Problem 7 - Advanced Engineering Mathematics

Question

Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=e^{-y}, \quad y(0)=0 ; y(0.5) $$

Step-by-Step Solution

Answer

Using RK4 with step size 0.1, \( y(0.5) \approx 0.4055 \).

Step 1: Understand the Problem

We are to use the Runge-Kutta 4th order method (RK4) to approximate the value of the function at a specific point. The function is defined by the differential equation \( y' = e^{-y} \) with an initial condition \( y(0) = 0 \). We need to find \( y(0.5) \) using a step size \( h = 0.1 \).

Step 2: Initial Setup

First, note the initial condition: \( x_0 = 0 \) and \( y_0 = 0 \). The step size is given as \( h = 0.1 \). We aim to find \( y(0.5) \), which means we will calculate \( y_n \) for \( x_n = 0.5 \). Thus, we need 5 steps (since \( 0.5/0.1 = 5 \)).

Step 3: Calculate RK4 Slopes for Step 1 (x=0)

For the first step, calculate the four intermediate slopes:- \( k_1 = h \cdot f(x_0, y_0) = 0.1 \cdot e^{-0} = 0.1 \)- \( k_2 = h \cdot f(x_0 + 0.5h, y_0 + 0.5k_1) = 0.1 \cdot e^{-(0 + 0.05)} = 0.1 \cdot e^{-0.05} \approx 0.0951 \)- \( k_3 = h \cdot f(x_0 + 0.5h, y_0 + 0.5k_2) = 0.1 \cdot e^{-0.0951} \approx 0.0907 \)- \( k_4 = h \cdot f(x_0 + h, y_0 + k_3) = 0.1 \cdot e^{-0.0907} \approx 0.0914 \)

Step 4: Update y-value for Step 1

Calculate the next \(y\)-value:\[y_1 = y_0 + \frac{1}{6} (k_1 + 2k_2 + 2k_3 + k_4)\]Substitute the values:\[y_1 = 0 + \frac{1}{6} (0.1 + 2 \cdot 0.0951 + 2 \cdot 0.0907 + 0.0914) \approx 0.0951\].

Step 5: Repeat RK4 Process for Subsequent Steps

Perform the same RK4 slope calculations and update rules for \( x = 0.1, 0.2, 0.3, 0.4 \), repeating the process similar to step 1. Each resulting value of \( y \) becomes the initial \( y \) for the next step.

Step 6: Final Step Calculation (x=0.5)

For the final calculation at \( x=0.5 \), perform the RK4 method one last time to compute \( y(0.5) \). After calculating, you'll find: \( y(0.5) \approx 0.4055 \).

Key Concepts

Numerical Methods for Differential Equations

When solving differential equations, particularly where analytic solutions are complex or impossible to find, numerical methods provide an invaluable tool. These methods convert continuous problems into discrete ones, making them easier to solve using computational algorithms.

Some well-known numerical methods include:

• Euler's Method: A straightforward approach that involves a simple step-by-step approximation of the function's value.
• Runge-Kutta Methods: More sophisticated and accurate methods like the 4th Order Runge-Kutta (RK4), offering better accuracy by calculating multiple slopes per step.
• Trapezoidal Rule and Predictor-Corrector Methods: These aim to improve precision by considering prior points in the solution process.

By tackling Initial Value Problems (IVPs) in differential equations, these numerical approaches break down complex systems into manageable pieces that can be approximated step by step.

Initial Value Problems

An initial value problem (IVP) in the context of differential equations involves finding a function that satisfies a given differential equation and also meets an initial condition. Initial conditions are specific values known at the start of a problem that the solution must pass through.

The general form of an IVP is:

• The differential equation: \[ y'(x) = f(x, y) \]
• Given condition: \[ y(x_0) = y_0 \]

Here, \( x_0 \) is the initial value of \( x \), and \( y_0 \) is the initial value of \( y \). Solving these problems usually involves numerically approximating the function \( y(x) \) over an interval, ensuring that the solution remains accurate and matches the initial conditions.

Euler's Method

Euler's Method is one of the simplest numerical approaches for solving ordinary differential equations (ODEs) and is particularly used in initial value problems. It starts at the initial point and moves along the tangent to reach the next point. Euler's method is easy to understand and quick to implement.

Here’s how it works:

• Start with the initial values \( (x_0, y_0) \) and step size \( h \).
• At each step, use the equation: \[ y_{n+1} = y_n + h imes f(x_n, y_n) \]
• Calculate the next point \( (x_{n+1}, y_{n+1}) \).
• Repeat the process for the required number of steps or until the desired \( x \) value is reached.

However, while Euler's Method is straightforward, it does come with limitations in terms of precision and stability, especially when larger step sizes are used or when dealing with complex differential equations. To improve accuracy, methods like the 4th Order Runge-Kutta are often preferred.