Stability of the Diffusion Equation with a Source

The stability problem for functional equations or differential equations started with the well-known question of Ulam [1]: Under what conditions does there exist an additive function near an approximately additive function? In 1941, Hyers [2] gave an affirmative answer to the question of Ulam 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)

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, Hegyi and Jung proved the generalized Hyers-Ulam stability of the diffusion equation on the restricted domain or with an initial condition (see [15,16]).
In this paper, applying ideas from [15,17], we investigate the generalized Hyers-Ulam stability of the (inhomogeneous) diffusion equation with a source   (, ) −  △  (, ) =  (, ) for  ∈ R  \ {(0, . . ., 0)} and  > 0, where  is a positive constant, △ =  2 / 2 1 +⋅ ⋅ ⋅+ 2 / 2  , and  is a positive integer.The main advantages of this present paper over the previous works [15,16] are that this paper deals with the inhomogeneous diffusion equation with a source and it describes the behavior of approximate solutions of diffusion equation in the vicinity of origin (roughly speaking, an approximate solution is a solution to a perturbed equation), while the previous works deal with domains not including the vicinity of origin or the homogeneous diffusion equation (without source term).

Preliminaries
If (, ) is a solution to the diffusion equation (4) with  = 1 and  is a positive constant, then the dilated function (, ) fl (√, ) satisfies the equality,   (, ) −   (, ) = (√, ), for all  > 0 and  > 0. When the source term (, ) satisfies the additional condition  (√, ) =  (, ) , (, ) is also a solution to (4) with  = 1.This property is called the invariance under dilation.Hence, it is worth searching for approximate solutions to (4), which are scalar functions of the form where  is a real parameter which will be determined later and V is a twice continuously differentiable function.That is, (, ) depends on  and  primarily through the term ||/ √ 4.We note that the intention of inclusion of the factor 1/ √ 4 in the above formula is to simplify our formulations later.Throughout this paper, let  be a fixed positive integer if there is no specification.Each point  in R  is expressed as  = ( 1 , . . .,   , . . .,   ), where   denotes the th coordinate of .Moreover, || denotes the Euclidean distance of  from the origin; i.e., Based on this argument, we define there exists a twice continuously differentiable function V : ) for all  ∈ R  \ {(0, . . ., 0)} ,  > 0} , (8) where  is a positive integer and  is a parameter.The proof of the following lemma runs in the usual and routine way.Hence, we omit the proof.
Let us define the second-order differential operator L 2  : where C(0, ∞) and C 2 (0, ∞) denote the set of all continuous real-valued functions and the set of all twice continuously differentiable real-valued functions defined on (0, ∞), respectively.
We now try to decompose the differential operator L 2  into differential operators L () and L () of first order such that for all V ∈ C 2 (0, ∞), where we define Then we have Comparing both ( 12) and ( 14), we obtain From the last system of equations, we get a Riccati equation one of whose solutions has the form ( − 1)/, where  ̸ = 0 is a real constant: If we put   () = ( − 1)/ in the Riccati equation ( 16), then we have Comparing ( 16) with (17) and considering that 4 is a constant, we conclude that for all integers  ≥ 2. (Even if  is not defined for  = 1, we can also verify the truth of the formulas for  and   () for  = 1 by a direct calculation.)Using this particular solution   () and in view of [18, § 1.2.1], the general solution of the Riccati equation ( 16) with 4 = 2 − 4 is given by where  0 is a nonnegative fixed real number,   is a constant, and we set   = ∞ for the particular solution   () = ( − 2)/.

Main Results
Before starting with our main theorem, we modify the theorem ([ for all  ∈ , then there exists a unique continuously differentiable function V 0 :  →  such that V  0 ()+()V 0 ()+() = 0 for all  ∈  and for all  ∈ .
On account of Lemma 2, we further have for all  > 0, where () is defined in (19).for all  > 0, where () is given in (19) with a positive real constant  0 .
We can now apply Theorem 3 to our inequality (33) by considering the substitutions as we see in Table 1.
Our hypothesis that  is an integer not less than 3 implies that It then follows from ( 19) and ( 27) that Hence, we have which implies that the condition () of Theorem 3 is satisfied.Moreover, it follows from the last inequality that exp and, by (26), we get for all  > 0, which means that the condition () of Theorem 3 is satisfied.Similarly, it also follows from (23) that by which we conclude that the condition () of Theorem 3 is satisfied.
We apply Theorem 3 to our inequality (45) by considering the substitutions as we see in Table 2.
First, in view of ( 19) and ( 35), we get and hence ln (   0 ) for all  > 0, by which we see that the condition () of Theorem 3 is satisfied.
Further, it follows from the last inequality that exp for any ,  0 > 0. By ( 42) and ( 48), we easily get It now follows from (26), (48), and (49) that which means that the condition () of Theorem 3 is satisfied.
Analogously, it follows from ( 23) and (48) that by which we conclude that the condition () of Theorem 3 is satisfied.

Discussions and Conclusions
The diffusion equation is sometimes called a heat equation or a continuity equation and it plays an important role in a number of fields of science.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.
The main advantages of this present paper over the existing results [15,16] are that this paper deals with the (inhomogeneous) diffusion equation with a source and it can describe the behavior of approximate solutions of (inhomogeneous) diffusion equation in the vicinity of origin, while the previous work [15] deals with the case of domain  = { ∈ R  |  < || < } with 0 <  <  ≤ ∞ and  ≥ 2, as we see that the domain  does not include the vicinity of origin and while the other existing result [16] deals with the generalized Hyers-Ulam stability of the homogeneous diffusion equation with an initial condition (but without source term).