Stability of the Wave Equation with a Source

The stability problem for functional equations or (partial) differential equations started with the question of Ulam [1]: Under what conditions does there exist an additive function near an approximately additive function? In 1941, Hyers [2] answered the question of Ulam in the affirmative for the Banach space cases. Indeed, Hyers’ theorem states that the following statement is true for all ε ≥ 0: if a function f satisfies the inequality ‖f(x + y) − f(x) − f(y)‖ ≤ ε for all x, then there exists an exact additive function F such that ‖f(x) − F(x)‖ ≤ ε for all x. In that case, the Cauchy additive functional equation, f(x + y) = f(x) + f(y), is said to have (satisfy) the Hyers-Ulam stability. Assume thatV is a normed space and I is an open interval of R. The nth-order linear differential equation an (x) y(n) (x) + an−1 (x) y(n−1) (x) + ⋅ ⋅ ⋅ + a1 (x) y󸀠 (x) + a0 (x) y (x) + h (x) = 0 (1)


Introduction
The stability problem for functional equations or (partial) differential equations started with the question of Ulam [1]: Under what conditions does there exist an additive function near an approximately additive function?In 1941, Hyers [2] answered the question of Ulam in the affirmative for the Banach space cases.Indeed, Hyers' theorem states that the following statement is true for all  ≥ 0: if a function  satisfies the inequality ‖( + ) − () − ()‖ ≤  for all , then there exists an exact additive function  such that ‖() − ()‖ ≤  for all .In that case, the Cauchy additive functional equation, ( + ) = () + (), is said to have (satisfy) the Hyers-Ulam stability.
Assume that  is a normed space and  is an open interval of R. The th-order linear differential equation is said to have (satisfy) the Hyers-Ulam stability provided the following statement is true for all  ≥ 0: if a function  :  →  satisfies the differential inequality for all  ∈ , then there exists a solution  0 :  →  to the differential equation ( 1) and a continuous function  such that ‖() −  0 ()‖ ≤ () for any  ∈  and lim →0 () = 0.
Prástaro and Rassias are the first authors who investigated the Hyers-Ulam stability of partial differential equations (see [14]).Thereafter, the first author [15], together with Lee, proved the Hyers-Ulam stability of the first-order linear partial differential equation of the form,   (, ) +   (, ) + (, ) +  = 0, where ,  ∈ R and ,  ∈ C are constants with R() ̸ = 0.As a further step, the first author proved the generalized Hyers-Ulam stability of the wave equation without source (see [16,17]).
One of typical examples of hyperbolic partial differential equations is the wave equation with a spatial variable  and a time variable , where  > 0 is a constant, whose solution is a scalar function  = (, ) describing the propagation of a wave at a speed  in the spatial direction.In this paper, applying ideas from [16,18], we investigate the generalized Hyers-Ulam stability of the wave equation ( 3) with a source, where  >  and  >  with ,  ∈ R ∪ {−∞}.The main advantages of this present paper over the previous papers [16,17] are that this paper deals with the wave equation with a source and it describes the behavior of approximate solutions of wave equation in the vicinity of origin while the previous one [17] can only deal with domains excluding the vicinity of origin.(Roughly speaking, a solution to a perturbed equation is called an approximate solution.)

Main Results
We know that if we introduce the characteristic coordinates then the wave equation,   (, ) =  2   (, ), is transformed into   (, ) = 0, which seems to be handled easily.
Considering the conditions in (6) and Figure 2, we can integrate each term of the last inequality from  to  with respect to the first variable and then we integrate each term of the resulting inequality from  to  with respect to the second variable to obtain  for all (, ) ∈ .Moreover, we get We now set V(, ) fl (, ) = ( + ,  − ) and, analogously to (11), we compute the partial derivatives: In view of ( 10), ( 12), (19), and (20), we get for all (, ) ∈ , that is, V is a solution to wave equation (3).
Remark 2. In general, it is somewhat tedious to estimate the upper bound of inequality (8).However, in view of ( 10 for all (, ) ∈ .