Question

Determine an approximate value of the solution at \(t=0.4\) and \(t=0.5\) using the specified method. For starting values use the values given by the Runge- Kutta method; see Problems 1 through 6 of Section 8.3 . Compare the results of the various methods with each other and with the actual solution (if available). $$ \begin{array}{l}{\text { (a) Use the fourth order predictor-corrector method with } h=0.1 . \text { Use the corrector }} \\ {\text { formula once at each step. }} \\ {\text { (b) Use the fourth order Adams-Moulton method with } h=0.1} \\ {\text { (c) Use the fourth order backward differentiation method with } h=0.1 .}\end{array} $$ $$ y^{\prime}=\left(t^{2}-y^{2}\right) \sin y, \quad y(0)=-1 $$

Short answer

Question: Compare the approximate values of the solution for the given differential equation \(y'(t) = (t^2 - y^2) \sin y,\ y(0) = -1\) at \(t=0.4\) and \(t=0.5\) using the (a) Fourth-order Predictor-Corrector method, (b) Fourth-order Adams-Moulton method, and (c) Fourth-order Backward Differentiation Formula method with an increment step of \(h=0.1\). Also, discuss the accuracy and efficiency of these methods.

Step-by-step solution

a) Fourth-order Predictor-Corrector Method

1. Start with the initial condition: \(y(0) = -1\) 2. Get the starting values using the Runge-Kutta method (from Section 8.3, Problems 1-6) Here, we will assume that the starting values are provided to us. 3. Set the step size: \(h=0.1\) 4. Apply the Predictor formula: $$\tilde{y}_{n+1} = y_n + \frac{h}{24}\left[55 f_n - 59 f_{n-1} + 37 f_{n-2} - 9 f_{n-3} \right]$$ 5. Now, apply the Corrector formula: $$y_{n+1} = y_n + \frac{h}{24}\left[9 f\left(t_{n+1},\tilde{y}_{n+1}\right) + 19 f_n - 5 f_{n-1} + f_{n-2} \right]$$ 6. Repeat steps 4 and 5 until we reach \(t=0.4\) and \(t=0.5\) 7. Record the approximate values for \(y(0.4)\) and \(y(0.5)\)

b) Fourth-order Adams-Moulton Method

1. Use the starting values from the Runge-Kutta method, provided from Section 8.3, Problems 1-6 2. Set the step size: \(h=0.1\) 3. Apply the Adams-Moulton formula: $$y_{n+1} = y_n + \frac{h}{720}\left[251 f\left(t_{n+1},y_{n+1}\right) + 646 f_n - 264 f_{n-1} + 106 f_{n-2} - 19 f_{n-3}\right]$$ 4. Iterate using explicit methods (such as the Adams-Bashforth method or the Euler method) to find \(y_{n+1}\) as an initial guess for the Adams-Moulton formula 5. Apply the formula iteratively until we reach \(t=0.4\) and \(t=0.5\) 6. Record the approximate values for \(y(0.4)\) and \(y(0.5)\)

c) Fourth-order Backward Differentiation Formula (BDF) Method

1. Use the starting values from the Runge-Kutta method, provided from Section 8.3, Problems 1-6 2. Set the step size: \(h=0.1\) 3. Apply the fourth-order BDF formula: $$ \frac{1}{h}\left[\frac{25}{12}y_{n+1}-\frac{48}{12}y_n+\frac{36}{12}y_{n-1}-\frac{16}{12}y_{n-2}+\frac{3}{12}y_{n-3}\right] = f\left(t_{n+1},y_{n+1}\right) $$ 4. Solve the implicit equation iteratively (using methods such as Newton-Raphson or the fixed-point iteration method) to find \(y_{n+1}\) 5. Apply the formula iteratively until we reach \(t=0.4\) and \(t=0.5\) 6. Record the approximate values for \(y(0.4)\) and \(y(0.5)\) Finally, compare the results of each method for \(y(0.4)\) and \(y(0.5)\) with each other and the actual solution if available. The accuracy and efficiency of the methods can then be determined.

