Solve the problems for your appropriate course track. Problems probe understanding of the course concepts. Formulate your answers clearly and cogently. Sketch out an approach on scratch paper first. Then briefly transcribe the approach to the answer you turn in, followed by appropriate calculations and conclusions, within allotted time. Use concise, complete English sentences in the description of your approach.

Each question is meant to be completely answered and transcribed from proof to final copy within thirty minutes. Concentrate foremost on clear exposition of the concept underlying your approach.

1Track 1

1. Consider the ballistic missile trajectory problem of national defense interest. From measurements of the positions $x_i=x\left(t_i\right)$ at successive times $t_i$, $i=0,...,n$ predict the target reached at time $T>t_n$. Formulate a procedure to predict $x\left(T\right)$, assuming the missile is known to follow a parabolic trajectory.

2. Construct a quadrature formula for integrals of the form

   ${\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\alpha t}f\left(t\right)dt.}$

3. Find the best approximant in the least squares sense of $\sin t$ within $\mathrm{span}\{1,t,t^2\}$.

4. Find the best inf-norm approximant of $f:[0,1]\to ℝ$, $f\left(t\right)=e^{-t}$ by a first-degree polynomial.

   Figure 1.

5. Propose a scheme to solve the integro-differential equation

   ${\displaystyle \frac{dy}{dt}+y=\underset{0}{\overset{t}{\int }}\sin (t-\tau )y\left(\tau \right)d\tau ,}$

2Track 2

1. Construct an approximant of $e^{𝑨\left(t\right)}$ where $𝑨\left(t\right)\in {ℝ}^{m\times m}$ is a symmetric positive definite matrix-valued function of $t\in ℝ$.

2. Construct a quadrature formula for integrals of the form

   ${\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{𝑨\left(t\right)}𝒇\left(t\right)dt,}$

   where $$$𝑨\left(t\right)\in {ℝ}^{m\times m}$ is a symmetric negative definite matrix-valued function of $t\in ℝ$, and $𝒇:ℝ\to {ℝ}^{}$ has Riemann integrable components.

3. Find the best approximant of $𝒚\in {ℝ}^m$ within $C\left(𝑨\right)$, $𝑨\in {ℝ}^{m\times n}$ in a space with scalar product

   ${\displaystyle (𝒖,𝒗)={𝒖}^T𝑷𝒗,}$

   and norm

   ${\displaystyle ||𝒖||={(𝒖,𝒖)}^{1/2},}$

   where $𝑷\in {ℝ}^{m\times m}$ is symmetric positive definite. Verify the correspondence principle that for $𝑷=𝑰$ standard least-squares projection is obtained.

4. Find the best inf-norm approximant of $f:[0,\infty )\to ℝ$, $f\left(t\right)=e^{-t}\cos t$ by a first-degree polynomial.

5. Consider the half-derivative operator $H$ defined as

   ${\displaystyle H^{2}f=H(Hf)=Df}$

   where $D=d/dt$ is the derivative operator. Propose a numerical scheme to evaluate $Hf$ that can be used to solve fractional differential equations.