A couple of weeks ago, we computed in class the sensitivity of a solution of an ODE IVP to the initial condition by co-solving another (coupled) IVP (the "variational equations"). Now consider the initial value $y_0$ to be fixed, but the differential equation depends on a parameter, $p$: $$ \frac{\partial y(t,p)}{\partial t} = f(y,p), \qquad y(0,p) = y_0 \qquad (1) $$ For simplicity we will consider an autonomous scalar equation depending on a scalar parameter. We want to know the derivative of the solution at time $t$ with respect to the parameter $p$.
(a) Call the derivative we are interested in knowing $v(t,p)$: $$v(t,p) = \frac{\partial y(t,p)}{\partial p}.$$
Determine a differential equation and initial condition for $v(t,p)$ that could be solved along with (1) to obtain $v(t,p)$. Assume as much smoothness as you like.
To avoid confusion or ambiguity, use the notation $ {\partial_i} f( y(t,p), p ) $ to denote the derivative of $f$ with respect to its $i^{th}$ argument ($i\in\{1,2\}$), evaluated at $( y(t,p), p )$; and the notation $$ \frac{\partial}{\partial p} f( y(t,p), p ) $$ for the derivative of the expression $ f( y(t,p), p )$ with respect to $p$.
(b) Specialize your answer for part (a) to the case $f(y,p) = py^3$, and write it the way we would if we were about to code it up for numerical solution.
Solve the following BVP by the method of finite differences: $$y^{\prime\prime}+20\sin(20x)y^\prime -\frac{6}{x^2}y = 1$$ for $x\in(2,4)$, and $y(2)=1$, $y(4)=8$.
Print your (best) estimate for y(3), and provide some empirical evidence that the maximum error is $O(h^2)$.
(a) Is it possible to obtain an approximation to 𝑦′′(0) with error = $𝑂(ℎ^2)$ using 𝑦(0), 𝑦(ℎ), 𝑦(2ℎ)? Either way, support your result with a calculation.
(b) How about with the 4 values 𝑦(0), 𝑦(ℎ), 𝑦(2ℎ), y(3h)?
(c) If yes in (b), how much worse (or better) is the approximation than the 3-point centered one with the same grid spacing?
Ackleh et al. Exercise 10.3.6 (Galerkin approximation)
Extra credit: Extend the basis to $\{ \sin(\pi x),\ \sin(2\pi x), ... , \sin(N\pi x) \}$ and explore how the error depends on $N$. Hint: a useful way to see what's going on as $N$ increases is to plot $N$ times the error, and/or $N^2$ times the error vs. $x$, for various $N$.
(a) For the BVP from Ackleh at the top of p547, perform a hand-calculation to find the Galerkin approximation to the solution using a single hat function as the basis for the approximation. TYPO WARNING: they mean $r(x)=1+x$, $s(x)=0$, $f(x)=0$.
(b) What is the derivative of the exact solution of this BVP?
(c) Validate the proposition that the Galerkin approximation minimizes the error in the norm $||e||=\int r {e^\prime}^2 + s e^2$, by explicitly minimizing this (using Calc I method) in this case.
(d) Is the norm defined in (c) really a norm for arbitrary integrable functions $r$ and $s$? If so, prove it. If not, elaborate.