Hyers-Ulam stability of first order linear differential equation using Fourier transform method

This research paper aims to identify and also tries to prove differential equation of first order that has Hyers-Ulam and Generalized Hyers-Ulam stability by inducing Fourier Transform method.The analysis of the Hyers-Ulam stability had resulted in providing a constant value for homogeneous and non-homogeneous differential equation with implementation of Fourier transform. In order to prove these facts, few application of non-homogeneous linear differential equation in connect with the main results are provided.


Introduction
In recent passed, differential equation have performed an unavoidable role in the feild of mathematics. The concept of stability of differential equation is nothing but a function satisfying the differential equation approximately is around the corner of an exact solution of differential equation. The history of Hyers -Ulam stability starts from middle of the 19th century. The class of stability we used in this paper was first formulated by S.M.Ulam [1] for functional equation which was solved by Hyers [2] for additive function defined on Banach space. After this result, the stability concept was investigated and generalized by M.Rassias [3] is called Hyers-Ulam Rassias or generalized Hyers-Ulam stability.
Further, C.Alsina and R.Ger [4] were established the Hyers-Ulam stability for differential equations by replacing functional equation. Rezaei and S.M.Jung and Rassias investigate the Hyers-Ulam stability of linear differential equation by applying Laplace Transform Method. In 2014, Q.H.Algifiary and S.M.Jung [8] also derived Hyers-Ulam stability of n-th order linear differential equation with the help of Laplace Transform method. The many researcher investigate the stability problems of functional, differential and fractional differential equation in various directions-see, for emample[ [5], [6], [7], [9]]. . Fourier Transform is one of the oldest and well known method to find solution of differential equation. The Fourier transform named after Joseph Fourier in 1807. It is a mathematical transform which convert a function in time domain into a function in frequency domain. In the literature, there are so many transform to solve differential equations such as Fourier, Laplace, Mellin and Hankel etc. One of the best integral transform for differential equation is Fourier transform. It converts differential equation into simple algebraic equation. After solving the algebraic equation, we can find the solution of the original equation by inverse Fourier technique. However, the Hyers-Ulam stability of first order differential equation has not been fully reported with the help of Fourier transform method.
Motivated by above results, the aim of this paper is to prove some results related to stability based on Fourier transform approach.

Preliminaries
Here, we give some basic definition, properties and theorems to prove our main results.
Definition 2.1 The linear differential equation has stability in sence of "Hyers-Ulam stability" on R if a differentiable mapping I: The inverse Fourier Transform is given by The properties of Fourier transform is given by (ii) (one-to-one)Let f,g: R → F be piecewise continuous and differential functions. If F (I(u)) = F (J (u)) then I(u) = J (u), ∀u ∈ R . (iii) (Differentiation)Let f and f' are continuous and absolutely integrable. Then, F (I (u)) = −iωF (I(u)) Remark:We can notice that differential operator is turned into "Multiplication" operator. Similar behavior is applied to higher number of dimensions. In general,

Definition 2.4 (Convolution) Given two function f and g are both Lebesgue integrable on R.
The convolution of f and g is defined as The Fourier transform of the convolution I(u) and J (u) is the product of the Fourier transform of functions I(u) and J (u). That is The convolution is mathematical operator which compute amount of overlap between two functions. A small

Main result
In this section, we are going to derive our results related to Hyers-Ulam stability of the homogeneous first order differential equation Where, I is a continuously differentiable function and a is a constant.
Where, I(ω) is Fourier Transform of I(u). We now set R (ω) = F (r(u)) = 1 −iω−a . By inverse Fourier transform We get, Z (u) − aZ (u) = 0 since Fourier transform is one to one operator. Hence Z(u) satisfies differential Eq.(3.1). Further, from Eq.(3.4) and (3.6) and convolution property of Fourier transform Thus, I (u) − Z (u) = q (u) * r (u) by one to one property of Fourier transform. We have The result of Hyers-Ulam stability for first order non-homogeneous differential equation is contained in the next theorem.
Where, I(ω) represents frequency value of I(u). We now set R (ω) = F (r(u)) = Which completes the proof.

Application to non-homogeneous linear differential equation
In this section, we provide example to illustrate our main results given above