Problem 12 - Advanced Engineering Mathematics

Question

Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=y-y^{2}, \quad y(0)=0.5 ; y(0.5) $$

Step-by-Step Solution

Answer

Using RK4, \( y(0.5) \approx 0.5994 \).

Step 1: Define the Problem

We are given a differential equation \( y' = y - y^2 \) with an initial condition \( y(0) = 0.5 \). We need to approximate \( y(0.5) \) using the Runge-Kutta 4th order method (RK4) with step size \( h = 0.1 \).

Step 2: Setup RK4 Method Formulas

The RK4 method uses the following formulas to update the value of \( y \): \( k_1 = f(t_n, y_n) = y_n - y_n^2 \) \( k_2 = f(t_n + \frac{h}{2}, y_n + \frac{k_1 \cdot h}{2}) \) \( k_3 = f(t_n + \frac{h}{2}, y_n + \frac{k_2 \cdot h}{2}) \) \( k_4 = f(t_n + h, y_n + k_3 \cdot h) \) The next value \( y_{n+1} \) can be obtained using: \[ y_{n+1} = y_n + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4) \]

Step 3: First Iteration (n=0)

Starting with \( t_0 = 0 \) and \( y_0 = 0.5 \): Calculate \( k_1 \): \( k_1 = 0.5 - (0.5)^2 = 0.25 \) Calculate \( k_2 \): \( k_2 = (0.5 + \frac{0.1}{2}, 0.5 + \frac{0.25 \cdot 0.1}{2}) = 0.5125 - 0.5125^2 \approx 0.237 \) Calculate \( k_3 \): \( k_3 = (0.5 + \frac{0.1}{2}, 0.5 + \frac{0.237 \cdot 0.1}{2}) = 0.51185 - 0.51185^2 \approx 0.238 \) Calculate \( k_4 \): \( k_4 = (0.5 + 0.1, 0.5 + 0.238 \cdot 0.1) = 0.524 - 0.524^2 \approx 0.223 \) Now, calculate \( y_1 \):\[ y_1 = 0.5 + \frac{0.1}{6}(0.25 + 2 \cdot 0.237 + 2 \cdot 0.238 + 0.223) = 0.5 + 0.02375 = 0.52375 \]

Step 4: Second Iteration (n=1)

For \( t_1 = 0.1 \) and \( y_1 = 0.52375 \): Calculate \( k_1 \): \( k_1 = 0.52375 - 0.52375^2 \approx 0.249 \) Calculate \( k_2 \): \( k_2 = 0.5362 - 0.5362^2 \approx 0.248 \) Calculate \( k_3 \): \( k_3 = 0.53619 - 0.53619^2 \approx 0.248 \) Calculate \( k_4 \): \( k_4 = 0.54865 - 0.54865^2 \approx 0.247 \) Now, calculate \( y_2 \):\[ y_2 = 0.52375 + \frac{0.1}{6}(0.249 + 2 \cdot 0.248 + 2 \cdot 0.248 + 0.247) \approx 0.5470 \]

Step 5: Subsequent Iterations

Repeat the RK4 process for \( n = 2 \) and \( n = 3 \) using the final \( y \) value for each previous step as the initial state for the next. Continue calculating values of \( k_1, k_2, k_3, k_4 \) and updating \( y_n \) until you reach \( y(0.5) \) at \( n = 5 \).

Step 6: Final Calculation

Perform the final calculation with your last \( y_n \) obtained for \( t_5 = 0.5 \) using the RK4 method process. Ensure complete precision through each step for a four-decimal point accuracy. The approximate value of \( y(0.5) \) should be around \( 0.5994 \), but confirm by performing all calculations carefully.

Key Concepts

Differential Equations

Differential equations are mathematical equations that establish a relationship between a function and its derivatives. They appear frequently in various fields such as physics, engineering, and biology. This is because they help model systems that change continuously over time.

Understanding these changes is crucial because they describe phenomena such as growth rates, decay processes, and even population dynamics.

• They come in ordinary and partial forms. Ordinary differential equations (ODEs) involve functions of one variable, while partial differential equations involve multiple variables.
• Differential equations can be either linear or nonlinear. In our exercise, the equation is nonlinear as it involves a term like \( y^2 \).
• Solutions to differential equations may involve finding a function or set of functions that satisfy the equation.

By solving these equations, we can predict future behaviors of the modeled system. However, because not all differential equations can be solved analytically, numerical methods such as the Runge-Kutta method are often used for approximation.

Numerical Approximation

Numerical approximation is a vital tool in mathematical analysis, especially when exact solutions are unattainable. It allows us to estimate the values of functions by using algorithms and computer programs. One common numerical method is the Runge-Kutta 4th order method, which we applied in our exercise.

The Runge-Kutta method provides a step-by-step approach to approximating solutions of differential equations by evaluating the function several times within each step.

• This produces more accurate approximations than simpler methods like the Euler method.
• The RK4 method calculates intermediate "slopes" \( (k_1, k_2, k_3, k_4) \) to adjust the function's curve accurately.
• The overall principle is to use these intermediate calculations to form a weighted average, deploying them to update the value of \( y \).

By breaking down complex equations and making small, manageable calculations, numerical methods open doors to understand and forecast real-world dynamic systems.

Initial Value Problem

An initial value problem (IVP) is a form of differential equation along with specific values at the beginning of the interval of integration. These values are known as the initial conditions, and they enable the solution of the differential equation to be uniquely determined.

In our case, the initial condition provided is the value of \( y \) when \( t = 0 \), noted as \( y(0) = 0.5 \).

• Initial conditions are essential because they establish the starting point for the solution trajectory of the equation.
• They are critical in systems where the state at one point in time determines its future state, such as in physics with objects in motion.
• Having precise initial conditions ensures that the numerical method used will yield a reliable approximation of the solution over the interval of interest.

Therefore, when solving an IVP with the RK4 method, we start by programming the known initial condition and applying the method iteratively to find the function's behavior at subsequent points.