$3$ MATLAB Project

$3\mathrm{.}1$ A Simple Two$-$Point Boundary Value Problem

Write a MATLAB script to divide the interval $0,1$ into $21(n+1)$ equal subintervals, and then

implement the finite difference method given by Equation $(10)$ in order to approximate the solution

of the two$-$point boundary value problem

$y^{\text{'}\text{'}}(t)=125t,y(0)=y(1)=0$

at the $20(n)$ interior grid points.

The following MATLAB commands generate a sparse tridiagonal representation of the second differ$-$

ence operator on $n$ points:

$e=$ones$(n,1)$;

$\{$:$A=$spdiags$([e,-2**e,e]],-1$:$1,n,n)$

Plot your approximate solution to the two$-$point boundary value problem in Equation $(11).$ Make sure

the boundary conditions are evident in your plot. You may download the file tpbvp$1.$m from the

course web page to do this part of the project.

Find the exact solution to the two$-$point boundary value problem in Equation $(11).$ On the same plot,

compare your approximate values at the grid points with the exact solution at the grid points. Your

plot should look similar to Figure $2.$ Comment on the accuracy of your approximations.

The augmented matrix associated with this system is shown below:

$\left[\begin{array}{cccccccc}2& -1& 0& \text{d}\text{o}\text{t}\text{s}& 0& 0& 0& -h^2{f}_1\\ -1& 2& -1& \text{d}\text{o}\text{t}\text{s}& 0& 0& 0& -h^2{f}_2\\ 0& -1& 2& \text{d}\text{o}\text{t}\text{s}& 0& 0& 0& -h^2{f}_3\\ \text{v}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}& \text{d}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}\\ 0& 0& 0& \text{d}\text{o}\text{t}\text{s}& 2& -1& 0& -h^2{f}_{n-2}\\ 0& 0& 0& \text{d}\text{o}\text{t}\text{s}& -1& 2& -1& -h^2{f}_{n-1}\\ 0& 0& 0& \text{d}\text{o}\text{t}\text{s}& 0& -1& 2& -h^2{f}_n\end{array}\right].$

By solving this system, approximate values of the unknown function $y$ at the grid points $t_i$ are obtained.

Larger values of $n$ produce smaller values of $h$ and hence better approximations to the exact values of the

$y_i\text{'}$s$.$

Matlab code to use as Basis:

$\%$ tpbvp$1.$m

$\%$ Solution to the two$-$point boundary value problem

$\%$ y$\text{'}\text{'}($t$)=125$t$,$ y$(0)=$ y$(1)=0$

n $=20$; $\%$ Number of interior grid points

a $=0$; $\%$ Initial time

b $=1$; $\%$ Final time

ya $=0$; $\%$ Boundary condition

yb $=0$; $\%$ Boundary condition

h $=($b $-$ a$)/($n $+1)$; $\%$ Step size

t $=[$a:h:b$]\text{'}$; $\%$ Time axis

$\%$ Tridiagonal matrix representation of second difference

e $=$ ones$($n$,1)$;

A $=$ spdiags$([-$e $2*$e $-$e$],-1$:$1,$n$,$n$)$;

$\%$ Function f$($t$)$

f $=125*$t;

fi $=-$h$^2*$f$(2$:end $-1)$; $\%$ Values of f at interior grid points

fi$(1)=$ fi$(1)+$ ya;

fi$($end$)=$ fi$($end$)+$ yb;

$\%$ Approximate solution

yi $=$ A$\backslash$fi;

yapp $=[$ya; yi; yb$]$;

$\%$ Exact solution: By integrating y$\text{'}\text{'}($t$)=125$t twice

$\%$ and matching the boundary conditions,

$\%$ y$($t$)=(125/6)($t$^3-$ t$)$

y $=(125/6)*($t$.^3-$ t$)$;

plot$($t$,$yapp,$\text{'}$o$\text{'},$t$,$y$)$

legend$(\text{'}$Approximate$\text{'},$'Exact'$)$

title$(\text{'}$Solution to TPBVP: y$"($t$)=125$t$,$ y$(0)=$ y$(1)=0\text{'})$

xlabel$(\text{'}$Time$,$ t$\text{'})$

ylabel$(\text{'}$Amplitude$,$ y$($t$)\text{'})$