Question

Carry out one step of the Euler method and of the improved Euler method using the step size \(h=0.1 .\) Suppose that a local truncation error no greater than 0.0025 is required. Estimate the step size that is needed for the Euler method to satisfy this requirement at the first step. $$ y^{\prime}=\sqrt{t+y}, \quad y(0)=3 $$

Short answer

Answer: After one step, the Euler's method gives an approximate value of \(y_1 \approx 3.173\), and the Improved Euler's method gives an approximate value of \(y_1 \approx 3.164\). To achieve a local truncation error no greater than 0.0025 at the first step, the estimated step size required for the Euler method is approximately \(h' \approx 0.0037\).

Step-by-step solution

Understand the Euler Method Formula and Steps

To perform one step of the Euler Method, use the following formula: $$y_{n+1} = y_n + h \, f(t_n, y_n)$$ Here, \(f(t_n, y_n) = y'(t_n) = \sqrt{t_n+y_n}\), and we have the given step size \(h=0.1\).

Use the Euler Method to Find the First Step Solution

Using the initial condition \(y(0) = 3\), we can calculate \(y_1\) as follows: $$y_1 = y_0 + h \, f(t_0, y_0) = 3 + 0.1 \, \sqrt{0 + 3} = 3 + 0.1 \times \sqrt{3}$$ Calculate the value: $$y_1 = 3 + 0.1 \times \sqrt{3} ≈ 3.173$$

Understand the Improved Euler Method Formula and Steps

To perform one step of the Improved Euler Method, use the following formula: $$y_{n+1} = y_n + \frac{h}{2} \left( f(t_n, y_n) + f(t_{n+1}, y_n + h \, f(t_n, y_n)) \right)$$ Here, \(f(t_n, y_n) = y'(t_n)=\sqrt{t_n+y_n}\) and we have the given step size \(h=0.1\).

Use the Improved Euler Method to Find the First Step Solution

Using the initial condition \(y(0) = 3\), we can calculate \(y_1\) as follows: $$y_1 = y_0 + \frac{h}{2} \left( f(t_0, y_0) + f(t_1, y_0 + h \, f(t_0, y_0)) \right)$$ $$y_1 = 3 + \frac{0.1}{2} \left( \sqrt{0 + 3} + \sqrt{0.1 + 3 + 0.1 \times \sqrt{3}} \right)$$ Calculate the value: $$y_1 ≈ 3 + \frac{0.1}{2} \left( \sqrt{3} + \sqrt{3.3} \right) ≈ 3.164$$

Estimate the Step Size Required to Achieve the Given Local Truncation Error

We want to find the step size \(h'\) such that the local truncation error for the Euler method is no greater than 0.0025. The local truncation error for the Euler method is approximately equal to the product of the step size (\(h'\)) and the second-order derivative evaluated at the initial point \((t_0, y_0)\): $$|h' \cdot y''(t_0, y_0)| \leq 0.0025$$ First, we need to find the second-order derivative of y with respect to t: $$y' = \sqrt{t+y}$$ $$y'' = \frac{1}{2 \sqrt{t+y}} (1 + y')$$ Since \(t_0=0\) and \(y_0=3\), and we found that \(y'(t_0) = \sqrt{t_0 + y_0} = \sqrt{3}\), we can now calculate \(y''(t_0, y_0)\): $$y''(t_0, y_0) = \frac{1}{2 \sqrt{3}} (1 + \sqrt{3})$$ Now, substitute into the inequality: $$|h' \cdot \frac{1}{2 \sqrt{3}} (1 + \sqrt{3})| \leq 0.0025$$ $$h' \geq \frac{0.0025 \cdot 2 \sqrt{3}}{1 + \sqrt{3}}$$ Calculate the value: $$h' ≈ 0.0037$$ So, the step size required for the Euler method to satisfy the local truncation error requirement of no greater than 0.0025 at the first step is approximately 0.0037.

Key concepts

Differential Equations

Differential equations are at the heart of much of modern science and engineering. They describe how a quantity changes in space and time, often in relation to its current state. For example, the equation
\[y' = \/(t + y), \quad y(0)=3\]
expresses a rate of change (\(y'\)) that's dependent on both the independent variable \(t\) and the unknown function \(y(t)\). This particular equation might model phenomena like population growth, chemical reactions, or motion under a force. Solving such an equation analytically can be challenging or even impossible in many cases, hence numerical methods like the Euler method are employed to find approximate solutions.

Local Truncation Error

When we approximate the solution of differential equations using numerical methods, we introduce a small error with each step we take. This is called the local truncation error, and it's a measure of how much the numerical solution deviates from the true solution at a single step, assuming that all previous steps were exact. In the context of the Euler method, the local truncation error after one step is proportional to the square of the step size (
\(h^2\)) times the second derivative of the true solution, as it is a first-order method. Reducing the step size \(h\) decreases this error, but at the cost of needing more computational steps. Our aim is usually to balance accuracy with computational efficiency.

Improved Euler Method

Also known as Heun's method, the Improved Euler Method is a simple and commonly used modification of the Euler method that improves accuracy by using an average of the slopes at the beginning and end of the interval. The formula for the Improved Euler Method is:
\[y_{n+1} = y_n + \frac{h}{2} (f(t_n, y_n) + f(t_{n+1}, y_n + h f(t_n, y_n)))\]
Notice how this incorporates an additional term compared to the standard Euler method. By taking into account the slope at the end of the interval as well as at the beginning, it can better approximate the true curve of the solution. This results in a smaller local truncation error for each step than the regular Euler method, leading to a more accurate solution over many steps.

Numerical Methods

Numerical methods are algorithms used for approximating the solutions to mathematical problems that cannot be solved exactly (analytically). They are essential in fields such as physics, engineering, and finance where precise analytical solutions are often unobtainable. Numerical methods for differential equations, such as the Euler method, the Improved Euler method, and others like the Runge-Kutta methods, transform the differential equations into algebraic equations which are solvable using standard computational techniques. One of the key goals in these methods is to minimize errors while maximizing computational efficiency.

Initial Value Problem

Our discussion centers on a specific type of differential equation problem known as an initial value problem (IVP). An IVP involves finding a function \(y(t)\) that satisfies a given differential equation and who also meets an initial condition, for example, \(y(0)=3\). The 'initial value' is the starting point, from which we use numerical methods to step through and predict subsequent values. It's essential in ensuring that the unique solution we are computing is correct for the specific scenario we're analyzing.