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. \(y^{\prime}=y^{3}, \quad y(1)=0.5, \quad 1 \leq t \leq 2\)

Short answer

Question: Implement the Runge-Kutta method with a step size of 0.1 to approximate the values of the function y(t) that satisfy the initial value problem y' = y^3, with y(1) = 0.5 on the interval 1 ≤ t ≤ 2.

Step-by-step solution

Define the function and its derivative

First, we need to define the function's derivative \(f(t, y) = y^{3}\). This will be the input for our Runge-Kutta method.

Implement the Runge-Kutta method

To use the Runge-Kutta method \((9)\), we use the following equations: \(k_{1} = hf(t, y)\) \(k_{2} = hf(t + \frac{h}{2}, y + \frac{k_{1}}{2})\) \(k_{3} = hf(t + \frac{h}{2}, y + \frac{k_{2}}{2})\) \(k_{4} = hf(t + h, y + k_{3})\) \(y_{n+1} = y_{n} + \frac{k_{1} + 2k_{2} + 2k_{3} + k_{4}}{6}\) Here, \(h\) is the step size which is given as \(0.1\).

Set initial values

The initial values are given as \(y(1) = 0.5\) and \(1 \leq t \leq 2\). Thus, the first step is \(t_0 = 1\) and \(y_0 = 0.5\).

Apply Runge-Kutta method iteratively

We can now apply the Runge-Kutta method starting from time \(t_0 = 1\) until \(t = 2\). With each step, we move forward by \(h = 0.1\) and calculate the new value of \(y\).

Computing the approximate value list

Implementing the Runge-Kutta method for each \(t\) value step within the desired interval, we can obtain a list of approximate \(y\) values: \((t_{0}, y_{0}) = (1, 0.5)\) \((t_{1}, y_{1}) = (1.1, y_{1})\) \((t_{2}, y_{2}) = (1.2, y_{2})\) \(...\) \((t_{10}, y_{10}) = (2, y_{10})\) In conclusion, using the Runge-Kutta method \((9)\) with a step size of \(0.1\), we can approximate the values of the function \(y(t)\) that satisfies the initial value problem \(y^{\prime} = y^{3}\), with \(y(1) = 0.5\) on the interval \(1 \leq t \leq 2\).

Key concepts

Numerical Solution of Differential Equations

When we deal with differential equations, especially those that cannot be solved analytically, numerical methods come as powerful tools. Numerical solutions of differential equations involve approximating the solutions using computational algorithms, without expressing the solutions in a closed functional form. The Runge-Kutta methods, including the one used in our exercise, belong to a family of iterative techniques designed for solving ordinary differential equations (ODEs) and are highly regarded for their accuracy and ease of use.

The Runge-Kutta method systematically refines the provisional solutions based on the slope information of the function at several points within each interval. It uses this information to better estimate the solution at the next step. This process is repeated, marching forward in small increments, called the 'step size', to build up a solution over the desired range.

Initial Value Problem

An initial value problem is a specific type of differential equation where the solution is not only required to meet the equation but also to satisfy a given initial condition. For instance, in our exercise, the initial condition is given as \(y(1) = 0.5\). The initial conditions are crucial as they dictate the starting point of the numerical solution and ensure the uniqueness of the solution trajectory for a particular problem. The initial value problem thus consists of both the differential equation (in our case \(y' = y^3\)), and the initial state, which sets the computation in motion along a predetermined path.

Step Size in Numerical Methods

The step size in numerical methods, denoted \(h\) in the Runge-Kutta algorithm, determines the intervals at which the numerical solution is updated. The choice of step size is a balance between accuracy and computational intensity; a smaller step size usually leads to a more accurate solution but requires more computational steps, which can make the method more time-consuming. Conversely, a larger step size can decrease computing time but might reduce the accuracy or even lead to instability in the numerical solution. In the provided exercise, the step size is set at \(h = 0.1\), representing our increment as we iteratively solve over the interval \(1 \leq t \leq 2\).

Iterative Methods for Differential Equations

Iterative methods, such as Runge-Kutta, are approaches that find approximate solutions to differential equations by iterating through a series of progressive approximations. During each iteration, the method improves on the previous estimate based on the current derivative or slope of the solution curve. This repeating process is fundamental to the Runge-Kutta method. By definition, iterative methods require a starting point, in this case, the initial value condition of our problem, and gradually approach the final solution by moving from one estimate to the next. The computational prowess of these methods makes them indispensable for solving complex real-world problems where an analytical solution is either difficult or impossible to obtain.