# Approximation of solution of IVP

Suppose that $$f$$ satisfies the Lipschitz condition.

For given initial values $$y_0, z_0\in \mathbb{R}$$, we consider the IVPs: $$y'=f(t,y), \ y(a)=y_0 \\ z'=f(t,z), \ z(a)=z_0$$ with $$a\leq t \leq b$$.



1) Show that the problem has unique solution, i.e. if $$y_0=z_0$$ then $$y(t)=z(t)$$ for all $$t$$. $$\max_{1\leq t\leq b}|y(t)-z(t)|\leq c|y_0-z_0|$$

2) Let $$\{y^n\}$$ and $$\{z^n\}$$ be aproximation of the problem that we get by the mean value method, let $$\epsilon^n:=y^n-z^n \\ \epsilon^{n+1/2}:=y^{n+1/2}-z^{n+1/2} \\ y^{n+1/2}=\frac{1}{2}(y^{n+1}+y^n)$$ show that $$(\epsilon^{n+1}-\epsilon^n)\epsilon^{n+1/2}=h[f(t^{n+1/2}, \frac{1}{2}(y^{n+1}+y^n))-f(t^{n+1/2}, \frac{1}{2}(z^{n+1}+z^n))]\epsilon^{n+1/2}$$



I have done the following:

1) I have shown that $$e^{-2Lt}(y(t)-z(t))^2$$ is descreasing for $$t\in [a,b]$$.

Then we have the following: $$0\leq e^{-2Lt}(y(t)-z(t))^2\leq e^{-2Lt_0}(y(t_0)-z(t_0))^2=e^{-2Lt_0}(y_0-z_0)^2$$ If $$y_0=z_0$$ we get $e^{-2Lt}(y(t)-z(t))^2=0 \Rightarrow y(t)-z(t)=0\Rightarrow y(t)=z(t)$\$ Therefore the problem has a unique solution.

For the inequality we can show that $$y(t)-z(t)$$ is decreasing, but then it follows that $$|y(t)-z(t)|\leq |y_0-z_0|$$, but how do we get that constant $$c$$ at the right side?

2) For this one I don't really have an idea. Could yyou give me a hint?

• In 2), what is the meaning of "approximation", is it a sequence of functions or the result of some numerical method? And what is the "mean value method"? Is it the implicit midpoint method? Aug 29, 2019 at 12:54
• It is meant the result of a numerical method.. @LutzL Aug 29, 2019 at 13:24

By the definition of the implicit midpoint method, you directly have \begin{align} y^{n+1}-y^n&=hf(t^{n+1/2},y^{n+1/2})\\ z^{n+1}-z^n&=hf(t^{n+1/2},z^{n+1/2})\\ \end{align} so that in the difference of both equations you get without any further transformation $$ϵ^{n+1}-ϵ^n=h[f(t^{n+1/2},y^{n+1/2})-f(t^{n+1/2},z^{n+1/2})].$$ Now simply multiply with $$ϵ^{n+1/2}$$ to get the claimed equation.