Aboodh transform and the stability of second order linear differential equations

In this paper, we introduce a new integral transform, namely Aboodh transform, and we apply the transform to investigate the Hyers–Ulam stability, Hyers–Ulam–Rassias stability, Mittag-Leffler–Hyers–Ulam stability, and Mittag-Leffler–Hyers–Ulam–Rassias stability of second order linear differential equations.

The theory of stability is an important branch of the qualitative theory of differential equations. During the last decades many interesting results have been investigated on different types differential equations (for more details, see [13][14][15][16][17][18]).
A generalization of Ulam's problem was proposed by replacing functional equations with differential equations: The differential equation φ(f , x, x , x , . . . , x (n) ) = 0 has the Hyers-Ulam stability if, for given > 0 and a function x such that there exists a solution x a of the differential equation such that |x(t)x a (t)| ≤ K( ) and that the corresponding differential equation has the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability.
Alsina and Ger [19] investigated the stability of the differential equation x (t)x(t). They proved the following celebrated theorem. Theorem 1.1 ([19]) Let f : I → R be a differentiable function, which is a solution of the following differential inequality x (t)x(t) ≤ , where I is an open interval of R. Then there is a solution g : I → R of x (t) = x(t) such that, for any t ∈ I, we have f (t)g(t) ≤ 3 . This result was generalized by Takahasi et al. [20], who proved the Hyers-Ulam stability for the Banach space-valued differential equation y (t) = λy(t). Furthermore, the Hyers-Ulam stability has been proved for the first order linear differential equations in more general settings [21][22][23][24][25].
In 2007, Wang, Zhou, and Sun [26] established the Hyers-Ulam stability of a class of first order linear differential equations.
Many different methods for solving differential equations have been used to study the Hyers-Ulam stability problem for various differential equations. But some initial conditions have more significant advantage for solving differential equations. In 2011, Gavruta, Jung, and Li [27] studied the Hyers-Ulam stability for the second order linear differential equation y + β(x)y = 0 with initial and boundary conditions using Taylor's formula.
In 2014, Alqifiary and Jung [28] investigated the generalized Hyers-Ulam stability of by using the Laplace transform method. In 2020, Murali and Selvan [29] established the Mittag-Leffler-Hyers-Ulam stability of the first order linear differential equation for both homogeneous and non-homogeneous cases by using Laplace transformation. The Hyers-Ulam stability of differential equations has been given attention, and it was established by many authors (see [30][31][32][33][34]). Recently, Murali, Selvan, and Park [35] investigated the Hyers-Ulam stability of various differential equations by using the Fourier transform method (also [36,37]).
In this paper, our main intention is to establish the Hyers-Ulam stability and the Mittag-Leffler-Hyers-Ulam stability of the following second order linear differential equations: and for all t ∈ I, u(t) ∈ C 2 (I) and q(t) ∈ C(I), I = [a, b], -∞ < a < b < ∞, by using a new integral transform method, i.e., Aboodh transform method.

Preliminaries
In this section, we introduce some standard notations and definitions which will be very useful to obtain our main results.
Throughout this paper, F denotes the real field R or the complex field C. A function f : (0, ∞) → F is said to be of exponential order if there exist constants A, B ∈ R such that |f (t)| ≤ Ae tB for all t > 0. Definition 2.1 ([38, 39]) The Aboodh integral transform is defined, for a function of exponential order f (t), by provided that the integral exists for some ξ , where ξ ∈ (k 1 , k 2 ). A is called the Aboodh integral transform operator.
Let f and g be Lebesgue integrable functions on (-∞, +∞). Let S denote the set of x for which the Lebesgue integral exists. This integral defines a function h on S called the convolution of f and g. We also write h = f * g to denote this function.
Let I, J ⊆ R be intervals. Throughout this paper, we denote the space of k continuously differentiable functions from I to J by C k (I, J) and denote C k (I, I) by C k (I). Furthermore, C(I, J) = C 0 (I, J) denotes the space of continuous functions from I to J. In addition, R + := [0, ∞). From now on, we assume that Here, we give some definitions of various forms of Hyers-Ulam stability and Mittag-Leffler-Hyers-Ulam stability of differential equations (1.1) and (1.2).

