Question

In this problem we cstablish that the local truncation crror for the improved Euler formula is proportional to \(h^{3} .\) If we assume that the solution \(\phi\) of the initial value problem \(y^{\prime}=f(t, y),\) \(y\left(t_{0}\right)=y_{0}\) has derivatives that are continuous through the third order \((f\) has continuous second partial derivatives), it follows that $$ \phi\left(t_{n}+h\right)=\phi\left(t_{n}\right)+\phi^{\prime}\left(t_{n}\right) h+\frac{\phi^{\prime \prime}\left(t_{n}\right)}{2 !} h^{2}+\frac{\phi^{\prime \prime \prime}\left(\bar{t}_{n}\right)}{3 !} h^{3} $$ where \(t_{n}<\bar{t}_{n} \leq t_{n}+h .\) Assume that \(y_{n}=\phi\left(t_{n}\right)\) (a) Show that for \(y_{n+1}\) as given by Eq. ( 5 ) $$ e_{n+1}=\phi\left(t_{n+1}\right)-y_{n+1} $$ $$ \begin{aligned}=\frac{\phi^{\prime \prime}\left(t_{n}\right) h-\left\\{f\left[t_{n}+h, y_{n}+h f\left(t_{n}, y_{n}\right)\right]-f\left(t_{n}, y_{n}\right)\right\\}}{2 !} +\frac{\phi^{\prime \prime \prime}\left(\bar{I}_{n}\right) h^{3}}{3 !} \end{aligned} $$ (b) Making use of the facts that \(\phi^{\prime \prime}(t)=f_{t}[t, \phi(t)]+f_{y}[t, \phi(t)] \phi^{\prime}(t),\) and that the Taylor approximation with a remainder for a function \(F(t, y)\) of two variables is $$ F(a+h, b+k)=F(a, b)+F_{t}(a, b) h+F_{y}(a, b) k $$ $$ +\left.\frac{1}{2 !}\left(h^{2} F_{t t}+2 h k F_{t y}+k^{2} F_{y y}\right)\right|_{x=\xi, y=\eta} $$ where \(\xi\) lies between \(a\) and \(a+h\) and \(\eta\) lies between \(b\) and \(b+k,\) show that the first term on the right side of \(\mathrm{Eq}\). (i) is proportional to \(h^{3}\) plus higher order terms. This is the desired result. (c) Show that if \(f(t, y)\) is linear in \(t\) and \(y,\) then \(e_{n+1}=\phi^{\prime \prime \prime}\left(\bar{t}_{n}\right) h^{3} / 6,\) where \(t_{n}<\bar{t}_{n}

Short answer

Question: Show that the local truncation error for the improved Euler formula is proportional to \(h^{3}\), and if the function \(f(t, y)\) is linear in \(t\) and \(y\), the error equals \(\phi^{\prime \prime \prime}\left(\bar{t}_{n}\right) h^{3} / 6\). Answer: We showed that the local truncation error for the improved Euler formula is proportional to \(h^3\) by calculating the error \(e_{n+1}\) using the Taylor expansion and the improved Euler formula. Moreover, we used facts related to partial derivatives of \(f(t, y)\) and Taylor approximation to analyze the error equation's first term. If the function \(f(t, y)\) is linear in \(t\) and \(y\), the error equals \(\phi^{\prime \prime \prime}\left(\bar{t}_{n}\right) h^{3} / 6\).

Step-by-step solution

Compute the error \(e_{n+1}\)

First, we need to compute the error \(e_{n+1}\). It is given by: $$ e_{n+1} = \phi(t_{n+1}) - y_{n+1} $$ To find \(e_{n+1}\), use the provided Taylor expansion of \(\phi(t)\) and the improved Euler formula to get: $$ e_{n+1}=\frac{\phi^{\prime \prime}(t_{n}) h -\left\\{f[t_{n}+h, y_{n}+h f(t_{n},y_{n})]-f(t_{n}, y_{n})\right\\}}{2 !}+\frac{\phi^{\prime \prime \prime}(\bar{I}_{n}) h^{3}}{3 !} $$

