Question

Prove that not only does the endpoint of the Euler broken line tend to \(\varphi(t)\), but also the whole sequence of piecewise-linear functions \(\varphi_{n}: I \rightarrow R^{n}\) whose graphs are the Euler broken lines tends uniformly to the solution \(\varphi\) on the interval \([0, t]\)

Short answer

Question: Prove that the endpoint of the Euler broken line tends to the solution function \(\varphi(t)\) and that the sequence of piecewise-linear functions \(\varphi_n\) tends uniformly to the solution \(\varphi\) on the interval \([0,t]\). Answer: The endpoint of the Euler broken line tends to the solution function \(\varphi(t)\) since the error at each step goes to zero as the number of partitions \(n \to \infty\). Similarly, the sequence of piecewise-linear functions \(\varphi_n\) tends uniformly to the solution \(\varphi\) on the interval \([0,t]\) because, for each \(t\in[0,t]\), the difference between \(\varphi_n(t)\) and \(\varphi(t)\) can be bounded by terms that also go to zero as \(n \to \infty\).

Step-by-step solution

Define the Euler broken line

Euler broken line is a piecewise linear interpolation of the solution to an initial value problem. Suppose we have given a first-order ordinary differential equation (ODE) of the form \(\frac{d\varphi(t)}{dt}=f(t, \varphi(t))\) with the initial condition \(\varphi(0)=\varphi_0\). Then Euler method involves subdividing the interval \([0, t]\) into \(n\) equally spaced intervals, and then estimating the solution function \(\varphi(t)\) using the Euler broken line \(\varphi_n\).

Show the endpoint tends to \(\varphi(t)\)

Let \(t_i = i \frac{t}{n}\) for \(i=0,1,2,...,n\). Then the Euler broken line is defined recursively as follows: - \(\varphi_n(t_0) = \varphi_0\) - \(\varphi_n(t_{i+1})=\varphi_n(t_i)+\frac{t}{n}f(t_i,\varphi_n(t_i)),\quad i=0,1,...,n-1\) To show that the endpoint of the Euler broken line tends to the solution \(\varphi(t)\), we need to look at the limit as \(n \to \infty\). The error at each step is of order \(\frac{t}{n}\left(\frac{1}{2}\frac{d^2\varphi}{dt^2}(t_i)\right)\), and since the ODE is assumed to be Lipschitz continuous in the second variable, this error goes to zero as \(n \to \infty\). Therefore, \(\lim_{n \to \infty} \varphi_n(t_n) = \varphi(t)\), which implies that the endpoint tends to the true solution.

Prove the uniform convergence of \(\varphi_n\) to \(\varphi\)

To prove that the sequence \(\varphi_n\) tends uniformly to \(\varphi\), we need to show that $$\lim_{n\to\infty}\sup_{t\in[0,t]}|\varphi_n(t)-\varphi(t)|=0$$ For each \(t\in[0,t]\), we have: $$|\varphi_n(t)-\varphi(t)|\leq|\varphi_n(t)-\varphi_n(t_i)|+|\varphi_n(t_i)-\varphi(t_i)|+|\varphi(t_i)-\varphi(t)|$$ As \(n\to\infty\), the first term can be bounded by \(\frac{t}{n}\sup_{t\in[0,t]}|f(t,\varphi(t))|\), which goes to zero. The second term can be controlled by the error estimation of Euler method, which we have shown to go to zero in the previous step. Finally, the third term can be bounded by a Lipschitz constant times \(|t-t_i|\), and since \(t_i\to t\) as \(n\to\infty\), this term goes to zero as well. Hence, we have proved that the sequence of piecewise-linear functions \(\varphi_n\) whose graphs are the Euler broken lines tends uniformly to the solution \(\varphi\) on the interval \([0,t]\).

Key concepts

Euler's Method

Euler's Method is a numerical technique to approximate solutions to ordinary differential equations (ODEs), more specifically, first-order initial value problems. It is a straightforward and practical method used widely for its simplicity. The essence of Euler's method lies in its step-by-step approach to approximating the solution. By dividing the interval of interest into small sub-intervals and using a simple linear approximation, we can estimate the values of the solution function.

• We start with a given initial value and a differential equation.
• The interval is subdivided into \( n \) equally spaced sub-intervals.
• The solution is recursively approximated using the slope (given by the derivative function in the ODE) at each step.

This results in a "broken line" or piecewise-linear function approximating the continuous solution.
By refining the subdivision (increasing \( n \)), this approximation becomes more accurate, eventually converging to the actual solution of the ODE.

Uniform Convergence

Uniform Convergence is a concept that describes a sequence of functions approaching a single function uniformly. In simpler terms, given a sequence of functions \( \varphi_n \), we say it converges uniformly to a function \( \varphi \) if, no matter how small the margin of error we choose, a point exists beyond which all functions in the sequence are within that margin from \( \varphi \).
For Euler's method, we're interested in proving that not just at a single point, but across an entire interval \([0, t]\), the piecewise-linear approximations \( \varphi_n \) uniformly converge to the solution \( \varphi \). This means:

• The maximum discrepancy between \( \varphi_n \) and \( \varphi \) over the interval can be made as small as desired by choosing a sufficiently large \( n \).
• This idea ensures the approximations become consistently accurate as \( n \) increases.

Uniform convergence is crucial because it guarantees consistency across the entire domain, not just pointwise convergence.

Piecewise-Linear Function

A piecewise-linear function is a function composed of straight-line segments. In the context of numerical methods like Euler's method, these functions approximate continuous solutions to differential equations by connecting points determined from an iterative process.

• The function is defined over an interval by using different linear equations for each sub-interval.
• Such functions can be seen as "broken lines" because, at each interval's boundary, the function changes directions, creating distinct line segments.

In Euler's method, the graph of \( \varphi_n \) forms a piecewise-linear function, connecting discrete approximations of the solution across intervals.
This approach is beneficial because it simplifies complex differential equations to a series of basic linear calculations, making them easier and faster to handle computationally.

Initial Value Problem

An Initial Value Problem (IVP) is a specific type of differential equation accompanied by a specified value at the starting point. In essence, you have an equation describing how things change over time (the differential equation) and a starting point to anchor that description (initial value).

• The general form of an IVP for a first-order ODE is \( \frac{d\varphi(t)}{dt} = f(t,\varphi(t)) \) with \( \varphi(0) = \varphi_0 \).
• This setup gives both the rule (ODE) and the starting condition necessary to determine the unique path or solution \( \varphi(t) \).

By defining the initial condition, we ensure there is a unique solution trajectory, which is paramount for both theoretical analysis and practical computations.
Analyzing and solving IVPs serves as the foundation for understanding how complex systems evolve over time from their initial states.