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Inhomogeneous Simple Harmonic Oscillator EquationĪ function is called a simple harmonic oscillator function if it satisfies the simple harmonic oscillator equation: ? ( ? ) + ? 2 ? ( ? ) = 0. Section 3 will be devoted to a partial solution of the Hyers-Ulam stability problem for the simple harmonic oscillator equation ( 2.1) in a subclass of analytic functions. In Section 2 of this paper, by using the ideas from, we investigate the general solution of the inhomogeneous simple harmonic oscillator equation of the form: ? ( ? ) + ? 2 ? ( ? ) = ∞ ? = 0 ? ? ? ?, ( 1. Subsequently, the authors investigated the Hyers-Ulam stability problem for Bessel differential equation by applying the same method.
#Harmonic oscillator equation series
Moreover, he could successfully apply the power series method to the study of the Hyers-Ulam stability of Legendre differential equation (see ). Jung also proved the Hyers-Ulam stability of various linear differential equations of first order. They also proved the Hyers-Ulam stability of linear differential equations of first order, ? ( ? ) + ? ( ? ) ? ( ? ) = 0, where ? ( ? ) is a continuous function. investigated the Hyers-Ulam stability of ?th order linear differential equation with complex coefficients. Indeed, it was proved in that the Hyers-Ulam stability holds true for the Banach space valued differential equation ? ( ? ) = ? ? ( ? ) (see also ). This result has been generalized by Takahasi et al. If a differentiable function ? ∶ ? → ℝ satisfies the inequality | ? ′ ( ? ) − ? ( ? ) | ≤ ?, where ? is an open subinterval of ℝ, then there exists a solution ? 0 ∶ ? → ℝ of the differential equation ? ( ? ) = ? ( ? ) such that | ? ( ? ) − ? 0 ( ? ) | ≤ 3 ? for any ? ∈ ?. Here, we will introduce a result of Alsina and Ger (see ). Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see ). For more detailed definitions of the Hyers-Ulam stability, we refer the reader to. 2 )Īnd ‖ ? ( ? ) − ? 0 ( ? ) ‖ ≤ ? ( ? ) for any ? ∈ ?, where ? ( ? ) is an expression of ? with l i m ? → 0 ? ( ? ) = 0, then we say that the above differential equation has the Hyers-Ulam stability. 1 )įor all ? ∈ ? and for a given ? > 0. Let ? be a normed space over a scalar field ? and let ? be an open interval, where ? denotes either ℝ or ℂ. This result to obtain a partial solution to the Hyers-Ulam stability problem for the simple harmonic oscillator equation. time derivative) of the oscillator.We solve the inhomogeneous simple harmonic oscillator equation and apply Since we have a second derivative in time of the position $x$, we need to specify two things: the initial position and the initial speed (i.e.
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This equation alone does not allow numerical computing unless we also specify initial conditions, which define the oscillator's state at the time origin. The above equation is the harmonic oscillator model equation. Let's simplify the notation in the following way: If the mass $m$ is located according to the coordinates $x(t)$, and the spring stiffness is $k$, then Newton's equations of momentum read: What's an harmonic oscillator? It's a relatively simple model for natural systems that finds wide applications.Ī classical example of such a system is a mass that is connected to a spring.