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Chapter 6: Problem 7

Use the Adams-Bashforth-Moulton method to approximate \(y(1.0)\), where \(y(x)\) is the solution of the given initial-value problem. First use \(h=0.2\) and then use \(h=0.1\). Use the RK4 method to compute \(y_{1}, y_{2}\), and \(y_{3}\). $$ y^{\prime}=(x-y)^{2}, \quad y(0)=0 $$

Short Answer

Expert verified
Approximate \( y(1.0) \) using Adams-Bashforth-Moulton with both \( h=0.2 \) and \( h=0.1 \).

Step by step solution

01

Initialize the Variables

Given the differential equation \( y' = (x - y)^2 \) with \( y(0) = 0 \). We aim to approximate \( y(1) \) using the Adams-Bashforth-Moulton method, starting with step size \( h = 0.2 \), followed by \( h = 0.1 \). Initialize \( x_0 = 0 \) and \( y_0 = 0 \).
02

RK4 Method to Compute Initial Values

We need \( y_1, y_2, \) and \( y_3 \) to initiate the Adams-Bashforth-Moulton method. Use the RK4 method for this purpose:- For each \( y_i \), calculate intermediate slopes \( k_1, k_2, k_3, k_4 \) using: \[ k_1 = h \cdot f(x_n, y_n), \] \[ k_2 = h \cdot f(x_n + \frac{h}{2}, y_n + \frac{k_1}{2}), \] \[ k_3 = h \cdot f(x_n + \frac{h}{2}, y_n + \frac{k_2}{2}), \] \[ k_4 = h \cdot f(x_n + h, y_n + k_3) \]- Update \( y_{i+1} \) using \( y_{i+1} = y_i + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) \).- Apply this for \( x = 0.2, 0.4, 0.6 \) to find \( y_1, y_2, \) and \( y_3 \).
03

Adams-Bashforth Predictor Step

Now use the three-step Adams-Bashforth Predictor, which is: \[ y^P_{n+1} = y_n + \frac{h}{12}(55f(x_n, y_n) - 59f(x_{n-1}, y_{n-1}) + 37f(x_{n-2}, y_{n-2}) - 9f(x_{n-3}, y_{n-3})) \].Apply it for steps at increasing values, starting from \( x = 0.6 \) to \( x = 0.8 \), and then from \( x = 0.8 \) to \( x = 1.0 \).
04

Adams-Moulton Corrector Step

Refine the predicted value using Adams-Moulton Corrector: \[ y_{n+1} = y_n + \frac{h}{24}(9f(x_{n+1}, y^P_{n+1}) + 19f(x_n, y_n) - 5f(x_{n-1}, y_{n-1}) + f(x_{n-2}, y_{n-2})) \].Update using each \( y^P_{n+1} \) to get final \( y_{n+1} \) at \( x = 0.8 \), and then \( x = 1.0 \).
05

Repeat with Smaller Step Size \( h = 0.1 \)

Repeat Steps 2 to 4 with a smaller step size \( h = 0.1 \). Re-calculate \( y_1, y_2, \) and \( y_3 \) using RK4 with new step size, then use Adams-Bashforth-Moulton method to predict and correct subsequent \( y \) values up to \( x = 1.0 \).

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Key Concepts

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

Adams-Bashforth-Moulton method
This method is a combination of the Adams-Bashforth predictor and the Adams-Moulton corrector. It is a popular multi-step method used to solve ordinary differential equations, especially initial value problems. The primary advantage of the Adams-Bashforth-Moulton method is its ability to provide accurate results by using information from several previous steps.

Predictor-Corrector Method
Adams-Bashforth serves as a predictor step, providing an initial estimate for the next value, while Adams-Moulton acts as a corrector, refining the prediction using the computed slope at the predicted point.
  • Predictor formula: It uses multiple precomputed points to predict the next point.
  • Corrector formula: It improves the estimate by averaging the slope at the predicted point and the points used by the predictor.
This two-step process helps in achieving better accuracy without a significant increase in computational cost.

