\Delta_{new}=\Delta-\delta$ using $h(t)$ of $\delta$ and $\Delta$.
\end{itemize}
\end{ans}
\begin{todo}
Use the analytical $a_n$ and $\lambda_n$ and $\phi_n(\bx)$ for a rectangle, compute $h_3(t)$.  This $h_3(t)$ depends on the number of eigenfunctions you use.  Plot $h_3(t)$ for several choices of $P$, where $P$ is the number of eigenfunctions.  Compare with $h_1(t)$ and $h_2(t)$.
\end{todo}
\begin{ans}
Give the Figure, caption, and explanation in text.
\end{ans}

\section{Information to include on the Figures}

In Matlab
\begin{itemize}
\item Give the Figure a title saying what the Figure is
\item x-axis label
\item y-axis label
\item Where there are multiple curves in a plot, use different markers/linestyles so the curves are easy to read. 
\item When there are multiple curves in a plot, use a legend to give a short description of the curves
\end{itemize}
In report
\begin{itemize}
\item Give the Figure a caption.  In the caption, describe the Figure with a few sentences.
\item In the text of the report, explain what you are trying to show with the Figure.
\end{itemize}

\section{Matlab files}

{Matlab functions that you should have in your project folder}

\begin{verbatim}
seqprofile.m (given, this is f(t))                     
seqintprofile.m  (given, this is F(t))   
project_inputs.in (given)
read_simulation_parameters.m  (given) 
Build_FE_Mesh.m (given)          
Plot_Figures.m (given)    
Solve_NeumannBC.m (given)

odefun_ForcingTerm.m   (need to write)     
odefun_NeumannBC.m  (need to write)              
ForcingTerm_notime.m (need to write)
NeumannBC_notime.m  (need to write)          
Solve_EigenFunctions.m (need to write)               
Solve_ForcingTerm.m (need to write)        

Any other functions you wrote...    
\end{verbatim}


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