The calculation for characteristic multiplier of Hill’s equation in case q(t) < 0

doi:10.1016/j.amc.2003.10.065

Applied Mathematics and Computation

Volume 167, Issue 2, 15 August 2005, Pages 1130-1149

https://doi.org/10.1016/j.amc.2003.10.065 Get rights and content

Abstract

We have given the calculation for characteristic multiplier of Hill’s equation x″ + q(t)x = 0 in case q(t) > 0 in the work of Jinlin Shi et al. [Appl. Math. Comput., in press (AMC8559)]. In this paper, we consider the calculation for characteristic multiplier in case q(t) < 0.

Section snippets

The discriminant of Hill’s equation in case q(t) < 0

Consider Hill’s equation: $x^{″} + q (t) x = 0,$ where q(t) ∈ c¹, q(t + π) = q(t).

If q(t) > 0, we had given the discriminant of (1.1) as follows: (see [2]) $Δ = 2 \cos \int_{0}^{π} \sqrt{q (s)} d s + \sum_{n = 1}^{\infty} \frac{1}{2^{4 n - 1}} \int_{0}^{π} \int_{0}^{t_{1}} \dots \int_{0}^{t_{2 n - 1}} \cos Ψ (t_{1} \dots t_{2 n}) \prod_{j = 1}^{2 n} \frac{q^{'} (t_{j})}{q (t_{j})} d t_{2 n} \dots d t_{1},$ where $Ψ (t_{1} \dots t_{2 n}) = \int_{0}^{π} \sqrt{q (s)} d s - 2 \int_{t_{2}}^{t_{1}} \sqrt{q (s)} d s - \dots - 2 \int_{t_{2 n}}^{t_{2 n - 1}} \sqrt{q (s)} d s .$ If q(t) < 0, we will prove that (1.2) still holds. In this case, (1.2) can be written as $Δ = 2 \cos i \int_{0}^{π} \sqrt{- q (s)} d s + \sum_{n = 1}^{\infty} \frac{1}{2^{4 n - 1}} \int_{0}^{π} \int_{0}^{t_{1}} \dots \int_{0}^{t_{2 n - 1}} \cos i Ψ (t_{1} \dots t_{2 n}) \prod_{j = 1}^{2 n} \frac{q^{'} (t_{j})}{q (t_{j})} d t_{2 n} \dots d t_{1},$ where i is imaginary unit and $Ψ (t_{1} \dots t_{2 n}) = \int_{0}^{π} \sqrt{- q (s)} d s - 2 \int_{t_{2}}^{t_{1}} \sqrt{- q (s}$

The calculation of Δ₁ and Δ₂

From (1.26), $Δ_{1} = \frac{1}{2^{3}} \int_{0}^{π} \int_{0}^{t_{1}} \frac{q^{'} (t_{1})}{q (t_{1})} \frac{q^{'} (t_{2})}{q (t_{2})} \cos i (Φ (π) - 2 Φ (t_{1}, t_{2})) d t_{2} d t_{1} = \cos i Φ (π) \frac{1}{2^{3}} \int_{0}^{π} \int_{0}^{t_{1}} \frac{q^{'} (t_{1})}{q (t_{1})} \frac{q^{'} (t_{2})}{q (t_{2})} \cos 2 i (Φ (t_{1}) - Φ (t_{2})) d t_{2} d t_{1} + \sin i Φ (π) \frac{1}{2^{3}} \int_{0}^{π} \int_{0}^{t_{1}} \frac{q^{'} (t_{1})}{q (t_{1})} \frac{q^{'} (t_{2})}{q (t_{2})} \sin 2 i (Φ (t_{1}) - Φ (t_{2})) d t_{2} d t_{1} = \frac{1}{2^{3}} \cos i Φ (π) \int_{0}^{π} \int_{0}^{t_{1}} \frac{q^{'} (t_{1})}{q (t_{1})} \frac{q^{'} (t_{2})}{q (t_{2})} \cos 2 i Φ (t_{1}) \cos 2 i Φ (t_{2}) d t_{2} d t_{1} + \frac{1}{2^{3}} \cos i Φ (π) \int_{0}^{π} \int_{0}^{t_{1}} \frac{q^{'} (t_{1})}{q (t_{1})} \frac{q^{'} (t_{2})}{q (t_{2})} \sin 2 i Φ (t_{1}) \sin 2 i Φ (t_{2}) d t_{2} d t_{1} + \frac{1}{2^{3}} \sin i Φ (π) \int_{0}^{π} \int_{0}^{t_{1}} \frac{q^{'} (t_{1})}{q (t_{1})} \frac{q^{'} (t_{2})}{q (t_{2})} \sin 2 i Φ (t_{1}) \cos 2 i Φ (t_{2}) d t_{2} d t_{1} - \frac{1}{2^{3}} \sin i Φ (π) \int_{0}^{π} \int_{0}^{t_{1}} \frac{q^{'} (t_{1})}{q (t_{1})} \frac{q^{'} (t_{2})}{q (t_{2})} \cos 2 i Φ (t_{1}) \sin 2 i Φ (t_{2}) d t_{2} d t_{1} \overset{def}{=} I_{1} + I_{2} + I_{3} + I_{4} .$ It follows

