The INITIAL VALUE PROBLEM (IVP)

ORDINARY DIFFERENTIAL EQUATIONS (ODE) Material for this section is taken from books by Kincaid & Cheney, Burden & Faires, Atkinson, Iserles, Isaacson & Keller, Coddington & Levinson, Stoer & Burlisch. The references to numerical analysis books can be found by clicking here.

The first part will consider the ``initial value problem.'' (IVP) in detail. The second part will present the ``boundary value problem'' (BVP) in a cursory way (see notes for 575B course, where we use variational methods to recast the BVP for its numerical solution, a powerful and elegant analytical technique that leads to a host of important numerical schemes).

An ordinary differential equation is a function that maps $x\mapsto y(x)\in R^n$, $n$ a natural number, and it involves $x$ the independent variable, $y(x)$, and finite set of derivatives of $y(x)$. It has the form

$$g\left( x,y,y',y'',\ldots,\frac{d^my}{dx^m}\right)=0.$$ (1)

where $g \in R^k$, $k$ a natural number. In many instances $k=1$. In fact, assume for now assume that $k=1$ in what follows. Equation (1) is an $m^{\mbox{th}}$ order ODE, since the highest non-zero derivative in $y$ is $$\frac{d^m y}{dx^m}$$.

We can recast (1) in ``normal form.'' Solving for $\frac{d^m y}{dx^m}$ we have

$$\frac{d^my}{dx^m}=f\left( x,y,y', \ldots, \frac{d^{m-1}y} {dx^{m-1}}\right).$$

$$\fbox{$y^{(j)}=\displaystyle \frac{d^j y}{dx^j}$} j=0, 1, 2, \ldots m \qquad \mbox {with }y^{(0)}\equiv y.$$

$$\frac{d}{dx}\left(\begin{array}{c} y^{(0)} y^{(1)} : y^{(... ...{(1)} y^{(2)} : f(x,y^{(0)},y^{(1)}\cdots y^{(m-1)})\end{array}\right),$$

$$\mbox {or } \frac{dY}{dx}=F(x,Y)$$

$$\mbox {with } Y=(y^{(0)},\cdots, y^{(m-1)})^{T}$$

$$\mbox {and } \displaystyle F \equiv (f^{(0)}, f^(1), \ldots, f^{(m-1)})^{T},$$

where the superscript $T$ stands for transpose.

Definition: Autonomous and Non-autonomous ODE'S:

$$\begin{array}{ll} \displaystyle \frac{dY}{dx}=F(Y) & \mbo... ...le \frac{dY}{dx}=F(Y,x) & \mbox{is non-autonomous}. \end{array}$$ (2)

Non-autonomous ODE's can be recast as an ODE by the following procedure:

$$\begin{array}{ll} \mbox{let }Y=\left(y^{(0)},\cdots, y^{(m)}\righ... ... \\ 1\end{array}\right), \mbox {is an $m+1$ dimensional vector},\end{array}$$

where

$$y^{(m)}=x$$

$$\frac{dy^{(m)}}{dx}=1$$

The Initial Value Problem is defined as:

$$\left\{\begin{array}{ll} \displaystyle \frac{dY}{dx}=F(Y,x) \\ \\ Y(x_0)=Y_0 & Y_0\mbox{ is an }m\mbox{ dimensional vector.} \end{array}\right.$$

Example) We recast the following initial value problem as a normalized autonomous system.

$$\sin t y'''+\cos(ty)+\sin(t^2+y'')+(y')^3=\log t$$

$$y(2)=7$$

$$y'(2)=3$$

$$y''(2)=-4$$

$$y^{(0)}=y,\quad y^{(1)} =\frac{dy^{(0)}}{dt},\quad y^{(2)} =\frac{dy^{(1)}}{dt}.$$ Let

Normalizing:

$$\quad y'''=-\frac{1}{\sin t}\left(\cos(ty)+\sin(t^2+y^{(2)}) +(y^{(1)})^3 \right)+\frac{\log t}{\sin t}$$

$$\left\{\begin{array}{l} \displaystyle \frac{d}{dt}y^{(2)}=-\frac{... ...^{(1)}=y^{(2)} \\ \displaystyle \frac{d}{dt}y^{(0)}=y^{(1)} \end{array}\right.$$

$$\mbox {Let }\quad Y=(y^{(2)},y^{(1)},y^{(0)})^T$$

$$\mbox {then }\quad \frac{dY}{dt}=F(t,Y)\equiv (f^{(2)},f^{(1)},f^{(0)})^T$$

$$\mbox {with }\quad F=\left\{\begin{array}{l} \displaystyle -\fra... ...+ (y^{{(1)}^3})+\frac{\log t}{\sin t} \\ y^{(2)} \\ y^{(1)}\end{array}\right.$$

$$\quad \underline {Y(2)=(-4,3,7)^T}$$

$$\quad \frac{dY}{dx}=f(Y)= (f^{(3)}, f^{(2)}, f^{(1)}, f^{(0)})$$ Made autonomous:

$$\mbox {where }\quad \qquad f^{(3)}=1$$

$$\mbox {and }\quad Y=(y^{(3)},y^{(2)}, y^{(1)}, y^{(0)}), \quad y^{(3)}=t\quad Y(2)=(2,-4, 3, 7)$$

$\Box$