Stability of the Wave Equation with a Source

Abstract

We prove the generalized Hyers-Ulam stability of the wave equation with a source, ( , ) − 2 ( , ) = ( , ), for a class of real-valued functions with continuous second partial derivatives in and .

A Dilation Invariance Method and the Stability of Inhomogeneous Wave Equations

“…In this paper, applying the method of dilation invariance (see [18,20]), we investigate the generalized Hyers-Ulam stability of the (inhomogeneous) wave Equation (2) with a source. The main advantages of this present paper over the previous works [18,19,21] are that this paper deals with the inhomogeneous wave equation and its time variable runs through the whole half line (0, ∞) while the previous one [18] deals with the homogeneous wave equation and its time variable is not allowed to run through the whole half line (roughly speaking, the relevant domain seems somewhat artificial) and the other previous ones [19,21] deal with the one-dimensional homogeneous case only. In addition, the present work gives a partial answer to the open problem raised in ( [18], Remark 3) concerning the domains of relevant functions.…”

“…for r ∈ (0, c), where c 1 is an arbitrary real constant and the last integral exists due to (12), (15) and (21). Now, let us define a function w 0 : (0, c) → R by…”

Hampiran Solusi Persamaan Gelombang Dua Dimensi Dengan Pendekatan Finite Difference

“…Bentuk persamaan gelombang satu dimensi adalah sebagai berikut dengan kondisi awal dan [4]. Sedangkan bentuk persamaan gelombang dua dimensi adalah Dengan kondisi awal dan .…”

Stability of the Diffusion Equation with a Source

“…For example, the diffusion equation describes the conduction of heat, the signal transmission in communication systems, and diffusion models of chemical diffusion phenomena and it is also connected with Brownian motion in probability theory. This paper was partially motivated by a previous work [21] in which the generalized Hyers-Ulam stability of the one-dimensional wave equation with a source, ( , ) − 2 ( , ) = ( , ) was investigated by using the method of characteristic coordinates. On the other hand, we prove in this paper the generalized Hyers-Ulam stability of the -dimensional diffusion equation with a source, ( , ) − △ ( , ) = ( , ), by applying a kind of method for decomposition of differential operators.…”