On the approximate solution obtained through Euler's method

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psie
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TL;DR Summary
I'm reading about Euler's method to construct approximate solutions to ODEs in Ordinary Differential Equations by Andersson and Böiers. I have questions about properties of the approximate solution.
This is a bit of a longer post. I have tried to be as brief as possible while still being self-contained. My questions probably do not have much to do with ODEs, but this is the context in which they arose. Grateful for any help.

In what follows ##|\cdot|## denotes either the absolute value of a scalar or the Euclidean norm of a vector (denoted in bold). First a definition of what it means to be an approximate solution:
Definition 1. Let ##I## be an interval on the real axis, and ##\Omega## an open set in ##\mathbf R\times\mathbf{R}^n##. Assume that the function ##\pmb{f}:\Omega\to\mathbf{R}^n## is continuous. A continuous function ##\pmb{x}(t),\ t\in I##, is called an ##\varepsilon##-approximate solution of the system ##\pmb{x}'=\pmb{f}(t,\pmb{x})## if ##(t,\pmb{x})\in\Omega## when ##t\in I## and $$\left|\pmb{x}(t'')-\pmb{x}(t')-\int_{t'}^{t''} \pmb{f}(s,\pmb{x}(s))ds\right|\leq \varepsilon|t''-t'|\quad \text{when } t',t''\in I.$$
Next, a theorem (for the sake of brevity, without proof) on an error estimate on the approximate solution:
Theorem 1. Assume that ##\pmb{f}(t,\pmb{x})## is continuous in ##\Omega\subseteq \mathbf{R}\times\mathbf{R}^n## and satisfies the Lipschitz condition $$|\pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})|\leq L|\pmb{x}-\pmb{y}|, \quad (t,\pmb{x}),(t,\pmb{y})\in\Omega.$$ Let ##\pmb{\tilde{x}}(t)## be an ##\varepsilon##-approximate and ##\pmb{x}(t)## an exact solution of ##\pmb{x}'=\pmb{f}(t,\pmb{x})## in ##\Omega## when ##t\in I##. For an arbitrary point ##t_0## in ##I## we then have $$|\pmb{\tilde{x}}(t)-\pmb{x}|\leq |\pmb{\tilde{x}}(t_0)-\pmb{x}(t_0)|e^{L|t-t_0|}+\frac{\varepsilon}{L}(e^{L|t-t_0|}-1),\quad t\in I.$$
The Euler's method works as follows. Consider the equation ##x'=f(t,x)## and consider an initial value ##(t_0,x(t_0))##. We know the slope of the tangent line through ##(t_0,x(t_0))##. Follow this tangent a bit to the right, to ##t=t_1=t_0+\delta## and repeat the procedure for the point ##(t_1,x(t_1))##. This way, one gets a broken curve of straight line segments resembling the solution.

To make the definition of the broken curve precise, consider the system ##\pmb{x}'=\pmb{f}(t,\pmb{x})## and an initial value ##(t_0,\pmb{x}_0)##. Make a division of the interval ##[t_0,t_0+a]## in equally long subintervals: ##t_0<t_1<\ldots<t_m=t_0+a##, and put ##\delta=t_j-t_{j-1}##. Then define the function ##\pmb{x}_{\delta}## recursively at the step points ##t_j## by \begin{align}
&\pmb{x}_{\delta}(t_0)=\pmb{x}_0, \nonumber \\
&\pmb{x}_{\delta}(t_{j+1})=\pmb{x}_{\delta}(t_j)+(t_{j+1}-t_j)\pmb{f}(t_j,\pmb{x}_{\delta}(t_j)),\quad j=0,1,\ldots,m-1. \tag1
\end{align}
Between the step points the curve of ##\pmb{x}_{\delta}## is supposed to be a straight line, so $$\pmb{x}_{\delta}(t)=\pmb{x}_{\delta}(t_j)+(t-t_j)\pmb{f}(t_j,\pmb{x}_{\delta}(t_j)),\quad t_j\leq t\leq t_{j+1}.\tag2$$ The function ##\pmb{x}_{\delta}## is defined correspondingly in the interval ##[t_0-a,t_0]##.

