Difference Equations

Mathematical Modeling · Axiom Academy

LESSON Difference Equations - Mathematical Modeling The Discrete Analog of Differential Equations for Modeling Sequential Change What is a Difference Equation? A difference equation (also called a recurrence relation ) is an equation that defines a sequence recursively: each term is defined as a function of preceding terms. Difference equations are the discrete counterpart of differential equations. While differential equations describe continuous change using derivatives, difference equations describe change that occurs in discrete steps. Continuous vs. Discrete: A Parallel First-Order Difference Equations The simplest difference equations express the next value in terms of only the current value: or equivalently, in terms of the index n : The function f defines the update rule that determines how the system evolves from one step to the next. Given x_ n+1 = 2x_n + 1 with x_0 = 1 , find the first few terms. The Difference Operator Notation Just as calculus uses derivative notation, difference equations have their own notational conventions: This measures the change in x from step n to step n+1 . Using this notation, we can write difference equations in a form that parallels differential equations: Higher-Order Difference Equations A difference equation is of order k if it involves terms up to x_ n-k or equivalently, if the value of x_ n+1 depends on the previous k terms. Second-Order Difference Equation Requires two initial conditions: x_0 and x_1 .

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