Question

(Difference equations) We consider functions \(f(\sigma)\) of an integer variable \(\sigma: \sigma=0\), \(\pm 1, \pm 2, \ldots\) Let \(f(\sigma)\) be defined for \(\sigma=m, \sigma=m+1, \ldots, \sigma=n\). Then the first difference \(\Delta_{+} f(\sigma)\) is the function \(f(\sigma+1)-f(\sigma)(m \leq \sigma4\), distinct numbers \(a_{1}\) and \(a_{2}\) can be chosen so that the functions \(c_{1} a_{1}^{\sigma}+c_{2} a_{2}^{\sigma}\) are solutions and that when \(p^{2}=4\), the functions \(c_{1}(-1)^{\sigma}+c_{2} \sigma(-1)^{\sigma}\) are solutions. c) Show that if \(f(0)\) and \(f(1)\) are given, then the difference equation of part (b) successively determines \(f(2) . f(3)\)..... Hence for such "initial conditions" there is one and only one solution. Determine the constants \(c_{1}, c_{2}\) for each of the cases of part (b) to match the given initial conditions. The fact that this is possible ensures that each expression gives the general solution for the corresponding value of \(p\). d) Show that the only solutions of the difference equation of part (b) which satisfy the boundary conditions: \(f(0)=0, f(N+1)=0\) are constant multiples of the \(N\) functions $$ \phi_{n}(\sigma)=\sin \left(\frac{n \pi}{N+1} \sigma\right) \quad(n=1, \ldots, N) $$ and that \(\phi_{n}(\sigma)\) is a solution only when $$ p^{2}=2\left(1-\cos \frac{n \pi}{N+1}\right) $$

Short answer

general solution is given by \(f(\sigma) = c_1 (-1)^{\sigma} + c_2 \sigma(-1)^{\sigma}\). Substituting the initial conditions: $$u_0 = c_1 (-1)^0$$ $$u_1 = c_1 (-1) + c_2 (1)(-1)$$ From these equations, we can find the constants \(c_1\) and \(c_2\): $$c_1 = u_0$$ $$c_2 = -(u_1 + u_0)$$ Thus, the expression for the general solution for \(p^2 = 4\) is given by: $$f(\sigma) = u_0 (-1)^{\sigma} - (u_1 + u_0) \sigma(-1)^{\sigma}$$ In all three cases, the expressions for the general solutions are fully determined by the initial conditions \(f(0)\) and \(f(1)\). Moreover, the general solutions are such that they uniquely determine the successive values of the function: For \(0 < p^2 < 4\), the successive values are determined by a linear combination of cosine and sine functions. For \(p^2 > 4\), the successive values are determined by a linear combination of exponential functions. For \(p^2 = 4\), the successive values are determined by a linear combination of powers of \((-1)^\sigma\). Regardless of the value of \(p^2\), the value of \(f(\sigma)\) can be uniquely determined based on the initial conditions, proving the claim that the initial conditions successively determine the entire function.

Step-by-step solution

Case (a) with a second-order difference equation

We are given the difference equation \(\Delta^2 f(\sigma) = 0\). Our goal is to show that the general solution is given by \(f(\sigma) = c_1 \sigma + c_2\), where \(c_1\), \(c_2\) are constants. To do this, we can rewrite the difference equation as: $$f(\sigma+1) - 2 f(\sigma) + f(\sigma-1) = 0$$ Let's assume that \(f(\sigma) = c_1 \sigma + c_2\). Then, we can substitute this into the equation and see if it solves the equation: $$c_1(\sigma + 1) + c_2 - 2(c_1 \sigma + c_2) + c_1(\sigma - 1) + c_2 = 0$$ Simplifying this equation, we get: $$c_1 = 0$$ Thus, our assumption is correct and the general solution to the difference equation is given by \(f(\sigma) = c_1 \sigma + c_2\), where \(c_1\), \(c_2\) are constants. Next, let's find the solution that satisfies the boundary conditions \(f(0) = u_0\) and \(f(N+1) = u_{N+1}\). By substituting \(\sigma = 0\) into the general solution, we get: $$f(0) = c_1 \cdot 0 + c_2 = u_0$$ So, \(c_2 = u_0\). By substituting \(\sigma = N + 1\) into the general solution, we get: $$f(N+1) = c_1 (N+1) + c_2 = u_{N+1}$$ So, $$c_1 = \frac{u_{N+1} - u_0}{N+1}$$ Thus, the solution that satisfies the boundary conditions is given by: $$f(\sigma) = \frac{u_{N+1} - u_0}{N+1} \cdot \sigma + u_0$$

Case (b) with a difference equation and different possible solutions

We are given the difference equation \(\Delta^2 f(\sigma) + p^2 f(\sigma) = 0\). Our goal is to analyze the possible solutions for different values of \(p^2\). First, let's rewrite the difference equation as: $$f(\sigma+1) - 2 f(\sigma) + f(\sigma-1) + p^2 f(\sigma) = 0$$ 1. For \(0 < p^2 < 4\), we assume that the solutions are of the form \(f(\sigma) = c_1 \cos(q\sigma) + c_2 \sin(q\sigma)\), where \(\cos q = 1 - \frac{1}{2}p^2\). Substituting this into the equation, we can verify that these functions are indeed solutions. 2. For \(p^2 > 4\), we assume that the solutions are of the form \(f(\sigma) = c_1 a_1^{\sigma} + c_2 a_2^{\sigma}\). To find the distinct numbers \(a_1\) and \(a_2\), we rewrite the difference equation as a characteristic equation: $$a^2 - 2a + 1 - p^2 = 0$$ The solutions to this equation are given by \(a_{1,2} = 1 \pm \sqrt{p^2 - 3}\), which are distinct for \(p^2 > 4\). Thus, the functions \(f(\sigma) = c_1 a_1^{\sigma} + c_2 a_2^{\sigma}\) are indeed solutions. 3. For \(p^2 = 4\), we assume that the solutions are of the form \(f(\sigma) = c_1 (-1)^{\sigma} + c_2 \sigma(-1)^{\sigma}\). Substituting this into the equation, we can verify that these functions are indeed solutions.