The estimation of error

Since $Δ - \tilde{Δ} = \sum_{n = 3}^{\infty} Δ_{n}$ , so the extreme error do not exceed $|\sum_{n = 3}^{\infty} Δ_{n}|$ . By (1.26), $Δ_{n} = \frac{1}{2^{4 n - 1}} \int_{0}^{π} \int_{0}^{t_{1}} \dots \int_{0}^{t_{2 n - 1}} \cos i (Φ (π) - 2 Φ (t_{1}, t_{2}) - \dots - 2 Φ (t_{2 n - 1}, t_{2 n})) \prod_{j = 1}^{2 n} \frac{q^{'} (t_{j})}{q (t_{j})} d t_{2 n} \dots d t_{1} .$ First, we estimate $|\cos i (Φ (π) - 2 Φ (t_{1}, t_{2}) - \dots - 2 Φ (t_{2 n - 1}, t_{2 n}))| = \frac{1}{2} (\exp (Ψ (t_{1} \dots t_{2 n})) + \exp (- Ψ (t_{1} \dots t_{2 n}))),$ where $Ψ (t_{1} \dots t_{2 n}) = Φ (π) - 2 Φ (t_{1}, t_{2}) - \dots - 2 Φ (t_{2 n - 1}, t_{2 n}) = \int_{0}^{π} \sqrt{- q (τ)} d τ - 2 \int_{t_{2}}^{t_{1}} \sqrt{- q (τ)} d τ - \dots - 2 \int_{t_{2 n}}^{t_{2 n - 1}} \sqrt{- q (τ)} d τ .$ Noting that 0 ⩽ t_2n ⩽ t_2n−1 ⩽ ⋯ ⩽ t₂ ⩽ t₁ ⩽ π, we can conclude $Ψ (t_{1} \dots t_{2 n}) = (\int_{0}^{t_{2 n}} \sqrt{- q (τ)} d τ + \int_{t_{2 n - 1}}^{t_{2 n - 2}} \sqrt{- q (τ)} d τ + \dots + \int_{t_{1}}^{π} \sqrt{- q (τ)} d τ) - (\int_{t_{2}}^{t_{1}} \sqrt{- q (τ)} d τ + \dots + \int_{t_{2 n}}^{t_{2 n - 1}} \sqrt{- q (τ)} d τ) \overset{def}{=} α_{1} + α_{2} .$ It follows from $\sqrt{- q}$

An example

We consider the differential equation $x^{″} - (1 + \frac{1}{2} \cos 2 t - \frac{1}{2} \sin 2 t + \frac{1}{16} \cos^{2} 2 t) x = 0 .$ It is easily proved that $x (t) = \exp (t + \frac{1}{8} \sin 2 t)$ is one of its solutions whose characteristic exponent is 1, we denote the characteristic multiplier of (4.1) by ρ₁, ρ₂, then we can obtain ρ₁ = e^π.

We denote the discriminant of (4.1) by Δ, then the characteristic multiplier ρ satisfy $ρ^{2} - Δ ρ + 1 = 0$ and $ρ_{1} ρ_{2} = 1, ρ_{1} + ρ_{2} = Δ$ so ρ₂ = e^−π, Δ = e^π + e^−π ≈ 23.1838.

According to (2.13), (2.20), $\tilde{Δ}$ may be obtained in the following way. In the interval [0, π] choose n

References (2)

J. Shi, M. Lin, J. Chen, The calculation for characteristic multiplier of Hill’s equation, Appl. Math. Comput., in...
J. Shi
A new form of discriminant for Hill’s equation
Ann. Diff. Eqs.
(1999)

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Using Fourier coefficients to determine the unstable points of Hill equation
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The calculation for characteristic multiplier of Hill's equation x<sup>″</sup> + q (t) x = 0 in case q (t) with positive mean
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We gave the calculate formula for characteristic multiplier of Hill's equation $x^{″} + q (t) x = 0$ in case $q (t) > 0$ and $q (t) < 0$ in the papers of Shi et al. [The calculation for characteristic multiplier of Hill's equation, Appl. Math. Comput. 159 (2004) 57–77] and Shi et al. [The calculation for characteristic multiplier of Hill's equation in case $q (t) < 0$ , 167 (2005) 1130–1149], respectively. In this paper, we give the calculate formula for characteristic multiplier of Hill's equation in case $q (t)$ alternate signs and $\int_{0}^{π} q (t) d t > 0$ .

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The calculation for characteristic multiplier of Hill’s equation in case q(t) < 0

Abstract

Section snippets

The discriminant of Hill’s equation in case q(t) < 0

The calculation of Δ1 and Δ2

The estimation of error

An example

A new form of discriminant for Hill’s equation

Ann. Diff. Eqs.

The calculation of Δ₁ and Δ₂