Forward Euler and First-Order Convergence

(20 pts) For the ODE \(u' = f(t,u)\) with initial condition \(u(0)\) and \(f\) globally Lipschitz continuous in \(u \in \mathbb {R}^M\) uniformly in \(t\), show that forward Euler with initial state \(u_0 = u(0)\) is convergent to first order. A possibly helpful definition: a scheme is 0-stable if, \[ \max _{n\in [N]}\|e_n\| \le C \max _{n\in [N]}\|R_n(u(t_n))\|, \] where \(e_n\) is the time-\(t_n\) error, and \(R_n(u(t_n))\) is the scheme residual using the exact solution.

(20 pts) For the ODE \(u' = f(t,u)\) with initial condition \(u(0)\) and \(f\) globally Lipschitz continuous in \(u \in \mathbb {R}^M\) uniformly in \(t\), show that forward Euler with initial state \(u_0 = u(0)\) is convergent to first order. A possibly helpful definition: a scheme is 0-stable if, \[ \max _{n\in [N]}\|e_n\| \le C \max _{n\in [N]}\|R_n(u(t_n))\|, \] where \(e_n\) is the time-\(t_n\) error, and \(R_n(u(t_n))\) is the scheme residual using the exact solution.

For the initial value problem \(u'(t) = f(u(t),t)\), \(u(0) = \eta \), where \(f(u,t)\) is continuous with respect to \(t\) and Lipschitz continuous with respect to \(u\), consider the scheme \[ \frac {U^{n+2}-\frac {4}{3}U^{n+1}+\frac {1}{3}U^n}{k} = \frac {2}{3}f(U^{n+2},t_{n+2}), \] where \(t_n = nk\), and \(U^n\) is supposed to approximate \(u(t_n)\).

1. Analyze the consistency, zero-stability, and convergence of this scheme.

2. If you apply this scheme to the initial value problem

   \[ u'(t) = -10^{12}(u(t)-\cos (t))-\sin (t),\qquad u(0) = 2, \]

   what issues should you consider in choosing the timestep \(k\)? Are there time intervals in which \(k\) must be very small to get a reasonable solution and others in which it does not? Explain your answers.

For the initial value problem \(u'(t) = f(u(t),t)\), \(u(0) = \eta \), where \(f(u,t)\) is continuous with respect to \(t\) and Lipschitz continuous with respect to \(u\), consider the scheme \[ \frac {U^{n+2}-\frac {4}{3}U^{n+1}+\frac {1}{3}U^n}{k} = \frac {2}{3}f(U^{n+2},t_{n+2}), \] where \(t_n = nk\), and \(U^n\) is supposed to approximate \(u(t_n)\).

1. Analyze the consistency, zero-stability, and convergence of this scheme.

2. If you apply this scheme to the initial value problem

   \[ u'(t) = -10^{12}(u(t)-\cos (t))-\sin (t),\qquad u(0) = 2, \]

   what issues should you consider in choosing the timestep \(k\)? Are there time intervals in which \(k\) must be very small to get a reasonable solution and others in which it does not? Explain your answers.

The following function \(f(x)\) [\(x\) is a real variable] is expanded in a series in powers of \((x+3)\). Find the radius of convergence of the series in each case. \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&e^{x^2},\\[1.4ex] \text {(b)}\quad f(x)&=&\left (\dfrac {\sin x-3}{\sin x-2}\right )^2,\\[1.8ex] \text {(c)}\quad f(x)&=&\dfrac {\sin (x+1)}{(x+1)(x+3)}. \end {array} \]

Function \(f(x)\) (see below) is expanded in powers of \(x-x_0\), i.e. in the Taylor or Laurent series with center \(x_0\) (\(x\) is a real variable, \(x_0\) is a real number). Find the radius of convergence of this series. \[ \begin {array}{rcll} \text {(a)}\quad f(x)&=&\dfrac {\sin x^2}{(x-3)^2(x^2+1)},& x_0=3,\\[1.4ex] \text {(b)}\quad f(x)&=&\dfrac {\sin x}{x},& x_0=5,\\[1.4ex] \text {(c)}\quad f(x)&=&\dfrac {\sin x}{\sin x+3},& x_0=8. \end {array} \]