Case (c) with initial conditions

We are given that \(f(0)\) and \(f(1)\) are given, and we want to show that the difference equation in case (b) successively determines \(f(2), f(3),\ldots\). Let's suppose we have a solution \(f(\sigma)\) that satisfies the initial conditions \(f(0) = u_0\) and \(f(1) = u_1\). Using these initial conditions, we can determine the possible forms for the solutions. 1. For \(0 < p^2 < 4\), the general solution is given by \(f(\sigma) = c_1 \cos(q\sigma) + c_2 \sin(q\sigma)\). Substituting the initial conditions: $$u_0 = c_1 \cos(0)$$ $$u_1 = c_1 \cos(q) + c_2 \sin(q)$$ From these equations, we can find the constants \(c_1\) and \(c_2\): $$c_1 = u_0$$ $$c_2 = \frac{u_1 - u_0\cos(q)}{\sin(q)}$$ Thus, the expression for the general solution for \(0 < p^2 < 4\) is given by: $$f(\sigma) = u_0 \cos(q\sigma) + \frac{u_1 - u_0\cos(q)}{\sin(q)}\sin(q\sigma)$$ 2. For \(p^2 > 4\), the general solution is given by \(f(\sigma) = c_1 a_1^{\sigma} + c_2 a_2^{\sigma}\). Substituting the initial conditions: $$u_0 = c_1 a_1^0 + c_2 a_2^0$$ $$u_1 = c_1 a_1 + c_2 a_2$$ From these equations, we can find the constants \(c_1\) and \(c_2\): $$c_1 = \frac{u_1 a_2 - u_0 a_2}{a_1 a_2 - a_2}$$ $$c_2 = \frac{u_0 a_1 - u_1 a_1}{a_1 - a_2}$$ Thus, the expression for the general solution for \(p^2 > 4\) is given by: $$f(\sigma) = \frac{u_1 a_2 - u_0 a_2}{a_1 a_2 - a_2} a_1^{\sigma} + \frac{u_0 a_1 - u_1 a_1}{a_1 - a_2} a_2^{\sigma}$$ 3. For \(p^2 = 4\), the

Key concepts

Integer Variable

In the study of difference equations, understanding the role of an integer variable is crucial. An integer variable, often denoted as \(\sigma\), can take on values that are whole numbers, such as 0, ±1, ±2, and so on.
These variables are key in discrete mathematics because they provide a fundamental unit of measurement in the analysis of sequences and difference equations.

The function \(f(\sigma)\) is defined where \(\sigma\) acts as the independent variable in a given range, such as from \(\sigma = m\) to \(\sigma = n\). In this context, an integer variable like \(\sigma\) allows us to analyze and solve problems where quantities change incrementally rather than continuously, making it a vital concept in difference equations.

First Difference

The concept of the first difference is integral to difference equations and helps describe how a function changes between integer points. If we consider a function \(f(\sigma)\) depending on an integer variable \(\sigma\), the first difference, denoted \(\Delta_{+} f(\sigma)\), is the change in the function as we increment \(\sigma\) by 1:
\(\Delta_{+} f(\sigma) = f(\sigma+1) - f(\sigma)\), applicable within a specified range like \(m \leq \sigma < n\).

There is also an alternate form known as \(\Delta_{-} f(\sigma)\), which measures the change going backward by comparing the current value with the previous one:
\(\Delta_{-} f(\sigma) = f(\sigma) - f(\sigma-1)\), used when \(m < \sigma \leq n\).
This concept is comparable to the derivative in calculus but tailored for discrete data, providing insight into the direction and rate of change of a sequence.

Second Difference

To go deeper into the analysis of how a sequence evolves, we apply the second difference. It is essentially the first difference of the first difference.
The second difference, denoted by \(\Delta^{2} f\), can be expressed as:
\(\Delta^{2} f = \Delta_{+} \Delta_{-} f = \Delta_{-} \Delta_{+} f = f(\sigma+1) - 2f(\sigma) + f(\sigma-1)\).

This calculation gives us information about the curvature of a sequence; for example, whether the changes in the sequence are accelerating or decelerating over time.
Similar to how the second derivative in calculus provides insights about concavity, the second difference in difference equations helps determine patterns and effects occurring between the points.

Linear Difference Equation

A linear difference equation is a key concept akin to linear differential equations found in continuous calculus. It involves an expression that relates a function with its differences to determine a sequence's behavior over discrete intervals.
A general form might look like \(\Delta^n f(\sigma) + a_{n-1}\Delta^{n-1} f(\sigma) + \ldots + a_0 f(\sigma) = 0\), where the coefficients \(a_{n-1}, \ldots, a_0\) are constants.

Linear difference equations require solutions that satisfy the equation under given conditions, and typically, these solutions are linear combinations of functions. For instance, functions of the form \(e^{r\sigma}\) often play an analogous role to power series solutions in differential equations, highlighting patterns in growth or decay based on the sequence defined.
These equations are instrumental in modeling numerous real-world scenarios, from iterative financial calculations to digital signal processing.