Problem 8 - A First Course in the Numerical Analysis of Differential Equations

Question

Let \(f\) be analytic. Prove that, for sufficiently small \(h>0\) and an analytic function \(\boldsymbol{x}\), the function $$ \boldsymbol{x}(t+h)-\boldsymbol{x}(t-h)-h \boldsymbol{f}\left(\frac{1}{2}(\boldsymbol{x}(t-h)+\boldsymbol{x}(t+h))\right) $$ can be expanded into power series in odd powers of \(h\). Deduce that the error in the implicit midpoint rule (1.13), when applied to autonomous ODEs \(y^{\prime}=f(y)\) also admits an expansion in odd powers of \(h\). (Hint: First try to prove the statement for a scalar function \(f\). Once you have solved this problem, a generalization should present no difficulties.)

Step-by-Step Solution

Answer

The function expands in odd powers of \(h\), and so does the error in the implicit midpoint rule for autonomous ODEs.

Step 1: Express the Problem

We need to show that for a sufficiently small \(h>0\), the function \(\boldsymbol{x}(t+h)-\boldsymbol{x}(t-h)-h \boldsymbol{f}\left(\frac{1}{2}(\boldsymbol{x}(t-h)+\boldsymbol{x}(t+h))\right)\) can be expanded in terms of odd powers of \(h\). This involves showing the Taylor expansion around \(t\) results only in terms of odd powers.

Step 2: Expand in Taylor Series

First, write the Taylor expansion for \(\boldsymbol{x}(t+h)\) and \(\boldsymbol{x}(t-h)\): \[ \boldsymbol{x}(t+h) = \boldsymbol{x}(t) + h \boldsymbol{x}'(t) + \frac{h^2}{2} \boldsymbol{x}''(t) + \cdots \] \[ \boldsymbol{x}(t-h) = \boldsymbol{x}(t) - h \boldsymbol{x}'(t) + \frac{h^2}{2} \boldsymbol{x}''(t) - \cdots \] Subtract these two expansions.

Step 3: Simplify the Subtraction

The subtraction gives: \[ \boldsymbol{x}(t+h) - \boldsymbol{x}(t-h) = 2h \boldsymbol{x}'(t) + \frac{2h^3}{6} \boldsymbol{x}^{(3)}(t) + \cdots \] This result contains only terms with odd powers of \(h\).

Step 4: Express the Function f

We need to expand \(\boldsymbol{f}\) around \(\boldsymbol{x}(t)\). Since \(\boldsymbol{x}\) is analytic and \(\boldsymbol{f}\) is as well, the expansion goes as: \[ \boldsymbol{f}\left(\frac{1}{2}(\boldsymbol{x}(t-h) + \boldsymbol{x}(t+h))\right) = \boldsymbol{f}(\boldsymbol{x}(t)) + \frac{h^2}{8} \boldsymbol{f}''(\boldsymbol{x}(t)) \boldsymbol{x}'(t) + \cdots \] Again, this expansion contains only terms with even powers of \(h\).

Step 5: Combine Expressions

Substituting back into \(\boldsymbol{x}(t+h) - \boldsymbol{x}(t-h) - h \boldsymbol{f}\), we have: \[ 2h \boldsymbol{x}'(t) + \frac{h^3}{3!} \boldsymbol{x}^{(3)}(t) - h \left(\boldsymbol{f}(\boldsymbol{x}(t)) + \frac{h^2}{8} \cdots \right)\] Simplifying further results in retaining only odd powers again.

Step 6: Deduce For ODE Error

The implicit midpoint rule's error when applied to the ODE \(y' = f(y)\) involves similar expansions in powers of \(h\): \[ y_{n+1} = y_n + h f \left( \frac{y_n + y_{n+1}}{2} \right) \] The error \(E(h)\) in this rule also expands into odd powers, similar to the previous derivation. Therefore, the statement holds true for autonomous ODEs.

Key Concepts

Taylor Expansion

In mathematics, a Taylor Expansion is a powerful tool that allows a function to be represented as an infinite sum of terms calculated from the values of its derivatives at a single point. This technique is mostly used for functions that are infinitely differentiable. The basic idea is to approximate a complex function with a polynomial, where more terms in the series result in a closer approximation. For a function \( f(x) \), its Taylor series centered at \( x = a \) is given by:\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots \] This expansion is especially useful in numerical methods for solving differential equations, where approximating complex expressions simplifies calculations and enables more manageable solutions.

Implicit Midpoint Rule

The Implicit Midpoint Rule is an important numerical method used to solve differential equations. It belongs to the family of implicit Runge-Kutta methods and is known for its simplicity and effectiveness in solving stiff equations. To apply the Implicit Midpoint Rule to a differential equation \( y' = f(y) \), the equation is approximated as:\[ y_{n+1} = y_n + hf\left(\frac{y_n + y_{n+1}}{2}\right) \] Here, \( h \) is the step size, and \( y_{n+1} \) is the estimated value at the next step. What makes this method implicit is the fact that \( y_{n+1} \) is involved in the function on the right-hand side, requiring iterative methods or special strategies to evaluate it.An important thing to note about the Implicit Midpoint Rule is its symmetry and second-order accuracy, which make it a preferred choice for certain types of differential equations, such as those arising in physics and engineering applications.

Autonomous Ordinary Differential Equations

Autonomous Ordinary Differential Equations (ODEs) are a specific class of differential equations where the independent variable, often time \( t \), doesn't explicitly appear in the function itself. Instead, these equations are expressed solely in terms of the dependent variable and its derivatives.For example, an autonomous ODE may look like:\[ y' = f(y) \] The key characteristic of autonomous ODEs is that the system's behavior is time-independent. This simplicity can lead to several convenient features, like easier integration and simpler stability analysis. These equations appear frequently in modeling natural systems where the conditions are assumed to be steady or time-invariant, such as in population dynamics or chemical reaction rates.

Analytic Functions

Analytic functions are a type of mathematical function that are infinitely differentiable and defined by a convergent power series about any point within their radius of convergence. They exhibit properties that make them very useful in complex analysis and in solving differential equations. The defining property of an analytic function \( f \) is that it can be represented as a power series:\[ f(z) = \sum_{n=0}^{\infty} a_n (z - z_0)^n \] within some radius around \( z_0 \), where \( a_n \) are the coefficients of the series. Such functions are smooth, lack sudden jumps or breaks, and can be elegantly expanded in series to approximate complex phenomena. These properties allow analytic functions to work nicely with Taylor expansions, making them integral to methods such as the one discussed in the original exercise.