Question

We ask you to use the fourth order Runge-Kutta method (14) to solve the problems in Exercises \(20-23\) of Section \(7.3\). \(y^{\prime}=\frac{1+y^{2}}{1+t}, y(0)=0 . \quad\) The exact solution is \(y(t)=\tan [\ln (1+t)]\)

Short answer

In this problem, we are solving the first-order differential equation \(y' = \frac{1+y^2}{1+t}\) using the fourth-order Runge-Kutta method, with the given initial condition \(y(0) = 0\). The exact solution is \(y(t) = \tan[\ln(1+t)]\). The steps to solve this problem are: 1. Understand the fourth-order Runge-Kutta Method formula. 2. Apply the formula to the given equation. 3. Implement the fourth-order Runge-Kutta Method, choosing an appropriate step size \(h\) and number of iterations. 4. Compare the obtained approximations \(y_i\) with the corresponding exact values, calculated from the exact solution \(y(t) = \tan[\ln(1+t)]\). The comparison will give you an indication of the accuracy of the approximation and help determine if adjustments to the step size or number of iterations are necessary.

Step-by-step solution

Understand the fourth-order Runge-Kutta Method formula

The fourth-order Runge-Kutta method is an iterative numerical technique used to solve ordinary differential equations of the form \(y' = f(t, y)\). The general formula for a single iteration is as follows: 1. \(k_1 = hf(t, y)\) 2. \(k_2 = hf(t + \frac{h}{2}, y + \frac{k_1}{2})\) 3. \(k_3 = hf(t + \frac{h}{2}, y + \frac{k_2}{2})\) 4. \(k_4 = hf(t + h, y + k_3)\) 5. Updated value \(y_{new} = y + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\) Here, \(h\) is the step size and \(y(t)\) is the value we are trying to approximate for the next step.

Apply the formula to the given equation

Now, we are going to apply the fourth-order Runge-Kutta method to the given equation. We use \(y' = \frac{1+y^{2}}{1+t}\), with initial condition \(y(0)=0\). Please note that depending on the desired accuracy, you can choose your \(h\) and the number of iterations accordingly.

Implement the fourth-order Runge-Kutta Method

Here, we present an example implementation of the fourth-order Runge-Kutta method for the given problem (with a sample step size \(h\) and number of iterations): 1. Choose the step size \(h\) and the number of iterations. 2. Set the initial values \(t_0 = 0\) and \(y_0 = 0\). 3. For each iteration \(1 \leq i \leq n\): a. Calculate \(k_1 = h \cdot \frac{1+y_{i-1}^{2}}{1+t_{i-1}}\) b. Calculate \(k_2 = h \cdot \frac{1+(y_{i-1}+\frac{k_1}{2})^2}{1+t_{i-1}+\frac{h}{2}}\) c. Calculate \(k_3 = h \cdot \frac{1+(y_{i-1}+\frac{k_2}{2})^2}{1+t_{i-1}+\frac{h}{2}}\) d. Calculate \(k_4 = h \cdot \frac{1+(y_{i-1}+k_3)^2}{1+t_{i-1}+h}\) e. Update the value \(y_i = y_{i-1} + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\) f. Update the value \(t_i = t_{i-1} + h\)

Compare the approximation with the exact solution

After completing the iterations, compare the obtained values \(y_i\) with the corresponding exact values, calculated from \(y(t) = \tan[\ln(1+t)]\). This will give you an indication of the accuracy of the fourth-order Runge-Kutta method for the given problem. Adjust the step size \(h\) and the number of iterations if necessary to achieve the desired accuracy.

Key concepts

Numerical Methods

Numerical methods are techniques used to find approximate solutions to mathematical problems that cannot be easily solved analytically. They are especially useful in applied mathematics and engineering, where real-world problems can often be complex. One of the most popular numerical methods for solving differential equations is the Runge-Kutta method. This method allows us to find approximate solutions to differential equations through calculations that involve incremental steps.

• The main advantage of numerical methods like Runge-Kutta is their ability to provide solutions where analytical methods fail.
• They help in solving equations that describe real-life processes, such as population growth or chemical reactions.
• These methods are often implemented in software, making it easier to predict behaviors in complex systems.

Understanding and applying numerical methods is crucial for handling a wide range of scientific and engineering problems, which are often modeled with ordinary differential equations.

Ordinary Differential Equations

Ordinary Differential Equations, often referred to as ODEs, are equations that contain functions of one independent variable and its derivatives. They play a crucial role in modeling the dynamic behavior of systems in various fields, such as physics, biology, and engineering.

An ODE can be written in the general form:\[ y' = f(t, y) \]Where:

• \( y' \) is the derivative of y concerning the independent variable \( t \).
• \( f(t, y) \) represents a relationship between the variables.

In the context of this exercise, we have the differential equation \( y' = \frac{1+y^2}{1+t} \), which describes how \( y \) changes with \( t \). Solving these equations analytically can be challenging, and this is where numerical methods like the Runge-Kutta method come into play.

Initial Value Problems

An initial value problem (IVP) is a type of differential equation together with a specified value at a starting point. These kinds of problems are a standard type of problem solved using numerical methods.

For the differential equation \( y' = \frac{1+y^2}{1+t} \), we are given the initial condition \( y(0)=0 \). This means that at time \( t=0 \), the value of \( y \) is 0. The goal is to find the value of \( y \) at a future point \( t \) using this initial condition.

• The initial value provides a specific starting point, which is essential for predicting future values in the system.
• With IVP, the aim is to extend the solution beyond the initial condition using a step-by-step approach.
• Using Runge-Kutta, we incrementally calculate \( y \) values step-by-step, starting from the initial value.

Initial value problems are crucial in modeling time-dependent processes, providing us a clear pathway to forecast the development of these systems over time.

Algorithm Implementation

The Runge-Kutta method is a powerful approach to implement when tackling initial value problems numerically. Implementing it involves following specific steps to approximate the solution over given intervals.

To apply the fourth-order Runge-Kutta method, you must:

• Select a suitable step size \( h \) and decide how many iterations \( n \) you wish to perform.
• Use the initial conditions, starting with \( t_0 = 0 \) and \( y_0 = 0 \).
• Iteratively calculate intermediate variables \( k_1, k_2, k_3, \) and \( k_4 \) as per the Runge-Kutta formulas.
• Update \( y \) values by incorporating these \( k \) terms and increment \( t \) by the step size \( h \).

After implementing the steps for each iteration, compare your approximations with the exact values if known. This helps assess the accuracy of your implementation.

Algorithm implementation not only involves coding and calculations but also deciding how fine-grained your intervals (step size \( h \)) should be, ensuring a balance between accuracy and computational efficiency.