“difference equation”的概念、定义、翻译、参考文献-科学参考

difference equation

Mathematics

Let u₀, u₁, u₂,…, u_n,… be a sequence. If the terms satisfy the first‐order difference equation u_n + 1 + au_n = 0, it is easy to see that u_n = A(−a)ⁿ, where A( = u₀) is arbitrary.

Suppose that the terms satisfy the second‐order difference equation u_n + 2 + au_n + 1 + bu_n = 0. Let α‎ and β‎ be the roots of the quadratic ‘auxiliary equation’ x² + ax + b = 0. If α‎ ≠ β‎, then u_n = Aα‎ⁿ + Bβ‎ⁿ, and if α‎ = β‎ ≠ 0, then u_n = (A + Bn)α‎ⁿ, where A and B are arbitrary constants. For example, the Fibonacci sequence is given by the difference equation u_n + 2 = u_n + 1 + u_n, with u₀ = 1 and u₁ = 1, and the above method gives Binet’s formula

$u_{n} = \frac{1}{2} {(\frac{1 + \sqrt{5}}{2})}^{n} - \frac{1}{2} {(\frac{1 - \sqrt{5}}{2})}^{n} .$

The above theory generalizes to linear constant coefficient difference equations (see linear equation) of higher orders, with the solutions forming a vector space with dimension equalling the order. The solutions to an inhomogeneous linear difference equation are a particular solution added to the homogeneous solutions. Difference equations, also called recurrence relations, do not necessarily have constant coefficients like those considered above. Such difference equations may be approached using generating functions.

Electronics and Electrical Engineering

1. An equation that defines an indexed sequence of numbers by giving some initial values and describing how subsequent members of the sequence are related to immediate neighbours. For example, the Fibonacci sequence defines f₀ and f₁ as 1 and then recursively defines f_n = f_n−1 + f_n−2 for n > 1.

2. A discrete form of differentiation applied to sequences of real numbers. For example, given the ordered sequence r₁, r₂, r₃, … we can define the first difference as the sequence Δ(r₁),Δ(r₂),Δ(r₃), … defined by
$Δ (r_{n}) = r_{n + 1} - r_{n}$
See also differential equation.

单词	difference equation
释义	difference equation Mathematics Let u₀, u₁, u₂,…, u_n,… be a sequence. If the terms satisfy the first‐order difference equation u_n + 1 + au_n = 0, it is easy to see that u_n = A(−a)ⁿ, where A( = u₀) is arbitrary. Suppose that the terms satisfy the second‐order difference equation u_n + 2 + au_n + 1 + bu_n = 0. Let α‎ and β‎ be the roots of the quadratic ‘auxiliary equation’ x² + ax + b = 0. If α‎ ≠ β‎, then u_n = Aα‎ⁿ + Bβ‎ⁿ, and if α‎ = β‎ ≠ 0, then u_n = (A + Bn)α‎ⁿ, where A and B are arbitrary constants. For example, the Fibonacci sequence is given by the difference equation u_n + 2 = u_n + 1 + u_n, with u₀ = 1 and u₁ = 1, and the above method gives Binet’s formula $u_{n} = \frac{1}{2} {(\frac{1 + \sqrt{5}}{2})}^{n} - \frac{1}{2} {(\frac{1 - \sqrt{5}}{2})}^{n} .$ The above theory generalizes to linear constant coefficient difference equations (see linear equation) of higher orders, with the solutions forming a vector space with dimension equalling the order. The solutions to an inhomogeneous linear difference equation are a particular solution added to the homogeneous solutions. Difference equations, also called recurrence relations, do not necessarily have constant coefficients like those considered above. Such difference equations may be approached using generating functions. Electronics and Electrical Engineering 1. An equation that defines an indexed sequence of numbers by giving some initial values and describing how subsequent members of the sequence are related to immediate neighbours. For example, the Fibonacci sequence defines f₀ and f₁ as 1 and then recursively defines f_n = f_n−1 + f_n−2 for n > 1. 2. A discrete form of differentiation applied to sequences of real numbers. For example, given the ordered sequence r₁, r₂, r₃, … we can define the first difference as the sequence Δ(r₁),Δ(r₂),Δ(r₃), … defined by $Δ (r_{n}) = r_{n + 1} - r_{n}$ See also differential equation.
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