Question

In each exercise, (a) Solve the initial value problem analytically, using an appropriate solution technique. (b) For the given initial value problem, write the Heun's method algorithm, $$ y_{n+1}=y_{n}+\frac{h}{2}\left[f\left(t_{n}, y_{n}\right)+f\left(t_{n+1}, y_{n}+h f\left(t_{n}, y_{n}\right)\right)\right] . $$ (c) For the given initial value problem, write the modified Euler's method algorithm, $$ y_{n+1}=y_{n}+h f\left(t_{n}+\frac{h}{2}, y_{n}+\frac{h}{2} f\left(t_{n}, y_{n}\right)\right) . $$ (d) Use a step size \(h=0.1\). Compute the first three approximations, \(y_{1}, y_{2}, y_{3}\), using the method in part (b). (e) Use a step size \(h=0.1\). Compute the first three approximations, \(y_{1}, y_{2}, y_{3}\), using the method in part (c). (f) For comparison, calculate and list the exact solution values, \(y\left(t_{1}\right), y\left(t_{2}\right), y\left(t_{3}\right)\). \(y^{\prime}=-y, \quad y(0)=1\)

Short answer

Question: Calculate the first three approximations of the given initial value problem using Heun's and Modified Euler's methods, and compare the approximations with the exact solution values. Initial value problem: $$ y^{\prime}=-y, \quad y(0)=1 $$ Analytical solution: $$ y(t) = e^{-t} $$ Approximations: 1. Heun's method: $$ y_1 \approx \phantom{-} $$ $$ y_2 \approx \phantom{-} $$ $$ y_3 \approx \phantom{-} $$ 2. Modified Euler's method: $$ y_1 \approx \phantom{-} $$ $$ y_2 \approx \phantom{-} $$ $$ y_3 \approx \phantom{-} $$ Exact solution values: $$ y(t_1) = e^{-t_1} $$ $$ y(t_2) = e^{-t_2} $$ $$ y(t_3) = e^{-t_3} $$ Compare the approximations from Heun's and Modified Euler's methods with the exact solution values and analyze their accuracy.

Step-by-step solution

To solve the given initial value problem, we can use separation of variables method: $$ \frac{dy}{dt} = -y $$ Separate variables and integrate: $$ \int \frac{1}{y}\,dy = - \int dt $$ By integrating the equation, we get $$ \ln |y| = -t + C $$ Taking the exponential of both sides and solving for the constant C by applying the initial condition y(0) = 1, the analytical solution is: $$ y(t) = e^{-t} $$ #b. Heun's method algorithm#

Based on the exercise, the Heun's method algorithm is as follows: $$ y_{n+1}=y_{n}+\frac{h}{2}\left[f\left(t_{n}, y_{n}\right)+f\left(t_{n+1}, y_{n}+h f\left(t_{n}, y_{n}\right)\right)\right] $$ Here, the function f(t, y) can be represented as: $$ f(t, y) = y^{\prime} = -y $$ #c. Modified Euler's method algorithm#

According to the exercise, the modified Euler's method algorithm is: $$ y_{n+1}=y_{n}+h f\left(t_{n}+\frac{h}{2}, y_{n}+\frac{h}{2} f\left(t_{n}, y_{n}\right)\right) $$ Similarly, f(t, y) can be represented as: $$ f(t, y) = y^{\prime} = -y $$ #d. Heun's method approximations#

We will use a step size h=0.1 to calculate the first three approximations using Heun's method. Applying the given function f(t, y) and the initial condition, we find the values as follows: $$ y_1 = y_0 + \frac{h}{2} [f(t_0, y_0) + f(t_1, y_0 + h f(t_0, y_0))] $$ $$ y_2 = y_1 + \frac{h}{2} [f(t_1, y_1) + f(t_2, y_1 + h f(t_1, y_1))] $$ $$ y_3 = y_2 + \frac{h}{2} [f(t_2, y_2) + f(t_3, y_2 + h f(t_2, y_2))] $$ #e. Modified Euler's method approximations#

