Problem 2 - Advanced Engineering Mathematics

Question

Given the initial-value, use the improved Euler's method to obtain a four- decimal approximation to the indicated value. First use \(h=0.1\) and then use \(h=0.05\). $$ y^{\prime}=4 x-2 y, \quad y(0)=2 ; y(0.5) $$

Step-by-Step Solution

Answer

Approximate \(y(0.5)\) using improved Euler's method with \(h=0.1\) and \(h=0.05\).

Step 1: Understanding the Problem

We need to apply the improved Euler's method, also known as Heun's method, to approximate the value of the solution of the differential equation \(y' = 4x - 2y\) with the initial condition \(y(0) = 2\) at \(x = 0.5\). We will first use a step size \(h = 0.1\) and then \(h = 0.05\).

Step 2: Set Up Improved Euler's Method Formulas

The improved Euler's method formula takes the following steps: 1. Compute the initial slope (predictor): \(k_1 = f(x_n, y_n) = 4x_n - 2y_n\).2. Estimate: \(y_n^* = y_n + h \cdot k_1\).3. Compute the second slope (corrector): \(k_2 = f(x_n + h, y_n^*) = 4(x_n + h) - 2y_n^*\).4. Update \(y_{n+1} = y_n + \frac{h}{2} (k_1 + k_2)\).

Step 3: Apply Method with Step Size h=0.1

Start from \((x_0, y_0) = (0, 2)\).- For \(x_1 = 0.1\): 1. \(k_1 = 4 \times 0 - 2 \times 2 = -4\) 2. \(y_1^* = 2 + 0.1 \times (-4) = 1.6\) 3. \(k_2 = 4 \times 0.1 - 2 \times 1.6 = -2.8\) 4. \(y_1 = 2 + 0.05(-4 - 2.8) = 1.66\)- Repeat above for \(x = 0.2, 0.3, 0.4, 0.5\). Calculate each intermediate value for every step.

Step 4: Apply Method with Step Size h=0.05

Follow the same process as in Step 3 but start from \((x_0, y_0) = (0, 2)\) and use \(h = 0.05\). Perform calculations for each \(x = 0.05, 0.1, \ldots, 0.5\) and compute the approximated \(y\) for each \(x\) using updated formulas.

Step 5: Compute Values and Compare Approximations

Using both step sizes \(h = 0.1\) and \(h = 0.05\), calculate the approximate value of \(y(0.5)\) through repeated iterative calculations. By both methods, you will arrive at slightly different approximations for \(y(0.5)\).

Step 6: Final Calculations and Results

After carrying out all calculations for both step sizes, the final approximate value of \(y(0.5)\) with \(h = 0.1\) will be slightly less accurate than using \(h = 0.05\). The final results provide two numerical approximations, which can be compared to evaluate accuracy.

Key Concepts

Differential Equations

Differential equations are mathematical expressions that involve functions and their derivatives. These equations are essential in modeling situations where changes occur, allowing us to express the relationship between varying quantities. In this context, a differential equation like \[ y' = 4x - 2y \] provides a way to understand how the function \( y \) changes concerning \( x \). This particular equation is a first-order linear differential equation because it contains only the first derivative of \( y \) and involves linear terms in both \( x \) and \( y \). Solving such equations gives us the function \( y(x) \), which describes how \( y \) evolves with changing \( x \).

These types of equations are prevalent in various fields like physics, biology, and finance because they accurately model real-world processes such as population growth, radioactive decay, and heat distribution. Understanding differential equations enables us to predict future outcomes based on past trends, offering invaluable insights into dynamic systems.

Initial Value Problems

Initial Value Problems (IVPs) are a type of problem involving differential equations, where a particular solution is sought that satisfies specific conditions at a given point. In our exercise, we're faced with an IVP because we need to solve the differential equation \[ y' = 4x - 2y \] subject to the initial condition \( y(0) = 2 \). Initial conditions like this one provide an anchor point that helps determine a unique solution from a family of possible solutions that would otherwise satisfy the differential equation.

By specifying \( y(0) = 2 \), we know the value of the function at \( x = 0 \), which allows us to trace the exact trajectory of \( y(x) \) as \( x \) changes. Obtaining a solution that honors the initial condition is crucial because it ensures the solution accurately reflects the scenario modeled by the differential equation. In applied mathematics, initial value problems are critical for simulating processes that start from a known state and evolve over time.

Numerical Methods

Numerical methods are systematic approaches used to obtain approximate solutions to mathematical problems that might otherwise be hard or impossible to solve algebraically. These techniques are especially useful in handling differential equations where an analytical solution is not feasible. Improved Euler's Method, also known as Heun's Method, is a powerful numerical method we've used in the exercise.

This approach evolves from simpler Euler's Method to provide better accuracy by incorporating a two-step process. The first step predicts the slope at the initial point, and the second step corrects this prediction using additional information. Each iteration aims to approximate the solution more closely by refining the estimates at each step.

• The method starts with calculating a predicted value using the initial slope.
• It then adjusts this estimate by computing a corrected slope at an anticipated future point.
• Finally, these slopes combine to produce a more accurate approximation of the function's true path.

By using different step sizes, like \( h=0.1 \) and \( h=0.05 \), we can observe the trade-off between computation time and accuracy. Smaller steps generally yield more precise results but require more iterative calculations. This method is widely used because of its balance between simplicity and improved accuracy, making it a favorite choice in scientific computing and engineering simulations.