[Ordinary differential equation]2014-2015-2 5th Teaching Week 2nd Lecture Lecture 3.1 Existence and uniqueness theorem of solution and stepwise approximation method

1. Existence of a uniqueness theorem. Consider Cauchy problem $$\bee\label{3.1.cauchy} \sedd{\ba{ll} \frac{\rd y}{\rd x}&=f (x, y), \ y (x_0) &=y_0, \ea} \eee$$ where

(1). $f (x, y) $ in rectangular area $$\bex R:\quad |x-x_0|\leq a,\quad |y-y_0|\leq b \eex$$ continuous;

(2). $f (x, y) $ on $R $ on $y $ meet Lipschitz Condition: $$\bex \exists\ l>0,\ |f (X,y_1)-F (x,y_2) |\leq L|y_1-y_2|,\quad (x,y_1), (X, y_2) \in R \eex$$ \eqref{3.1.cauchy} There is a unique solution $\phi (x) $, defined on $[x_0-h,x_0+h]$, $$\bex H=\min\sed{a,\frac{b}{m}},\quad M=\ max_{(x, y) \in r}|f (x, y) |. \eex$$ Proof:

(1). It is only necessary to prove the conclusion on the $[x_0,x_0+h]$, which is similar to the $[x_0-h,x_0]$.

(2). We use Picard step approximation method to prove the conclusion, the general idea is: $$\bex \mbox{ode}\eqref{3.1.cauchy} \mbox{solution}\lra \mbox{ide}y=y_0+\int_{x_0}^x F (t,y The solution of \rd t\mbox{}; \eex$$ $$\bee\label{3.1.picard} \phi_0 (x) =y_0,\quad \phi_n (x) =y_0+\int_{x_0}^x F (t,\phi_{n-1} (t)) \rd T,\quad n=1,2, \cdots. \eee$$ if $\sed{\phi_n (x)}$ converge on $[x_0-h,x_0+h]$ uniformly on $\phi (x) $, then $\sed{f (\phi_n (x))}$ also converges uniformly to $f (T,\phi (t)) $ (why?), while in \eq Ref{3.1.picard} In order $n \to\infty$ have $$\bex \phi (x) =y_0+\int_{x_0}^x F (t,\phi (t)) \rd T, \eex$$ $\phi (x) $ is required. Here, $\phi_n (x) $ is referred to as the first approximate solution $n $. This method is called the stepwise approximation method.

(3). $$\bex \mbox{ode}\eqref{3.1.cauchy} \mbox{solution}\lra \mbox{ide}y=y_0+\int_{x_0}^x F (t,y) \rd t\mbox{Solution}; \eex$$

(4). $|\phi_n (x)-y_0|\leq b$.

(5). $\sed{\phi_n (x)}$ converge uniformly on the $[x_0-h,x_0+h]$.

(6). $\phi (x) $ is the continuous solution of the IDE on the $[x_0-h,x_0+h]$.

(7). Uniqueness.

2. Notes.

(1). Error estimate $$\bee\label{3.1.error} |\phi_n (x)-\phi (x) |\leq \frac{ml^n}{(n+1)!} H^{n+1}. \eee$$

(2). ' Lipschitz condition ' ' common ' $f $ on $R $ on $y $ has a continuous partial derivative ' instead.

(3). If \eqref{3.1.cauchy} is linear, $$\bee\label{3.1.linear} \frac{\rd y}{\rd x}=p (x) y+q (x), \eee$$ when $P (x), Q (x) $ in $[\al,\be ta]$ on a continuous, the solution of the arbitrary initial value $ (x_0,y_0), \ x_0\in (\al,\beta) $, \eqref{3.1.linear} is defined throughout the $[\al,\beta]$.

(4). For the first-order implicit ode $F (X,y,y ') =0$, we have the theorem of page 86 in the book.

3. Homework. Page 3, page, T 6.

[Ordinary differential equation]2014-2015-2 5th Teaching Week 2nd Lecture Lecture 3.1 Existence and uniqueness theorem of solution and stepwise approximation method