Application
The Adams-Bashforth-Moulton method is particularly effective for small step sizes, which helps in reducing errors in predictions. By iterating this process over a series of steps, one can achieve reliable approximations of the solution to differential equations, such as given in the original exercise.
Runge-Kutta methods
Runge-Kutta methods are a group of iterative techniques for solving ordinary differential equations. They are single-step methods, meaning they calculate the next state using only information from the current state, unlike multi-step methods, which leverage prior states.

Fourth-order Runge-Kutta (RK4)
The RK4 method is among the most well-known due to its balance of complexity and accuracy. It estimates the value of a function one step ahead by considering the weighted average of four increment estimates (slopes).
  • Intermediate Slopes Calculation:
    • First slope (\(k_1\)): Using the current point.
    • Second & third slopes (\(k_2, k_3\)): Using midpoints for better accuracy.
    • Fourth slope (\(k_4\)): At the endpoint of the interval.
  • Weighted Average: Uses all these slopes in a specific formation for accurate results.
Because of this accuracy and efficiency, RK4 is often used to obtain initial values for more complex, multi-step methods like the Adams-Bashforth-Moulton method in solving initial-value problems.
Initial Value Problems
Initial value problems (IVPs) are a subclass of differential equations where the solution is required to satisfy certain initial conditions. In general, IVPs take the form:\[\frac{dy}{dx} = f(x, y), \quad y(x_0) = y_0\]

Understanding IVPs
The solution seeks to find a function \(y(x)\) that meets the given differential equation, starting from a known initial point \((x_0, y_0)\).
  • Challenges: Solving IVPs involves deriving a function that changes as specified by the differential equation, throughout the domain starting from the initial point.
  • Applications: These problems are common in real-world scenarios such as modeling population growth, predicting time-dependent system behaviors, and more.
IVPs are crucial in numerical analysis as they provide a starting point for computational methods that approximate the solutions to differential equations, like the Adams-Bashforth-Moulton and Runge-Kutta methods. They form the basis for simulations and predictive modeling across various scientific and engineering disciplines.

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Most popular questions from this chapter

Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=x+y^{2}, \quad y(0)=0 ; y(0.5) $$

The RK4 method for solving an initial-value problem over an interval \([a, b]\) results in a finite set of points that are supposed to approximate points on the graph of the exact solution. In order to expand this set of discrete points to an approximate solution defined at all points on the interval \([a, b]\), we can use an interpolating function. This is a function, supported by most computer algebra systems, that agrees with the given data exactly and assumes a smooth transition between data points. These interpolating functions may be polynomials or sets of polynomials joined together smoothly. In Mathematica the command \(\mathbf{y}=\) Interpolation[data] can be used to obtain an interpolating function through the points data \(=\left\\{\left\\{x_{0}, y_{0}\right\\},\left\\{x_{1}, y_{1}\right\\}, \ldots,\left\\{x_{n}, y_{n}\right\\}\right\\} .\) The interpolating function \(\mathbf{y}[\mathbf{x}]\) can now be treated like any other function built into the computer algebra system. (a) Find the analytic solution of the initial-value problem \(y^{\prime}=-y+10 \sin 3 x ; y(0)=0\) on the interval \([0,2] .\) Graph this solution and find its positive roots. (b) Use the RK4 method with \(h=0.1\) to approximate a solution of the initial- value problem in part (a). Obtain an interpolating function and graph it. Find the positive roots of the interpolating function on the interval \([0,2]\).

Use the Runge-Kutta method to approximate \(x(0.2)\) and \(y(0.2) .\) First use \(h=0.2\) and then use \(h=0.1\). Use a numerical solver and \(h=0.1\) to graph the solution in a neighborhood of \(t=0\). $$ \begin{aligned} &x^{\prime}=6 x+y+6 t \\ &y^{\prime}=4 x+3 y-10 t+4 \\ &x(0)=0.5, y(0)=0.2 \end{aligned} $$

Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=4 x-2 y, \quad y(0)=2 ; y(0.5) $$

Consider the boundary-value problem $$ y^{\prime \prime}+x y=0, \quad y^{\prime}(0)=1, \quad y(1)=-1 $$ (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.
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