Analyze the first term of the right side of the error equation

In order to show that the first term on the right side of the error equation is proportional to \(h^{3}\), we have to make use of facts related to partial derivatives of \(f(t,y)\) (i.e. \(f_t, f_y, f_{tt}, f_{ty},\) and \(f_{yy}\)) and the Taylor approximation for a function \(F(t, y)\). Recall that the Taylor approximation with remainder for a function \(F(t, y)\) of two variables is: $$ F(a+h, b+k)=F(a, b)+F_{t}(a, b) h+F_{y}(a, b) k $$ $$ +\left.\frac{1}{2 !}\left(h^{2} F_{t t}+2 h k F_{t y}+k^{2} F_{y y}\right)\right|_{x=\xi, y=\eta} $$ Use the Taylor approximation for \(f(t, y)\) to find the expression for \(\{f[t_{n}+h, y_{n}+h f(t_{n},y_{n})]-f(t_{n}, y_{n})\}\): $$ f[t_{n}+h, y_{n}+h f(t_{n},y_{n})]-f(t_{n}, y_{n})=f_t(t_n, y_n)h + f_y(t_n, y_n)(hf(t_n, y_n)) $$ $$ +\left.\frac{1}{2 !}\left(h^{2} f_{t t}+2 h (hf(t_{n}, y_{n})) f_{t y}+(hf(t_{n}, y_{n}))^{2} f_{y y}\right)\right|_{t=\xi, y=\eta} $$ Substitute the value of \(\phi^{\prime \prime}(t)\) from the given facts into the error equation: $$ \begin{aligned} \phi^{\prime \prime}(t_{n}) h &=\left[f_{t}(t_{n},\phi(t_{n}))+f_{y}(t_{n},\phi(t_{n}))f(t_{n},\phi(t_{n}))\right] h^2 \end{aligned} $$ Now substitute these expressions in the error equation: $$ e_{n+1}= \frac{\left[f_{t}(t_{n},\phi(t_{n}))+f_{y}(t_{n},\phi(t_{n}))f(t_{n},\phi(t_{n}))\right] h^{2} - \left(f_t(t_n, y_n)h + f_y(t_n, y_n)hf(t_n, y_n)\right)}{2 !} $$ $$ +\frac{\phi^{\prime \prime \prime}(\bar{I}_{n}) h^{3}}{3 !} + \frac{\left(h^{2} f_{tt}+2 h (hf(t_{n}, y_{n})) f_{ty}+(hf(t_{n}, y_{n}))^{2} f_{yy}\right)}{2 !} $$ Since \(y_n = \phi(t_n)\), the resulting error is: $$ e_{n+1}\approx \frac{\left[h^3 f_{t y}\right]}{2 !} +\frac{\phi^{\prime \prime \prime} (\bar{I}_{n}) h^3}{3 !}, $$ which is proportional to \(h^3\).

Show that for linear function \(f(t, y)\), the error equals \(\phi^{\prime \prime \prime}\left(\bar{t}_{n}\right) h^{3} / 6\)

If \(f(t, y)\) is linear in \(t\) and \(y\), we have: $$ f_{t t} = f_{t y} = f_{y y} = 0 $$ In this case, the expression for the first term in the error equation becomes: $$ \frac{\left[h^3 f_{t y}\right]}{2 !} = 0 $$ Thus, the error is: $$ e_{n+1}=\frac{\phi^{\prime \prime \prime}(\bar{t}_{n}) h^3}{3 !} = \frac{1}{6}\phi^{\prime \prime \prime}(\bar{t}_{n}) h^{3} $$ Hence, we have established that the local truncation error for the improved Euler formula is proportional to \(h^3\), and if the function \(f(t, y)\) is linear in \(t\) and \(y\), the error equals \(\phi^{\prime \prime \prime}\left(\bar{t}_{n}\right) h^{3} / 6\).

Key concepts

Improved Euler Formula

The Improved Euler Formula, also known as Heun’s method, is a straightforward yet powerful technique used to numerically solve ordinary differential equations (ODEs). Unlike the simple Euler's method, which approximates the solution by taking a step forward using the derivative at the beginning of the interval, the improved Euler formula does a predefined 'correction' step.

After predicting a new value with Euler's step, it takes a second look - averaging the derivative at the beginning and at the end of the interval before taking the final step. This results in a much more accurate estimate of the function's value at the next point, thus reducing the local truncation error compared to the standard Euler method.

By accounting for the changes in the derivative over the step, you essentially capture some curvature of the function between the old and new time points, making the improved Euler a second-order method.

Taylor Series Expansion

The Taylor series expansion is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. It is widely used in approximating functions when dealing with differential equations and numerical analysis.

The beauty of the Taylor series lies in its ability to approximate nonlinear functions using a series of polynomials, which are much easier to compute. In the context of solving differential equations, this expansion allows us to express a function value at a future time based on its current value and derivatives, which are linked to the function's behavior and rate of change.

For example, in the solution provided, the Taylor series is used to expand the solution \( \phi(t_{n}+h) \) of a differential equation around the current point \( t_{n} \). This provides an expression for the local truncation error, which is an estimation of the error introduced when we approximate the true solution with a numerical method like the improved Euler formula.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They are pivotal in science, engineering, and mathematics because they describe various phenomena such as the rates of chemical reactions, motion of objects under force, heat conduction, and population dynamics, among many others.

When presented with a differential equation, the goal is often to find a function (or functions) that satisfy the equation, which represents the behavior of a physical system. These equations can be either ordinary differential equations (ODEs), which involve functions of a single variable and their derivatives, or partial differential equations (PDEs), involving functions of multiple variables and their partial derivatives.

The improved Euler formula is a technique specifically used for approximating solutions to ODEs. The local truncation error in such methods provides insight into how much error we are incurring in our approximation at each step of the numerical solution process. This error analysis is crucial for understanding the accuracy and stability of the numerical method being applied.