Definition 2.4
We say that differential equation (1.1) has the Hyers-Ulam stability if there exists a constant L > 0 satisfying the following condition: For every > 0 and some u(t) ∈ C 2 (I) satisfying the inequality u (t) + μ 2 u ≤ for all t ∈ I, there exists a solution v ∈ C 2 (I) satisfying the differential equation v (t)+μ 2 v = 0 and |u(t)v(t)| ≤ L for all t ∈ I. We call such L the Hyers-Ulam stability constant for (1.1).

Definition 2.5
We say that differential equation (1.1) has the Hyers-Ulam-Rassias stability with respect to φ ∈ C(R + , R + ) if there exists a constant L φ > 0 with the following property: For every > 0 and some u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I, there exists a solution v ∈ C 2 (I) satisfying the differential equation v (t)+μ 2 v = 0 and for all t ∈ I. We call such L the Hyers-Ulam-Rassias stability constant for (1.1).

Definition 2.6
We say that differential equation (1.2) has the Hyers-Ulam stability if there exists a constant L > 0 satisfying the following condition: For every > 0 and some u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I, there exists some v ∈ C 2 (I) satisfying v (t) + μ 2 v = q(t) and for all t ∈ I. We call such L the Hyers-Ulam stability constant for (1.2).

Definition 2.7
We say that differential equation (1.2) has the Hyers-Ulam-Rassias stability with respect to φ ∈ C(R + , R + ) if there exists a constant L φ > 0 such that, for every > 0 and some u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I, there exists some v ∈ C 2 (I) satisfying the differential equation v (t) + μ 2 v = q(t) and for all t ∈ I. We call such L the Hyers-Ulam-Rassias stability constant for (1.2).

Definition 2.8
We say that differential equation (1.1) has the Mittag-Leffler-Hyers-Ulam stability if there exists a positive constant L satisfying the following condition: For every > 0 and some u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I. We call such L the Mittag-Leffler-Hyers-Ulam stability constant for (1.1).

Definition 2.10
We say that differential equation (1.2) has the Mittag-Leffler-Hyers-Ulam stability if there exists a positive constant L satisfying the following condition: For every > 0 and some u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I, there exists a solution v ∈ C 2 (I) satisfying the linear differential equation v (t)+ μ 2 v = q(t) and for all t ∈ I. We call such L the Mittag-Leffler-Hyers-Ulam stability constant for (1.2).

Definition 2.11
We say that differential equation (1.2) has the Mittag-Leffler-Hyers-Ulam-Rassias stability with respect to φ : (0, ∞) → (0, ∞) if there exists a positive constant L φ satisfying the following condition: For every > 0 and some u(t) ∈ C 2 (I) satisfying the inequality for all t ∈ I, there exists a solution v ∈ C 2 (I) satisfying the linear differential equation v (t)+ We call such L φ the Mittag-Leffler-Hyers-Ulam-Rassias stability constant for (1.2).
for all t ∈ I. We prove that there exists a real number L > 0 which is independent of and u such that and thus In view of (3.2), a function u 0 : (0, ∞) − → F is a solution of (1.1) if and only if If there exist constants l and m in F such that ξ 2 + μ 2 = (ξl)(ξm) with l + m = 0 and lm = μ 2 , then (3.2) becomes We have v(0) = u(0) and u (0) = v (0). Taking the Aboodh transform to v(t), we obtain On the other hand, is a solution of (1.1). It follows from (3.3) and (3.4) that The above equalities show that Taking modulus on both sides and using |p(t)| ≤ , we get By Definition 2.4, linear differential equation (1.1) has the Hyers-Ulam stability. This finishes the proof.
By using the same technique as in Theorem 3.1, we can also prove the Hyers-Ulam-Rassias stability of differential equation (1.1). The method of the proof is similar, but we include it for the sake of completeness.

Corollary 3.4 For every > 0, let u(t) be a twice continuously differentiable function on I which satisfies the inequality
for all t ∈ I. Then there exists a real number L φ > 0 which is independent of and u such that

Theorem 4.1 Non-homogeneous linear differential equation (1.2) has Hyers-Ulam stability.
Proof For every > 0 and for each solution u(t) ∈ C 2 (I) satisfying for all t ∈ I, we prove that there exists a real number L > 0 which is independent of and u such that Taking the Aboodh transform to p(t), we have Equality (4.2) shows that a function u 0 : (0, ∞) − → F is a solution of (1.2) if and only if If there exist constants l and m in F such that ξ 2 + μ 2 = (ξl)(ξm) with l + m = 0 and lm = μ 2 , then (4.2) becomes Set r(t) = e lt -e mt l-m and v(t) = u(0) le ltme mt lm + u (0)r(t) + (r * q)(t) .
Then v(0) = u(0) and u (0) = v (0). Once more, taking the Aboodh transform to v(t), we have On the other hand, By (4.4), the last equality becomes A{v (t) + μ 2 v} = A{q}. Since A is one-to-one and linear, v (t) + μ 2 v = q(t), which shows that v(t) is a solution of (1.

