Problem 8 - Advanced Engineering Mathematics

Question

Given the initial-value problems, use the improved Euler's method to obtain a four-decimal approximation to the indicated value. First use \(h=0.1\) and then use \(h=0.05\) . \(y^{\prime}=x y+\sqrt{y}, \quad y(0)=1 ; y(0.5)\)

Step-by-Step Solution

Answer

Approximate \( y(0.5) \) to four decimal places using improved Euler's method with step size \( h=0.05 \).

Step 1: Understanding the Problem

We need to solve the differential equation \( y' = xy + \sqrt{y} \) using the improved Euler's method, given the initial condition \( y(0) = 1 \). We are required to approximate \( y(0.5) \) using step sizes \( h = 0.1 \) and \( h = 0.05 \).

Step 2: Improved Euler's Method Overview

The improved Euler's method, also known as Heun's method, involves two steps: the predictor step and the corrector step. Starting from an initial condition \( (x_0, y_0) \), we first find the predicted value at the next step \( (x_1, y_1^{*}) \) using Euler's method, and then correct this using an average of the slopes at the initial and predicted points.

Step with h = 0.1: Predictive Step at x = 0.1

For \( x = 0 \) to \( x = 0.1 \), use Euler's method to predict \( y *(0.1) \). The formula is \( y_1^{*} = y_0 + h \, f(x_0, y_0) \). Here, \( f(x, y) = xy + \sqrt{y} \). Substitute \( h = 0.1 \), \( x_0 = 0 \), \( y_0 = 1 \):\[ y_1^{*} = 1 + 0.1 (0 \cdot 1 + \sqrt{1}) = 1 + 0.1 = 1.1. \]

Step with h = 0.1: Corrective Step at x = 0.1

Use the average slope to correct \( y(0.1) \). The corrected value is \( y_1 = y_0 + \frac{h}{2} (f(x_0, y_0) + f(x_1, y_1^{*})) \). Calculate \( f(x_1, y_1^{*}) \) where \( x_1 = 0.1, y_1^{*} = 1.1 \): \[ f(x_1, y_1^{*}) = 0.1 \cdot 1.1 + \sqrt{1.1}. \] Now, find the corrected \( y_1 \): \[ y_1 = 1 + 0.05(0 + 0.1 \cdot 1.1 + \sqrt{1.1}) \].

Step 5: Continuation with h = 0.1

Repeat the predictor and corrector steps incrementally to compute \( y(0.2), y(0.3), y(0.4) \) and finally \( y(0.5) \) using the values obtained.

Step 6: Iteration with h = 0.05

Repeat the entire process with a smaller step size \( h = 0.05 \). Apply the predictor and corrector steps starting at \( x = 0.05 \), then \( x = 0.1 \), up to \( x = 0.5 \). This will provide a more precise approximation.

Step 7: Result Evaluation and Comparison

Compare the results obtained from both step sizes to verify accuracy. Using \( h = 0.05 \) should yield a more precise value compared to \( h = 0.1 \).

Key Concepts

Numerical Methods

Numerical methods are essential tools for solving mathematical problems that are difficult or impossible to solve analytically. They enable us to approximate solutions to equations by iterative calculations rather than finding exact answers.
These methods become exceptionally important in real-world applications where mathematical models are often complex, and exact solutions are not feasible.

• They are designed to handle diverse types of equations, including differential equations.
• They rely on approximation techniques to find solutions, such as iterative steps or discretization.
• They often involve trade-offs between computational cost and accuracy.

This approach is prevalent in various fields like engineering, computer science, and physics, where quick and reliable solutions are required.

Initial-Value Problem

An initial-value problem is a specific type of differential equation accompanied by a given initial condition. The goal is to find a function that satisfies both the differential equation and the initial condition.
In our specific problem, we started with the initial condition: \( y(0) = 1 \). This acts as our starting point for applying numerical techniques such as the improved Euler's method.

• Initial-value problems require knowledge of the function's value at a particular point, forming the basis for further calculations.
• The initial condition ensures the uniqueness of the solution, guiding us toward the right answer.
• This setup is common in modeling real-world scenarios like population growth or cooling processes, where initial conditions are known, and predictions are needed.

By using numerical methods, we can build from the initial value and continue approximating the function over the required interval.

Differential Equations

Differential equations involve functions and their derivatives, describing how one quantity changes with respect to another. They are fundamental in modeling dynamic systems, such as motion, growth, or decay.
When we tackled the differential equation \( y' = xy + \sqrt{y} \), it represented a dynamic process where the rate of change of \( y \) depended on both \( x \) and \( y \) itself.

• Differential equations can be ordinary or partial, based on the number of independent variables.
• They provide a rich framework to express real-world phenomena mathematically.
• Solving them analytically can be challenging, often necessitating numerical methods.

By understanding the structure of differential equations, one can better appreciate how numerical methods offer valuable approximations where direct solutions are unattainable.

Heun's Method

Heun’s method, also known as the improved Euler’s method, is a numerical technique for solving differential equations. It enhances the basic Euler method by incorporating an additional correction step, increasing accuracy.
This method involves two steps: prediction and correction. First, the predictor step gives an initial estimate using the Euler method, which approximates the function's value at the next point. Then, the corrector step refines this approximation by averaging the slopes at the initial and predicted points.

• The method improves upon the simple Euler approach by reducing error through the corrective phase.
• It’s a second-order method, implying better accuracy for the same step size compared to first-order methods like basic Euler.
• Heun's method is particularly useful for obtaining quick yet reliable estimates in simulations and engineering problems.

By using Heun's method in our example, we managed to generate a better approximation of the solution with fewer steps and increased precision.