The following theorem shows ##\pmb{x}_{\delta}## is an ##\varepsilon##-approximate solution to ##\pmb{x}'=\pmb{f}(t,\pmb{x})## under certain conditions on ##\pmb{f}##.
Theorem 2. [Let ##B,L## and ##C## be positive constants.] Assume that
  1. the function ##\pmb{f}(t,\pmb{x})## is continuous in ##\Omega\subseteq\mathbf{R}\times\mathbf{R}^n##, that ##|\pmb{f}(t,\pmb{x})|\leq B## in ##\Omega##, and that ##a## is so small that ##\Lambda(a)\subseteq\Omega##, where ##\Lambda(a)## denotes the double cone $$\Lambda(a)=\{(t,\pmb{x})\in \mathbf{R}\times\mathbf{R}^n;|t-t_0|\leq a, \ |\pmb{x}-\pmb{x}_0|\leq B|t-t_0|\},$$
  2. ##|\pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})|\leq L|\pmb{x}-\pmb{y}| \qquad \ \ \ (t,\pmb{x}),(t,\pmb{y})\in\Omega##,
  3. ##|\pmb{f}(t',\pmb{x})-\pmb{f}(t'',\pmb{x})|\leq C|t'-t''| \quad (t',\pmb{x}),(t'',\pmb{x})\in\Omega##
Then ##\pmb{x}_{\delta}## is an ##\varepsilon##-approximate solution to the system ##\pmb{x}'=\pmb{f}(t,\pmb{x})## in the interval ##[t_0-a,t_0+a]##, with ##\varepsilon=\delta(C+LB)##.

Proof. All line segments in the definition of ##\pmb{x}_{\delta}## have a slope at most ##B##. Therefor ##(t,\pmb{x}_{\delta}(t))\in\Lambda(a)## when ##t\in[t_0-a,t_0+a]##. Moreover, if ##|t'-t''|\le\delta## then \begin{align} |\pmb{f}(t',\pmb{x}_{\delta}(t'))-\pmb{f}(t'',\pmb{x}_{\delta}(t''))|&\leq |\pmb{f}(t',\pmb{x}_{\delta}(t'))-\pmb{f}(t'',\pmb{x}_{\delta}(t'))| \nonumber \\ &+|\pmb{f}(t'',\pmb{x}_{\delta}(t'))-\pmb{f}(t'',\pmb{x}_{\delta}(t''))| \nonumber \\ &\leq C|t'-t''|+L|\pmb{x}_{\delta}(t')-\pmb{x}_{\delta}(t'')| \nonumber \\ &\leq C|t'-t''|+LB|t'-t''| \nonumber \\ &\leq \delta(C+LB).\tag3 \end{align}
[The first inequality is the triangle inequality, the second follows from 2. and 3. in the assumptions and the third by the fact that each line segments of ##\pmb{x}_{\delta}## have a slope at most ##B##, so ##|\pmb{x}_{\delta}(t')-\pmb{x}_{\delta}(t'')|\leq B|t'-t''|##.] We must prove that $$\left|\pmb{x}_{\delta}(t'')-\pmb{x}_{\delta}(t')-\int_{t'}^{t''} \pmb{f}(s,\pmb{x}_{\delta}(s))ds\right|\leq \delta(C+LB)|t'-t''|.\tag4$$
If ##t'## and ##t''## belong to the same subinterval ##[t_j,t_{j+1}]## then [from ##(2)##] $$\pmb{x}_{\delta}(t'')-\pmb{x}_{\delta}(t')=(t''-t')\pmb{f}(t_j,\pmb{x}_{\delta}(t_j))=\int_{t'}^{t''} \pmb{f}(t_j,\pmb{x}_{\delta}(t_j))ds,$$ and ##(4)## follows from ##(3)##. (In particular, ##(4)## is valid when ##t'=t_j## and ##t''=t_{j+1}##). If ##t'## and ##t''## belong to different subintervals then use ##(4)## on each one of the intervals ##[t',t_{j+1}],[t_{j+1},t_{j+2}],\ldots,[t_{k-1},t_k],[t_k,t'']## and add the results.
Now comes the part I have some questions about. The authors claim that if we drop assumptions 1. and 2. in theorem 2, it is still possible to show that there is a number ##\varepsilon(\delta)##, tending to zero as ##\delta\to 0##, such that ##\pmb{x}_{\delta}## is an ##\varepsilon(\delta)##-approximate solution in the interval ##I(a)=[t_0-a,t_0+a]##. They write:
We know that ##(t,\pmb{x}_{\delta}(t))\in\Lambda(a)## when ##t\in I(a)## and that $$|\pmb{x}_{\delta}(t')-\pmb{x}_{\delta}(t'')|\leq B|t'-t''|\quad t',t''\in I(a).\tag5$$ Put $$\varepsilon(\delta)=\sup\limits_{|t'-t''|\leq\delta}|\pmb{f}(t',\pmb{x}_{\delta}(t'))-\pmb{f}(t'',\pmb{x}_{\delta}(t''))|.$$ The computations in the proof of theorem 2 show that ##\pmb{x}_{\delta}## is an ##\varepsilon(\delta)##-approximate solution. Furthermore, ##\pmb{f}## is uniformly continuous on ##\Lambda(a)##. From ##(5)## it accordingly follows that $$\lim_{\delta\to0}\varepsilon(\delta)=0.\tag6$$
Question 1: I simply do not understand how ##(6)## follow from ##(5)##. How does it?