2). Now, relations (4.3) and (4.4) necessitate that
Taking modulus on both sides of the last equality and using |p(t)| ≤ , we get where the integrals In analogous way to Theorem 4.1, we have the following result which proves the Hyers-Ulam-Rassias stability of differential equation (1.2).
By using the same technique as in Theorem 3.1, we can also prove the Mittag-Leffler-Hyers-Ulam stability of differential equation (1.2). The method of the proof is similar, but we include it for the sake of completeness.
In analogous way to Theorem 4.3, we have the following corollary which proves the Mittag-Leffler-Hyers-Ulam-Rassias stability of differential equation (1.2).

Corollary 4.4 For every > 0, let u(t) be a twice continuously differentiable function on I which satisfies the inequality
for all t ∈ I. Then there exists a real number L φ > 0 which is independent of and u such that for some v ∈ C 2 (I) satisfying v (t) + μ 2 v = q(t) for all t ∈ I.

Examples and remarks
In this section, we provide some examples to make it easier to understand the main results of this paper.
Example 5.1 We consider the following homogeneous linear differential equation of second order: where μ 2 = 1, with the initial conditions u(0) = u (0) = 1. Letting p(t) = u (t) + u(t) in Theorem 3.1 and taking the Aboodh transform, we get By the initial conditions, we have A{u} = ξ P(ξ )+ξ +1 ξ (ξ 2 +1) . If a continuously differentiable function u : [0, ∞) → F of exponential order satisfies for all t ≥ 0 and for some > 0, then, by Theorem 3.1, there exists a solution v : [0, ∞) → F of differential equation (5.1) such that for all t ≥ 0. In fact, v(t) = c 1 cos t + c 2 sin t for some constants c 1 , c 2 ∈ F. Example 5.2 Let us take the non-homogeneous linear differential equation with the initial conditions u(0) = u (0) = -1. Here q(t) = t is a function of exponential order and μ 2 = 3. If a continuously differentiable function u : [0, ∞) → F of exponential order satisfies for all t ≥ 0 and some ε > 0, then, by Theorem 4.1, there exists a solution v : [0, ∞) → F of differential equation (5.2) such that v(t) is of exponential order and for all t ≥ 0. In fact, v(t) = c 1 cos √ 3t + c 2 sin √ 3t + 1 3 t for some constants c 1 , c 2 ∈ F. where q(t) = 4e 3t is a function of exponential order with μ 2 = 2.
By the initial conditions, we have If a continuously differentiable function u : [0, ∞) → F of exponential order satisfies u (t) + 2u(t) -4e 3t ≤ for all t ≥ 0 and some > 0, then, by Theorem 4.1, there exists a solution v : [0, ∞) → F of differential equation (5.3) such that where q(t) = 2 cos t is a function of exponential order with μ 2 = 2.
Remark 5. 5 The above examples are also true when we replace and Kε with φ(t)ε and Kφ(t) , respectively, where φ(t) is an increasing function. In this case, we see that the corresponding differential equations have the Hyers-Ulam-Rassias stability.

Conclusion
In this paper, we introduced a new integral transform, namely Aboodh transform, and we applied the transform to investigate the Hyers-Ulam stability, the Hyers-Ulam-Rassias stability, the Mittag-Leffler-Hyers-Ulam stability, and the Mittag-Leffler-Hyers-Ulam-Rassias stability of second order linear differential equations with constant coefficients.
In other words, we established sufficient criteria for the Hyers-Ulam stability of second order linear differential equations with constant coefficients by using the Aboodh transform method. Moreover, this paper provides a new method to investigate the Hyers-Ulam stability of differential equations. This is the first attempt to use the Aboodh transformation to prove the Hyers-Ulam stability for linear differential equations of second order. Furthermore, this paper shows that the Aboodh transform method is more convenient for investigating the stability problems for linear differential equations with constant coefficients. Readers can also apply this terminology to various problems on differential equations.