Question

Show that the third order Adams-Moulton formula is $$ y_{x+1}=y_{x}+(h / 12)\left(5 f_{x+1}+8 f_{n}-f_{x-1}\right) $$

Short answer

Question: Show that the third order Adams-Moulton formula is given by the equation: $$ y_{x+1}=y_{x}+(h / 12)\left(5 f_{x+1}+8 f_{x}-f_{x-1}\right) $$ Answer: In order to derive the third order Adams-Moulton formula, we first find a three-point Lagrange interpolating polynomial using points (\(x_{x-1}\), \(y_{x-1}\)), (\(x_x\), \(y_x\)), and (\(x_{x+1}\), \(y_{x+1}\)). After differentiating the Lagrange polynomial, we evaluate the derivative at \(x_x\). Then, we set the derivative equal to the ODE and solve for the desired formula. This process results in the following equation: $$ y_{x+1}=y_{x}+(h / 12)\left(5 f_{x+1}+8 f_{x}-f_{x-1}\right) $$

Step-by-step solution

Understand the problem

We are given an initial value problem (IVP), which is a type of differential equation problem with an initial condition. The third order Adams-Moulton formula should contain information from 3 points: the current point, the next point, and a past point. We need to derive this formula as the exercise states.

Obtain a three-point Lagrange interpolating polynomial

To achieve this, let's first find the three-point Lagrange interpolating polynomial using points (\(x_{x-1}\), \(y_{x-1}\)), (\(x_x\), \(y_x\)), and (\(x_{x+1}\), \(y_{x+1}\)). The Lagrange interpolating polynomial is given by: $$ L(t) = \frac{(t-x_x)(t-x_{x+1})}{(x_{x-1}-x_x)(x_{x-1}-x_{x+1})}y_{x-1} + \frac{(t-x_{x-1})(t-x_{x+1})}{(x_x-x_{x-1})(x_x-x_{x+1})}y_x + \frac{(t-x_{x-1})(t-x_x)}{(x_{x+1}-x_{x-1})(x_{x+1}-x_x)}y_{x+1} $$

Differentiate the Lagrange polynomial

Next, differentiate the Lagrange interpolating polynomial with respect to t: $$ L'(t) = \frac{(t-x_{x+1}) + (t-x_x)}{(x_{x-1}-x_x)(x_{x-1}-x_{x+1})} f_{x-1} + \frac{(t-x_{x+1}) + (t-x_{x-1})}{(x_x-x_{x-1})(x_x-x_{x+1})} f_x + \frac{(t-x_x) + (t-x_{x-1})}{(x_{x+1}-x_{x-1})(x_{x+1}-x_x)} f_{x+1} $$

Evaluate the derivative at \(x_x\)

Now we will evaluate the derivative at \(x_x\). Let's denote \(h = x_{x+1} - x_x = x_x - x_{x-1}\). Therefore, \(t=x_x\) will be substituted in \(L'(t)\) giving: $$ L'(x_x) = \frac{(-h) + h}{(-2h)(-h)} f_{x-1} + \frac{(h) + (-h)}{(h)(-h)} f_x + \frac{h}{(h)(2h)} f_{x+1} \\ L'(x_x) = -\frac{1}{12}f_{x-1} + \frac{8}{12}f_x + \frac{5}{12}f_{x+1} $$

Set the derivative equal to the ODE and solve

We set the derivative \(L'(x_x) = f_x\) , so that we have: $$ f_x = -\frac{1}{12}f_{x-1} + \frac{8}{12}f_x + \frac{5}{12}f_{x+1} $$ Now, integrate both sides of the equation w.r.t \(x\): $$ \int_{x_x}^{x_{x+1}}f_x dx = -\frac{1}{12}\int_{x_x}^{x_{x+1}}f_{x-1} dx + \frac{8}{12}\int_{x_x}^{x_{x+1}}f_x dx + \frac{5}{12}\int_{x_x}^{x_{x+1}}f_{x+1} dx $$

Simplify the equation and obtain the final result

Let's simplify the equation and express it in terms of \(y_i\)'s: $$ y_{x+1}-y_x = -\frac{1}{12}h( f_{x-1} - 8f_x + 5f_{x+1}) $$ Thus, we have derived the third order Adams-Moulton formula as follows: $$ y_{x+1}=y_{x}+(h / 12)\left(5 f_{x+1}+8 f_{x}-f_{x-1}\right) $$

Key concepts

Lagrange Interpolating Polynomial

The Lagrange Interpolating Polynomial is a powerful method in numerical analysis used for polynomial interpolation. Think of it as a way of estimating a function using a polynomial that passes exactly through a set of given points. This interpolation is crucial for approximating functions because it helps in predicting unknown values by using known data points.

In simple terms, if you have a few known data points from a function, the Lagrange polynomial helps you find a smooth curve that goes through all those points. The formula involves constructing a weighted sum of the given data points, where each term in the sum is a Lagrange basis polynomial. Each basis polynomial is built so that it equals 1 at its corresponding data point and 0 at all other data points.

For example, when we deal with three points, say \((x_{x-1}, y_{x-1})\), \((x_x, y_x)\), and \((x_{x+1}, y_{x+1})\), we use these to build a polynomial through those points. The Lagrange polynomial becomes particularly useful in numerical methods such as the Adams-Moulton method, where it helps approximate solutions to differential equations by estimating the function between known data points.

Numerical Methods for Differential Equations

Numerical methods for differential equations are a core area of study in mathematics and engineering. They allow us to find approximate solutions to complex differential equations where analytical solutions are impossible or impractical. Key methods in this field include Euler's method, Runge-Kutta methods, and Adams-Bashforth-Moulton methods.

The Adams-Moulton method is a predictor-corrector method specifically used for solving ordinary differential equations (ODEs). It belongs to the class of linear multistep methods, where the current and several previous steps are used to estimate the solution at future steps. In practice, this method takes calculated approximations and improves them using subsequent corrections, leading to higher accuracy. The third-order Adams-Moulton formula, which accounts for three points, provides a balance between computational complexity and accuracy.

These numerical techniques are invaluable in the real world, especially in fields like physics, engineering, and finance, where systems are often modeled by differential equations that do not have simple, solvable forms.

Differential Equations Analysis

Differential equations form the backbone of many scientific disciplines as they describe how something changes over time or space. Analysis of these equations helps us to understand dynamic systems, predicting behavior based on initial conditions.

When analyzing differential equations, we often face initial value problems (IVPs). An IVP requires us to find a function that satisfies a differential equation and also meets an initial condition. This initial condition ensures the uniqueness of the solution, making it a critical step in the process.

In the context of the step-by-step solution above, using Lagrange interpolating polynomial within the Adams-Moulton framework allows for an enhanced understanding of the ODE being worked on. By calculating and setting derivatives and then integrating, we make a thorough analysis to derive meaningful approximations of the solutions. This blend of theoretical analysis and practical computation is central to effectively solving differential equations.