The following function \(f(x)\) [\(x\) is a real variable] can be represented by a series (Taylor or Laurent) in powers of \(x\). Find the radius of convergence of the series in each case \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&\left (\dfrac {\sin x}{x}\right )^2,\\[1.4ex] \text {(b)}\quad f(x)&=&\dfrac {\sin x}{x^2},\\[1.4ex] \text {(c)}\quad f(x)&=&\left (\dfrac {x}{\sin x}\right )^2,\\[1.4ex] \text {(d)}\quad f(x)&=&\dfrac {x^2}{\sin x},\\[1.4ex] \text {(e)}\quad f(x)&=&\dfrac {\sin x}{x^2+1}. \end {array} \] [You do not need to find the series themselves.]

The following function \(f(x)\) [\(x\) is a real variable] can be represented by a series (Taylor or Laurent) in powers of \((x-7)\). Find the radius of convergence of the series in each case \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&e^{(x-7)^{10}},\\[0.6ex] \text {(b)}\quad f(x)&=&\left (\dfrac {\sin x-3}{\sin x-2}\right )^2,\\[1.4ex] \text {(c)}\quad f(x)&=&\dfrac {\sin x-2}{x^2-49}. \end {array} \] [You do not need to find the series themselves.]

The following functions \(f(x)\) [\(x\) is a real variable] can be represented by series (Taylor or Laurent) in powers of \((x-2)\). Find the radius of convergence of the series in each case \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&e^{x^2},\\[0.6ex] \text {(b)}\quad f(x)&=&\dfrac {\sin x-3}{\sin x-2},\\[1.2ex] \text {(c)}\quad f(x)&=&\dfrac {e^x}{(x-2)(1+e^x)}. \end {array} \]

Find the radius of convergence of the Taylor series with the center at the origin (\(x_0=0\)) for the following function

1. \[ f(x)=\frac {\sqrt {x+3}}{x^2+3} \]
2. \[ f(x)=\frac {1}{(\cos x+3)^2} \]

(\(x\) is a real variable; you do not need to find the series themselves).

Find the radius of convergence of the Taylor series with the center at \(x_0=1\) for the following function

1. \[ f(x)=x^{1/3} \]
2. \[ f(z)=\frac {(\sin x+2)^2}{(\sin x-2)^2} \]

(\(x\) is a real variable; you do not need to find the series themselves).

Find the radius of convergence of the Taylor series \[ \text {for the function } f(x)=\frac {1}{\cos ^2 x+4} \quad \text {around the origin } x_0=0. \] {You do not need to find the Taylor series itself.}

Find the radius of convergence of the Taylor series for the function \[ \frac {1}{2+\cos x} \qquad (x\text { is a real variable}) \] around the origin (\(x_0=0\)). {You do not need to find the Taylor series itself.}

Convergence Consider the one-dimensional heat equation initial boundary value problem \[ u_t=u_{xx},\quad x\in [0,1],\quad t\geq 0,\quad u(x,0)=f(x),\quad u(0,t)=u(1,t)=0 \tag {7} \] and the discretization \[ x_m=mh,\qquad t_n=nk,\qquad \text {and}\qquad U_m^n\approx u(x_m,t_n) \tag {8} \] where \[ U_m^{n+1}=U_m^n+r\left (U_{m-1}^n-2U_m^n+U_{m+1}^n\right ) \tag {9} \] and \(r=k/h^2\) is the grid constant. Define what it means that this method is convergent, say for what values of \(r\) it is convergent, and show that your statement is correct.

Newton’s Method Suppose \(f(x)\) is a real-valued function defined for all values of the real variable \(x\), and that \(f'\) is Lipschitz continuous. State and prove a local convergence theorem for Newton’s method for finding a simple zero of \(f\). How does this result change if the root is a double root? Is Newton’s method globally convergent? If you answered yes, prove your claim. If you answered no, sketch an example which illustrates your claim, and outline a method that has the Newton method’s local convergence properties but which is globally convergent.

Discretization and Convergence. What is a discretization? What does it mean for a discretization to be convergent? Give a precise definition for a differential equation problem and a discretization of your choice.