Key concepts

Predictor-Corrector Methods

Predictor-corrector methods are a class of iterative techniques used in numerical analysis to solve ordinary differential equations (ODEs). These methods enhance accuracy by refining estimates of a function's solution over small intervals, known as steps. In a predictor-corrector approach, the solution at the next step is initially approximated by a predictor method. This value is then refined by a corrector method.

A basic example is using a simple Euler method as a predictor and then refining it with a trapezoidal rule as a corrector. This dual-step process compensates for errors and provides greater accuracy. In detail, the predictor forecasts the solution using past values, while the corrector adjusts this prediction using an average or weighted average.

During implementation, especially with methods like the Fourth-order predictor-corrector method, the predictor uses more straightforward formulas to estimate, while the corrector employs more rigorous checks. This iterative adjustment leads to more precise results, especially when there are potential errors due to larger step sizes.

Adams-Moulton Method

The Adams-Moulton method is an implicit method used for solving ODEs. As a member of the linear multistep family, it utilizes previous points and values to improve the current estimate. Specifically, the Adams-Moulton method has an advantage in stability, which is particularly beneficial for stiff equations.

This implicit technique means that the new function value is part of the equation being solved, requiring iterative methods such as Newton's method for each step. The given formula in the exercise involves both current and several previous points, weighting them to achieve greater accuracy. It often requires an initial guess from explicit methods like Euler's to start the iteration process.

• Uses historical data for better prediction
• Stable with stiff equations
• Relies on implicit rather than explicit solutions

The combination of these attributes makes it very apt for problems where stability is crucial over long intervals.

Backward Differentiation Formula

Backward Differentiation Formulas (BDFs) are another class in the stable family of implicit methods. These are particularly powerful for stiff differential equations. Instead of looking forward to predict future values, BDFs look backward at previous points, providing a strong level of integration accuracy.

The fourth-order BDF exercises in the given problem rely on several past values, integrating them with different coefficients. The strength of BDF lies in its ability to handle stiff problems reliably. By solving an implicit equation at each step through iterative solutions, such as Newton-Raphson, it manages potential large differences in step sizes and function behavior.

• Particularly effective for stiff equations
• Relies on preceding values
• Combines with iterative solvers for implicit equations

These features make BDFs robust tools in numerical methods, particularly when tackling equations that require careful treatment of stiffness.

Runge-Kutta Method

The Runge-Kutta methods are a group of iterative techniques used to approximate solutions of ODEs. They offer a great balance between complexity and accuracy. The most commonly used is the fourth-order Runge-Kutta method (RK4). It uses four calculations per step, incorporating multiple function evaluations to improve accuracy without excessive complexity.

By combining these evaluations in a weighted average, RK4 can achieve high levels of precision, similar to adaptive step-size methods. Often, Runge-Kutta is used to generate initial conditions for other methods, as seen in the exercise where it provides starting values.

• Popular for their balance of complexity and precision
• Involves multiple evaluations per step
• Useful for generating initial conditions for other solutions

This method is perfectly suited for equations requiring meticulous precision, where the complexity of higher-order methods might not be necessary.

Step Size in Numerical Analysis

Step size, denoted as \( h \) in numerical methods, is a critical parameter that affects both the accuracy and efficiency of solving differential equations. Essentially, the step size signifies how much to move forward in the independent variable's domain, typically time or space.

Choosing the right \( h \) is crucial. A small step size generally offers more accurate results but demands more computational effort and time. Conversely, a large \( h \) might be computationally efficient but can introduce significant errors or even instability into the results.

• Small \( h \): More accurate, less efficient
• Large \( h \): More efficient, potentially less accurate
• Balancing \( h \) is key for optimal results

In practice, adaptive step size methods can adjust \( h \) dynamically to match the problem's needs best, providing a balance between accuracy and efficiency.