Question

Obtain a formula for the local truncation error for the Euler method in terms of \(t\) and the solution \(\phi\) $$ y^{\prime}=2 t+e^{-t y}, \quad y(0)=1 $$

Short answer

Answer: The local truncation error for the Euler method, denoted by τ, can be represented in terms of t and the solution φ as: $$ \tau = \phi(t_{n+1}) - \phi(t_n) - h(2t_n + e^{-t_n \phi(t_n)}) $$ where h is the step size, and φ(t) is the true solution to the given ODE.

Step-by-step solution

Derive the Euler method formula

For a first-order ordinary differential equation (ODE) given by: $$ y^{\prime}=f(t,y) $$ The Euler method approximates the next value of the solution, \(y_{n+1}\), using the following formula: $$ y_{n+1} = y_n + hf(t_n, y_n) $$ where \(h\) is the step size, \(t_n\) is the current time step, and \(y_n\) is the current solution.

Identify the function f(t, y) for the given ODE

In our given ODE, the function \(f(t,y)\) is: $$ f(t, y) = 2t + e^{-ty} $$

Apply Euler method formula to calculate the next step approximation

To obtain a formula for the local truncation error in terms of \(t\) and the solution \(\phi\), we should input the given ODE in Euler method formula: $$ y_{n+1} = y_n + h(2t_n + e^{-t_n y_n}) $$

Calculate the true value of y

To evaluate the truncation error, we need the true solution of the ODE: $$ \phi(t) $$ Alternatively, we can represent the true value at the time step \(t_{n+1}\) as: $$ \phi(t_{n+1}) $$

Calculate the local truncation error

The local truncation error, \(\tau\), is the difference between the true value of y at the next step and the approximate value of y at the next step: $$ \tau = \phi(t_{n+1}) - y_{n+1} $$ Using the equation for \(y_{n+1}\) found in Step 3, we have: $$ \tau = \phi(t_{n+1}) - (y_n + h(2t_n + e^{-t_n y_n})) $$

Express the local truncation error in terms of t and the solution

As \(\phi(t)\) is the true solution to the ODE, we can substitute \(\phi(t_n) = y_n\) and express the truncation error as: $$ \tau = \phi(t_{n+1}) - \phi(t_n) - h(2t_n + e^{-t_n \phi(t_n)}) $$ Now, we have expressed the local truncation error for the Euler method in terms of \(t\) and the solution \(\phi\).

Key concepts

Local Truncation Error

In numerical analysis, the local truncation error is crucial to understanding how closely our numerical methods, like the Euler method, approximate the true solution of a differential equation. In essence, the local truncation error measures the error made in one step of the method. It is defined as the difference between the true solution at the next step and the calculated solution using an approximation method at that same step.

For the Euler method, consider it as:

• True Solution: \( \phi(t_{n+1}) \)
• Approximation: \( y_{n+1} = y_n + h f(t_n, y_n) \)

The local truncation error \( \tau \) is given by: \[ \tau = \phi(t_{n+1}) - y_{n+1} \] This error depends on the step size \( h \) used in the method. The smaller the step size, the smaller the local truncation error, making the approximation more precise.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are mathematical expressions involving functions and their derivatives. They are called 'ordinary' because they deal with one independent variable, often representing time. In this exercise, the ODE is given by: \[ y^{\prime} = 2t + e^{-ty} \] ODEs are powerful tools for modeling a wide range of phenomena in science and engineering due to their ability to represent how rates of change are influenced by various factors. Solving ODEs analytically can be complex, hence numerical methods like the Euler method are often used to find approximate solutions.

Numerical Approximation

Numerical approximation refers to techniques that help approximate solutions of mathematical problems that are difficult to solve analytically. The Euler method is one such technique used to approximate solutions of ordinary differential equations. It is a simple yet effective way of incrementally building up an approximate solution over a range of values.

The central idea in numerical approximation using Euler's method is to use known values at a given point and the slope of the tangent line (as estimated by the differential equation) to estimate the value at the next point. This method offers an easy-to-understand approach to solving complex differential equations numerically, especially when you're just getting started.

Step Size

Step size, represented as \( h \), is a fundamental component in Euler's method and other numerical approximation methods. It determines the interval at which approximations are calculated. A smaller step size means calculating more intermediate points and generally leads to a more accurate approximation. However, it also means more computations and increased processing time.

Consider these points when choosing a step size:

• Smaller Step Size: Decreases truncation error, improves accuracy, increases computations.
• Larger Step Size: Increases truncation error, decreases accuracy, reduces computations.
• Balanced Choice: Aim for a balance between computational efficiency and accuracy.

The choice of step size plays a critical role in finding a good trade-off between the precision of results and the computational resources available.