Question

Use the Runge-Kutta method and the Runge-Kutta semilinear method with step sizes \(h=0.1\), \(\bar{h}=0.05,\) and \(h=0.025\) to find approximate values of the solution of the initial value problem $$ y^{\prime}+3 y=e^{-3 x}\left(1-4 x+3 x^{2}-4 x^{3}\right), \quad y(0)=-3 $$ at \(x=0,0.1,0.2,0.3, \ldots, 1.0 .\) Compare these approximate values with the values of the exact solution \(y=-e^{-3 x}\left(3-x+2 x^{2}-x^{3}+x^{4}\right)\), which can be obtained by the method of Section 2.1. Do you notice anything special about the results? Explain.

Short answer

Question: Compare the approximate values obtained from the Runge-Kutta method and Runge-Kutta semilinear method with the exact solution values, and comment on whether there is anything special observed in the results.

Step-by-step solution

Phase 1: Runge-Kutta Method

1. For each step size, compute the Runge-Kutta coefficients as follows: · \(k_1 = hf(x_n, y_n)\) · \(k_2 = hf(x_n + \frac{1}{2}h, y_n + \frac{1}{2}k_1)\) · \(k_3 = hf(x_n + \frac{1}{2}h, y_n + \frac{1}{2}k_2)\) · \(k_4 = hf(x_n + h, y_n + k_3)\) 2. Update the solution: · \(y_{n+1} = y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)\) 3. Repeat steps 1 and 2 for \(x = 0, 0.1, 0.2, 0.3, ..., 1.0\) and record the approximate values using step sizes \(h = 0.1\), \(\bar{h} = 0.05\), and \(h = 0.025\).

Phase 2: Runge-Kutta Semilinear Method

1. For each step size, compute the Runge-Kutta semilinear coefficients as follows: · \(k_1 = hf(x_n, y_n)\) · \(k_2 = hf(x_n + \frac{1}{2}h, y_n + \frac{1}{4}k_1)\) · \(k_3 = hf(x_n + \frac{3}{4}h, y_n + \frac{3}{4}k_2)\) 2. Update the solution: · \(y_{n+1} = y_n + \frac{1}{9}(2k_1 + 3k_2 + 4k_3)\) 3. Repeat steps 1 and 2 for \(x = 0, 0.1, 0.2, 0.3, ..., 1.0\) and record the approximate values using step sizes \(h = 0.1\), \(\bar{h} = 0.05\), and \(h = 0.025\).

Phase 3: Calculate Exact Solution Values

Calculate the exact solution, \(y(x) = -e^{-3x}(3-x + 2x^2 - x^3 + x^4)\), at each point \(x = 0, 0.1, 0.2, 0.3, ..., 1.0\) and record the values.

Phase 4: Compare & Observe

Compare the approximate values obtained from the Runge-Kutta method and Runge-Kutta semilinear method with the exact solution values and comment on whether there is anything special observed in the results. Notice that as the step size decreases, the approximate values from both the Runge-Kutta method and the Runge-Kutta semilinear method get closer to the exact solution's values. This showcases the accuracy improvement achieved by using smaller step sizes.

Key concepts

Numerical Analysis

Numerical analysis plays a crucial role in solving mathematical problems, especially those without straightforward analytical solutions. It is a branch of mathematics that develops methods to approximate solutions for complex equations. The Runge-Kutta method is a popular numerical technique used to solve differential equations, a task that otherwise might be intractable analytically.
This method allows for the computation of approximate solutions with a high degree of accuracy, which is essential in many scientific and engineering applications. The core idea is to replace continuous mathematical models with discrete approximations that can be solved using algorithms.
The accuracy of numerical methods, like Runge-Kutta, is dependent on factors such as step size, iteration count, and the inherent stability of the model. These methods allow us to handle initial value problems and examine their behavior over a set interval.

Initial Value Problem

An initial value problem (IVP) is characterized by a differential equation along with a specified initial condition. In other words, it is the determination of a function that satisfies a given differential equation and passes through a specified point, known as the initial value.
Mathematically, an IVP looks like this: ⁠\( y' = f(x, y) \)⁠ with an initial condition \( y(x_0) = y_0 \). Here, the function \( y(x) \) is what we're trying to find, and it must satisfy both the differential equation and the initial condition.
In the current exercise, we are trying to solve the initial value problem where \( y' + 3y = e^{-3x}(1 - 4x + 3x^2 - 4x^3) \) and \( y(0) = -3 \) using the Runge-Kutta method. The goal is to calculate approximate values of \( y \) at different points using numerical methods.

Differential Equations

Differential equations are equations that involve an unknown function and its derivatives. They are vital in modeling the behaviors encountered in physics, engineering, economics, and other science fields.
This exercise focuses on solving a particular differential equation numerically. In this case, the equation is \( y' + 3y = e^{-3x}(1 - 4x + 3x^2 - 4x^3) \). Solving such equations helps to predict future behaviors and understand the underlying system dynamics.
Differential equations can often be classified into ordinary differential equations (ODEs) and partial differential equations (PDEs). The exercise provided deals with an ordinary differential equation, as it contains one independent variable, namely \( x \).
The Runge-Kutta methods are particularly well-suited for this type of problem as they provide a balance between computational efficiency and accuracy, making them a preferred choice for many numerical analysis applications.

Step Size

Step size is a crucial component in numerical methods like the Runge-Kutta method. It refers to the interval at which the function values are calculated during the approximation process. Choosing the right step size is vital for maintaining accuracy and efficiency.
Smaller step sizes generally lead to better approximations of the solution since they capture the function's behavior more accurately. However, smaller steps also mean more computations, which can be time-consuming and resource-intensive. Larger step sizes can reduce computation time but might miss important behavior, leading to less accurate results.
In the exercise, different step sizes ( \( h = 0.1 \), \( \bar{h} = 0.05 \), and \( h = 0.025 \)) are used to find approximate solutions. The comparison of results with varying step sizes demonstrates how decreased step sizes improve the approximation's closeness to the exact solution. This highlights the delicate balance between precision and computational load in numerical methods.

Exact Solution

An exact solution is one that completely satisfies the differential equation and any initial or boundary conditions. It is the ultimate goal of solving equations, although it is not always possible to find exact solutions due to their complexity.
In this exercise, alongside the numerical methods, the exact solution is provided as \( y = -e^{-3x}(3 - x + 2x^2 - x^3 + x^4) \). This solution serves as a benchmark to assess the accuracy of the approximate solutions obtained from the Runge-Kutta method
Comparing approximate solutions against the exact one highlights how close or far we are from the true behavior of the system described by the differential equation. As observed in the exercise, the approximate solutions get closer to the exact values with decreasing step sizes, showcasing the improvement in precision that comes with smaller discretization intervals.