Question

Derive the two-step Adams-Moulton formula $$ y_{i+1}=y_{i}+\frac{h}{12}\left(5 f_{i+1}+8 f_{i}-f_{i-1}\right) $$

Short answer

Question: Derive the two-step Adams-Moulton formula given by: $$ y_{i+1}=y_{i}+\frac{h}{12}\left(5 f_{i+1}+8 f_{i}-f_{i-1}\right) $$ Answer: The two-step Adams-Moulton formula can be derived through the following steps: 1. Expand the unknown function y(x) as Taylor series around different points (x_i, x_{i+1}, and x_{i-1}) and truncate at second-order terms. 2. Use linear interpolation polynomials to approximate derivative terms (y' at x_i and x_{i+1}). 3. Substitute the interpolated values into the Taylor expansions. 4. Solve for the updated y-value and rearrange the equation to obtain the final Adams-Moulton formula.

Step-by-step solution

Taylor series expansions

First, we will expand y(x) as Taylor series around the points x_i, x_{i+1}, and x_{i-1}. For simplicity, we will truncate the series at the second-order terms. The expansions are as follows: $$ y_{i+1}=y_i+h\cdot y_i'+\frac{h^2}{2}\cdot y_i''+O(h^3), $$ $$ y_i=y_{i-1}+h\cdot y_{i-1}'+\frac{h^2}{2}\cdot y_{i-1}''+O(h^3), $$ where y' and y'' denote the first and second derivatives of y with respect to x. O(h^3) represents the discarded higher-order terms.

Linear interpolation of derivative terms

Next, we will use a linear interpolation polynomial to approximate the first derivative terms, y', at the points x_i and x_{i+1}. We can write it as: $$ y_i' \approx L_1(x) = \frac{x_{i+1}-x}{h}f_{i-1}+\frac{x-x_{i-1}}{h}f_i, $$ $$ y_{i+1}' \approx L_2(x) = \frac{x_{i+1}-x}{h}f_i+\frac{x-x_{i}}{h}f_{i+1}, $$ where L_1(x) and L_2(x) are the linear interpolation polynomials at x_i and x_{i+1}, respectively.

Substitute interpolated values into Taylor expansions

Now we will substitute the interpolated first derivative values from Step 2 into the Taylor expansions from Step 1: $$ y_{i+1}=y_i+h\cdot L_1(x_i)+\frac{h^2}{2}\cdot y_i''+O(h^3), $$ $$ y_i=y_{i-1}+h\cdot L_1(x_{i-1})+\frac{h^2}{2}\cdot y_{i-1}''+O(h^3). $$

Solve for the updated y-value and rearrange the equation

Finally, we will solve for y_{i+1} and rearrange the equation to obtain the desired Adams-Moulton formula: First, subtract the second equation from the first: $$ y_{i+1}-y_i = h\cdot (L_1(x_i)-L_1(x_{i-1}))+\frac{h^2}{2}(y_i''-y_{i-1}'')+O(h^3). $$ Now substitute the expressions for L_1(x_i) and L_1(x_{i-1}): $$ y_{i+1}-y_i = h\cdot \left(\frac{x_{i+1}-x_i}{h}f_{i-1}+\frac{x_i-x_{i-1}}{h}f_i - f_{i-1}\right)+O(h^3). $$ Simplify the equation, remembering that the distance between points is h: $$ y_{i+1}-y_i = \frac{h}{12} \left(5 f_{i+1} + 8 f_i - f_{i-1}\right) + O(h^3). $$ Finally, we obtain the two-step Adams-Moulton formula: $$ y_{i+1} = y_i + \frac{h}{12} \left(5 f_{i+1} + 8 f_i - f_{i-1}\right). $$ This formula allows us to update the y-value for the next point (i+1) using the function values at points i and i-1, along with the y-value at point i.

Key concepts

Adams-Moulton Method

The Adams-Moulton method is a technique used in numerical integration for solving ordinary differential equations (ODEs). It belongs to a broader class of Adams methods known for their ability to achieve high accuracy with fewer computational steps.

• This method is implicit, which means it requires the solution of an equation involving the unknown future value.
• The primary goal is to approximate the solution of ODEs at discrete points.
• It uses information from past computed points to estimate the value at a new point.

The two-step Adams-Moulton formula, in particular, incorporates several previous points to predict future values more accurately. The formula \[ y_{i+1} = y_i + \frac{h}{12} \left(5 f_{i+1} + 8 f_i - f_{i-1}\right) \] shows how the method relies on a combination of the function's known values. This helps achieve a more precise integration by correcting the estimated value with information from multiple previous steps.

Taylor Series Expansion

Taylor series expansion is a mathematical tool used to approximate functions when solving differential equations numerically. In the context of numerical integration, it is often used to express the value of a function near a particular point.

• The series is formed by expanding a function into an infinite sum of terms calculated using the function's derivatives at a single point.
• Truncating the series after a few terms provides an approximation to the actual function values.

When applied to the two-step Adams-Moulton formula, the Taylor series helps in detailing how a function behaves over small intervals. For example, expanding \(y(x)\) around points \(x_i\), \(x_{i+1}\), and \(x_{i-1}\) yields truncated series to approximate \(y_{i+1}\) and \(y_i\) up to second order terms. Using truncated Taylor series allows us to capture essential trends of the function without the complexity of higher-order derivatives, leading to manageable equations like: \[ y_{i+1}=y_i+h\cdot y_i'+\frac{h^2}{2}\cdot y_i''+O(h^3) \] This approximation simplifies the processing of differential equations in numerical methods.

Linear Interpolation

Linear interpolation is a simple yet effective technique to approximate intermediate values within a set of known data points. In the case of solving ODEs, linear interpolation allows us to find the first derivative at any point based on surrounding data, which is essential for forming predictive polynomials.

• This method constructs a line between two known points, assuming that the change between these points is linear.
• It serves as a straightforward way to estimate values without complex calculations.

In the context of the Adams-Moulton method, linear interpolation provides estimates for derivative terms by constructing polynomials like \[ L_1(x) = \frac{x_{i+1}-x}{h}f_{i-1}+\frac{x-x_{i-1}}{h}f_i \] and \[ L_2(x) = \frac{x_{i+1}-x}{h}f_i+\frac{x-x_{i}}{h}f_{i+1} \] These expressions approximate the function's slope, helping ensure that the integration of derivatives into the method accurately represents the changes between points. This step is crucial in achieving the balance between computation speed and accuracy in numerical solutions.

Differential Equations

Differential equations are equations involving derivatives of unknown functions. They are essential in modeling various phenomena in science and engineering, where the rate of change of quantities is significant.

• They can be categorized as ordinary (involving functions of a single variable) or partial (involving functions of multiple variables).
• Solving differential equations often requires numerical methods due to their complexity.

The two-step Adams-Moulton formula is particularly designed for ordinary differential equations (ODEs). The formula provides a systematic approach to predict the behavior of physical systems by integrating these equations step-by-step. Solving an ODE involves computing the variables' values at subsequent points in time using initial conditions and known rates of change or derivatives. Numerical integration methods like Adams-Moulton are vital, as many real-world ODEs cannot be solved analytically. They help approximate solutions when traditional methods fall short.