We will use a step size h=0.1 to calculate the first three approximations using the Modified Euler's method. Applying the given function f(t, y) and the initial condition, we find the values as follows: $$ y_1 = y_0 + h f\left(t_0 + \frac{h}{2}, y_0 + \frac{h}{2} f(t_0, y_0)\right) $$ $$ y_2 = y_1 + h f\left(t_1 + \frac{h}{2}, y_1 + \frac{h}{2} f(t_1, y_1)\right) $$ $$ y_3 = y_2 + h f\left(t_2 + \frac{h}{2}, y_2 + \frac{h}{2} f(t_2, y_2)\right) $$ #f. Comparing with the exact solution#

To compare the approximations from parts (d) and (e) with the exact solution, we need to calculate the exact solution values at \(t_1\), \(t_2\), and \(t_3\). We have the analytical solution of the problem: $$ y(t)= e^{-t} $$ Now we need to evaluate the function at \(t_1\), \(t_2\), and \(t_3\) (keeping in mind that the step size was 0.1): $$ y(t_1) = e^{-t_1} $$ $$ y(t_2) = e^{-t_2} $$ $$ y(t_3) = e^{-t_3} $$ With these exact solution values, the approximations from Heun's method and Modified Euler's method can be compared and analyzed for their accuracy.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They are crucial in modeling real-world phenomena where change is a constant factor, such as in physics, engineering, and biology. The primary goal when working with differential equations is to find a function or set of functions that satisfy the equation given particular conditions.
The specific equation presented in this exercise is a first-order linear differential equation: \[ \frac{dy}{dt} = -y \]Here, the function's derivative with respect to time \(t\) is proportional to the negative of the function itself. Solutions to this type of differential equation often involve separation of variables, a method where you can solve it by integrating each side independently, as shown in the problem's solution. By integrating and using initial conditions, we can solve for the constant of integration and find a unique solution.

Initial Value Problems

An initial value problem in the realm of differential equations specifies the value of the unknown function at a particular point. This additional information is crucial as it helps to determine a unique solution among the possibly infinite solutions to a differential equation. For the equation \( \frac{dy}{dt} = -y \), with the initial condition \( y(0) = 1 \), it means we are seeking a function \( y(t) \) that not only satisfies the differential equation but also that \( y(0) \) equals 1.
This initial condition is applied after finding the general solution to the differential equation, helping pinpoint the constant of integration, as demonstrated in the solution where the specific solution \( y(t) = e^{-t} \) was derived from the general solution by applying the initial condition.

Heun's Method

Heun's method, also known as the improved Euler's method, is a numerical approach to solving differential equations. It's a predictor-corrector method that provides a more accurate estimate by taking the average of two predicted slopes for each time step.
The algorithm, explained through the equation provided:\[ y_{n+1} = y_{n} + \frac{h}{2} \left[ f(t_{n}, y_{n}) + f(t_{n+1}, y_{n} + h f(t_{n}, y_{n})) \right] \]The idea here is to first predict the slope at the beginning of the interval \( f(t_{n}, y_{n}) \) and then again at the end of the interval using the predicted "next" value \( y_{n} + h f(t_{n}, y_{n}) \). These predictions are averaged to give an improved estimate of the slope, reducing the error from Euler's method.
Heun's method is especially useful when seeking to balance computational efficiency with accuracy, making it ideal for simple models where some degree of error is permissible, but something more precise than Euler's method is desired.

Euler's Method

Euler's method is the simplest and most straightforward numerical method for approximating solutions to differential equations. It works by taking a small step along the direction given by the slope (derivative) at the current point, effectively drawing a tangent line to estimate the next point.
The equation is given by:\[ y_{n+1} = y_{n} + h f \left(t_n + \frac{h}{2}, y_n + \frac{h}{2} f(t_n, y_n)\right) \]This version of Euler’s method, often referred to as the modified or improved Euler's method, includes a midpoint to predict the slope, thus improving accuracy over the basic Euler iteration. By using information from the midpoint, this method attempts to "see" further in the function, providing a slightly more accurate step than a simple calculation at the endpoint would.
Despite its simplicity, Euler's method can be quite powerful for small step sizes or when a quick, rough estimate is needed. However, it's important to note that the accuracy decreases with larger step sizes, prompting the use of more sophisticated methods like Heun's when precision is needed.