Question

Prove that \(\left(\Delta^{2} \varphi\right)(t)=\varphi(t+2 h)-2 \varphi(t+h)+\varphi(t)\) The nth difference is defined similarly: \(\Delta^{n} \varphi=\Delta\left(\Delta^{n-1} \varphi\right)\).

Short answer

Question: Prove that the second forward difference of a function \(\varphi(t)\) is equal to \(\varphi(t+2h) - 2\varphi(t+h) + \varphi(t)\). Solution: We calculated the first and second forward differences of the function \(\varphi(t)\) using the definition of the nth difference. After simplifying the expression for the second forward difference, we found that it equals \(\varphi(t+2h) - 2\varphi(t+h) + \varphi(t)\). This proves the given statement.

Step-by-step solution

Calculate the first forward difference

To calculate the first forward difference, we will use the definition of the nth difference, where n = 1. In this case, $$\Delta\varphi(t) = \varphi(t+h) - \varphi(t).$$

Calculate the second forward difference

To calculate the second forward difference, we will again use the definition of the nth difference, where n = 2. In this case, $$\Delta^2\varphi(t) = \Delta (\Delta\varphi(t)).$$ Since we already found the first forward difference in Step 1, we can substitute \(\Delta\varphi(t) = \varphi(t+h) - \varphi(t)\) into this equation. Thus, we have $$\Delta^2\varphi(t) = \Delta(\varphi(t+h) - \varphi(t)).$$ Now, we will apply the definition of the nth difference with n = 1. $$\Delta^2\varphi(t) = [\varphi((t+h)+h) - \varphi(t+h)] - [\varphi(t+h) - \varphi(t)].$$

Prove the second forward difference formula

Let's simplify the expression we obtained in Step 2. $$\Delta^2\varphi(t) = [\varphi(t+2h) - \varphi(t+h)] - [\varphi(t+h) - \varphi(t)]$$ Distribute the negative sign in the second term: $$\Delta^2\varphi(t) = \varphi(t+2h) - \varphi(t+h) - \varphi(t+h) + \varphi(t)$$ Combine the like terms, \(\varphi(t+h)\) and \(\varphi(t)\): $$\Delta^2\varphi(t) = \varphi(t+2h) - 2\varphi(t+h) + \varphi(t)$$ We have proved that the second forward difference of a function \(\varphi(t)\) is equal to \(\varphi(t+2h) - 2\varphi(t+h) + \varphi(t)\).

Key concepts

Ordinary Differential Equations

Ordinary differential equations (ODEs) are fundamental in mathematics, forming the cornerstone of many scientific and engineering disciplines. An ODE is an equation that relates a function to its derivatives, capturing the rate of change and the relationship of the function to itself over time. It often takes the form \( \frac{dy}{dx} = f(x,y) \) where \( y \) is the function of the independent variable \( x \) and \( dy/dx \) represents its derivative. Finding a function that satisfies this equation means you’ve solved the ODE, which is equivalent to uncovering a law describing a physical system’s behavior.
ODE applications are vast, including modeling the dynamics of celestial bodies, the behavior of electrical circuits, and the pattern of population growth. Each type of ODE, whether it be first-order, second-order, or higher, has specific methods for finding solutions. Sometimes, these equations can't be solved analytically and numerical methods, like the forward difference method, are employed to approximate a solution.

Forward Difference Method

The forward difference method serves as a numerical technique for approximating the derivatives of a function. This approach is particularly useful when dealing with ODEs where analytical solutions are challenging or impossible to find. At its core, the forward difference method uses the concept of finite differences—a sequence of values that represents the approximate derivatives of a function.
Imagine a function \( \varphi(t) \) where \( h \) is a small step size; the first forward difference \( \Delta\varphi(t) = \varphi(t+h) - \varphi(t) \) estimates the derivative of \( \varphi \) at \( t \) by measuring the change in \( \varphi \) over a small interval \( h \). By iteratively applying this process, higher-order differences can be computed, such as the second forward difference \( \Delta^2\varphi(t) \) as shown in the exercise, providing a way to approximate the behavior of a function's derivatives to solve ODEs numerically.
The effectiveness of this method largely depends on the step size \( h \)—too large, and the approximation might be too coarse; too small, and computational errors may become significant. Therefore, striking the right balance is key when employing the forward difference method in practice.

Difference Equations

Difference equations are discrete counterparts to differential equations and describe the evolution of sequences over time. They are often derived from approximating the derivatives in ordinary differential equations using finite differences, as demonstrated in the forward difference method. Unlike ODEs that deal with continuous functions, difference equations work with discrete sequences which can be easier to analyze and simulate on a computer.
A simple example of a first-order difference equation is \( y_{n+1} = ay_n + c \) where \( a \) and \( c \) are constants, and \( y \) represents the sequence at discrete time intervals \( n \). This provides a recursion relation that can be iterated to produce the next term in the sequence. As seen in the textbook exercise, the second forward difference falls into this category, and it's a way to relate the values of a sequence at different points in time, offering insight on how the sequence evolves.
By understanding the relationship between ODEs and difference equations through numerical methods like the forward difference, we bridge the continuous world of calculus with the discrete realm of sequences and digital computation, expanding our toolbox to explore and solve complex real-world problems.