Question 2: The authors go on to claim that when the IVP ##\pmb{x}'=\pmb{f}(t,\pmb{x}), \pmb{x}(t_0)=\pmb{x}_0## has a solution ##\pmb{x}(t)##, then it follows from ##(6)## and theorem 1, that ##\pmb{x}_{\delta}## converges uniformly to ##\pmb{x}## as ##\delta\to 0##. The definition of uniform convergence I'm used to is ##\lVert f_n-f\rVert\to 0## as ##n\to\infty##, where ##\lVert\cdot\rVert## denotes the sup-norm, but here they are claiming that the sup-norm should tend to ##0## as ##\delta\to 0##. This makes me wonder; what is ##\pmb{x}_{\delta}##? Is it a sequence? If not, what definition of uniform convergence are the authors using?

EDIT: The last quoted passage is just prior to presenting Peano's existence theorem. They note that:
If you only assume that ##\pmb f## is continuous, you can not use [Picard-Lindelöf's theorem]. In fact, you can not even be certain that ##\pmb{x}_{\delta}(t)## converges when ##\delta\to 0##. But what you can prove is that there is a sequence ##\delta_p##, tending to zero as ##p\to\infty##, such that the functions ##\pmb {x}_{\delta_p}(t)## converge uniformly on ##I(a)##.
 
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Question 2: The authors go on to claim that when the IVP ##\pmb{x}'=\pmb{f}(t,\pmb{x}), \pmb{x}(t_0)=\pmb{x}_0## has a solution ##\pmb{x}(t)##, then it follows from ##(6)## and theorem 1, that ##\pmb{x}_{\delta}## converges uniformly to ##\pmb{x}## as ##\delta\to 0##. The definition of uniform convergence I'm used to is ##\lVert f_n-f\rVert\to 0## as ##n\to\infty##, where ##\lVert\cdot\rVert## denotes the sup-norm, but here they are claiming that the sup-norm should tend to ##0## as ##\delta\to 0##. This makes me wonder; what is ##\pmb{x}_{\delta}##? Is it a sequence? If not, what definition of uniform convergence are the authors using?

Let [itex]X[/itex] be a compact space and [itex]f_\alpha : X \to \mathbb{R}^n[/itex] a family of functions defined for [itex]\alpha \in U \subset \mathbb{R}[/itex]. Then [itex]f_\alpha \to f[/itex] uniformly on [itex]X[/itex] as [itex]\alpha \to \alpha_0 \in \bar{U}[/itex] if and only if for every [itex]\epsilon > 0[/itex] there exists [itex]\delta > 0[/itex] such that for all [itex]\alpha \in U[/itex], [tex]|\alpha - \alpha_0| < \delta \quad \Rightarrow\quad
\sup_{x \in X} \|f_\alpha(x) - f(x)\| < \epsilon.[/tex] Note that this has the same relation to the defintiion of uniform convergence of a sequence of functions as the definition of the limit of a function at a point has to the definition of the limit of a sequence.
 
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