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Chapter 9: Problem 15

Consider the initial-value problem $$ y^{\prime}=x^{2}+y^{3}, \quad y(1)=1 \text {. } $$ (See Problem 15 in Exercises 9.2.) (a) Compare the results obtained from using the Runge-Kutta formula over the interval \([1,1.4]\) with step sizes \(h=0.1\) and \(h=0.05\). (b) Use an ODE solver to obtain a graph of the solution on the interval \([1,1.4]\).

Short Answer

Expert verified
Use Runge-Kutta for both step sizes, compare results, and graph the solution using an ODE solver.

Step by step solution

01

Initial Setup for Runge-Kutta Method

The Runge-Kutta method is a numerical technique used for solving differential equations. We will apply this method to the given initial-value problem over the interval \([1, 1.4]\) for two different step sizes: \(h=0.1\) and \(h=0.05\). Runge-Kutta is particularly useful for problems where analytical solutions are difficult to find.
02

Apply Runge-Kutta with Step Size h=0.1

Using the fourth-order Runge-Kutta method, calculate the approximate values of \(y\) at each step of size \(h=0.1\) starting from \(x=1\) to \(x=1.4\). We use the equations:\[K_1 = h f(x_n, y_n), \quadK_2 = h f(x_n + \frac{h}{2}, y_n + \frac{K_1}{2}), \quadK_3 = h f(x_n + \frac{h}{2}, y_n + \frac{K_2}{2}), \quadK_4 = h f(x_n + h, y_n + K_3)\]Then update with:\[y_{n+1} = y_n + \frac{1}{6}(K_1 + 2K_2 + 2K_3 + K_4)\]Repeat this process up to \(x=1.4\). Record every computed value.
03

Apply Runge-Kutta with Step Size h=0.05

Repeat the process from Step 2, but with a smaller step size \(h=0.05\). This will involve more calculations since we are taking smaller steps, but it should provide more accurate results. Follow the same procedure to compute \(y\) values up to \(x=1.4\).
04

Compare Results from Different Step Sizes

Compare the results obtained from using \(h=0.1\) and \(h=0.05\). Generally, a smaller step size like \(h=0.05\) provides a more accurate solution, but with increased computational effort. Assess the differences in solutions and the computational trade-offs involved.
05

Graph the Solution Using an ODE Solver

Utilize an ODE solver like Python's SciPy library to compute and graph the solution over the interval \([1, 1.4]\). Specify the initial condition and generate the graph to visualize the solution curve. This will provide insight into the behavior of the solution over the specified interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial-Value Problem
An initial-value problem in the context of differential equations involves finding a function, often denoted as \( y(x) \), that satisfies a given differential equation along with specified initial conditions at a particular point. For example, in this exercise, we have the differential equation \( y' = x^2 + y^3 \) with an initial condition \( y(1) = 1 \). This tells us the starting value of the function when \( x = 1 \).
  • The differential equation dictates how \( y \) changes with \( x \).
  • The initial condition tells us where \( y \) starts.
Understanding the initial-value problem is crucial as it sets the stage for applying numerical methods like the Runge-Kutta to estimate solutions when an analytical solution is not easily attainable.
Numerical Methods in Differential Equations
Numerical methods in differential equations provide powerful tools for approximating solutions to problems that are difficult or impossible to solve analytically. These methods transform continuous mathematical problems into discrete approximations that can be solved with the aid of a computer. The Runge-Kutta method is one such numerical technique, specifically designed to solve ordinary differential equations (ODEs).
  • They are indispensable for complex equations without straightforward solutions.
  • These methods approximate the true solution by taking small, discrete steps.
  • The Runge-Kutta method is widely used because of its accuracy and stability.
Thanks to numerical methods, engineers and scientists can tackle real-world problems that involve complex systems.
ODE Solver
An ODE solver is a computational tool designed to find approximate solutions to ordinary differential equations. Solvers like those found in Python鈥檚 SciPy library offer a variety of methods for approaching these problems. For the initial-value problem mentioned, an ODE solver can graphically represent how the solution behaves over a given interval.
  • ODE solvers automate the solution process, alleviating manual calculations.
  • They can handle different types of equations and initial conditions.
  • Visual outputs such as graphs help convey solution trends and behaviors.
Using an ODE solver not only saves time but also enhances understanding by providing a clear graphical view.
Step Size Analysis
Step size is a critical concept in numerical methods like the Runge-Kutta method, affecting both the accuracy of the solution and the computational time required. By breaking down the interval into steps, each step size defines the amount of progress we make with each iteration.
  • A smaller step size typically results in a more accurate solution.
  • However, smaller steps require more calculations, increasing computational effort.
  • Larger step sizes might reduce accuracy but make computations faster.
In the exercise, different step sizes \( h=0.1 \) and \( h=0.05 \) demonstrate this trade-off. Smaller steps lead to finer approximations of the solution curve, which is crucial for precision in engineering and scientific applications.

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Most popular questions from this chapter

Use the Adams-Bashforth/Adams-Moulton method to approximate the value of \(y(0.4)\), where \(y(x)\) is the solution of $$ y^{\prime}=4 x-2 y, \quad y(0)=2 \text {. } $$ Use the Runge-Kutta formula and \(h=0.1\) to obtain the values of \(y_{1}\), \(y_{2}\), and \(y_{3}\).

Consider the boundary-value problem \(y^{\prime \prime}+x y=0, y^{\prime}(0)=1\), \(y(1)=-1\). (a) Find the difference equation corresponding to the differential equation. Show that for \(i=0,1,2, \ldots, n-1\) the difference equation yields \(n\) equations in \(n+1\) unknowns \(y_{-1}, y_{0}, y_{1}\), \(y_{2}, \ldots, y_{n-1} .\) Here \(y_{-1}\) and \(y_{0}\) are unknowns since \(y_{-1}\) represents an approximation to \(y\) at the exterior point \(x=-h\) and \(y_{0}\) is not specified at \(x=0\). (b) Use the central difference approximation (5) to show that \(y_{1}-y_{-1}=2 h\). Use this equation to eliminate \(y_{-1}\) from the system in part (a). (c) Use \(n=5\) and the system of equations found in parts (a) and (b) to approximate the solution of the original boundary-value problem.

In Problems 5-8 use the Adams-Bashforth/Adams-Moulton method to approximate \(y(1.0)\), where \(y(x)\) is the solution of the given initial-value problem. Use \(h=0.2\) and \(h=0.1\) and the Runge-Kutta method to compute \(y_{1}, y_{2}\), and \(y_{3}\). \(y^{\prime}=y+\cos x, \quad y(0)=1\)

Given the initial-value problems in Problems use the Runge-Kutta method with \(h=0.1\) to obtain a four-decimal-place approximation to the indicated value.\(y^{t}=1+y^{2}, y(0)=0 ; \quad y(0.5)\)

In Problems 1-10 use the finite difference method and the indicated value of \(n\) to approximate the solution of the given boundary-value problem. \(x^{2} y^{\prime \prime}-x y^{\prime}+y=\ln x, \quad y(1)=0, y(2)=-2 ; \quad n=8\)
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