Question

Exercises 23-27: A computer or programmable calculator is needed for these exercises. For the given initial value problem, use the Runge-Kutta method \((9)\) with a step size of \(h=0.1\) to obtain a numerical solution on the specified interval. \(y^{\prime}=y^{2}, \quad y(0)=1, \quad 0 \leq t \leq 0.9\)

Short answer

Question: Using the 4th order Runge-Kutta method with a step size of \(h=0.1\), solve the initial value problem \(y'(t) = y(t)^2\), \(y(0)=1\), on the interval \(0\leq t \leq 0.9\), and provide the numerical solution at each time instance.

Step-by-step solution

1. Rewrite the equation for convenience

We can rewrite our problem as an initial value problem with \(t\) and \(y\). Let \(y'(t) = f(t, y) = y^2\). We have been given the initial value \(y(0)=1\).

2. Define the step size and the interval

Set the step size \(h = 0.1\) and the interval \(0\leq t \leq 0.9\). We have to find the approximate value of \(y\) at these time instances \(t\): \(0, 0.1, 0.2, \dots, 0.9\).

3. Apply the 4th Order Runge-Kutta method

To implement the 4th order Runge-Kutta method, we will follow these steps for each time step \(t_i\): 1. Compute \(k_1 = h \cdot f(t_i, y_i)\) 2. Compute \(k_2 = h \cdot f(t_i + \frac{h}{2}, y_i + \frac{k_1}{2})\) 3. Compute \(k_3 = h \cdot f(t_i + \frac{h}{2}, y_i + \frac{k_2}{2})\) 4. Compute \(k_4 = h \cdot f(t_i + h, y_i + k_3)\) 5. Update \(y_{i+1} = y_i + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\)

4. Calculate the values of \(y\) for the given time instances

Initialize \(y_0 = 1\) and \(t_0 = 0\). First iterate with \(i=0\), then \(i=1\), and so on, until \(i=8\). Compute the values of \(k_1\), \(k_2\), \(k_3\), \(k_4\) and \(y_{i+1}\) in each iteration as described in step 3.

5. Present the approximation values for the given interval

After all iterations, we will have approximate values for \(y\) at the time instances \(t\): \(0, 0.1, 0.2, \dots, 0.9\). These approximate values are the numerical solution to the given initial value problem using the 4th order Runge-Kutta method with step size \(h=0.1\).

Key concepts

Initial Value Problem

An initial value problem is a type of differential equation equipped with a specific starting value or condition. It's like having a puzzle where you know the starting piece and you use it to figure out the rest. In our problem, we begin with a differential equation:

• The differential equation: \( y'(t) = y^2 \)
• The initial condition: \( y(0) = 1 \)

The goal is to determine the function \( y(t) \) that satisfies both the differential equation and the initial condition. Knowing the starting value allows us to "march" into the future, calculating approximate values at various points, which help us see the solution evolve over time.
Initial value problems are fundamental in modeling scenarios such as population growth, heating and cooling processes, and motion of objects.

Numerical Solution

A numerical solution involves approximating the answer to a problem through computational methods, especially when an exact solution is difficult or impossible to find. In the context of differential equations, we track how a function evolves over increments of time or space.
Most differential equations don't have simple solutions, making numerical methods like Runge-Kutta indispensable. These methods allow computers and calculators to "approximate" solutions by performing repeated calculations.
In our exercise, we apply the 4th order Runge-Kutta method to achieve a series of values for \( y(t) \) across specified intervals. By iterating through a series of steps much like a recipe, we obtain values that are quite close to what we might find with exact calculus solutions. This process is crucial in scientific simulations, engineering applications, and financial modeling where time-series predictions are essential.

Differential Equations

Differential equations form the backbone of modeling changes in various scientific and engineering systems. They describe relationships between a function and its rates of change, represented by derivatives.
In our problem, the equation \( y'(t) = y^2 \) establishes how the rate of change of \( y \), the derivative \( y' \), is related to the function itself.

• Types of differential equations: there are ordinary (ODE) and partial (PDE) differential equations.
• Our case is an ordinary differential equation (ODE) because it involves derivatives of a function of a single variable \( t \).

Through differential equations, one can model physical phenomena such as the growth of populations, the motion of planets, and the transfer of heat. Applying solutions like Runge-Kutta, we can tackle these equations even when they are complex, providing valuable insights into how systems evolve.

Step Size

Step size, often denoted as \( h \), is a crucial parameter in numerical methods used to approximate solutions to differential equations. It determines how much we "move forward" in the independent variable between consecutive calculations.
A smaller step size provides a more precise approximation but requires more computations, increasing computational effort. Conversely, a larger step size speeds up calculations but may lose accuracy.
In the Runge-Kutta method, the step size aids in calculating intermediate \( k \) values, crucial for determining the next value of \( y \).

• In this exercise, we use a step size of \( h = 0.1 \).
• This means that each calculated \( y \) value represents an increment of 0.1 in time or space.

Choosing the right step size balances accuracy with computational demands, essential in simulations where too coarse steps might overlook important behavior, and too fine steps take prohibitively longer to compute.