Ulam Stability for Fractional Differential Equation in Complex Domain

Abstract

The present paper deles with a fractional differential equation 𝑧^𝛼𝐷^𝛼_𝑧𝑢(𝑧)+𝑧𝑢′(𝑧)+(𝑧²−𝑎²)𝑢(𝑧)=∑^∞_𝑛=0𝑎_𝑛𝑧^𝑛+𝛼, 1<𝛼≤2, where 𝑧∈𝑈∶={𝑧∶|𝑧|<1} in sense of Srivastava-Owa fractional operators. The existence and uniqueness of holomorphic solutions are established. Ulam stability for the approximation and holomorphic solutions are suggested.

1. Introduction

Fractional calculus is a rapidly growing subject of interest for physicists and mathematicians. The reason for this is that problems may be discussed in a much more stringent and elegant way than using traditional methods. Fractional differential equations have emerged as a new branch of applied mathematics which has been used for many mathematical models in science and engineering. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations [1–7].

The class of fractional differential equations of various types plays important roles and tools not only in mathematics but also in physics, control systems, dynamical systems, and engineering to create the mathematical modeling of many physical phenomena. Naturally, such equations required to be solved. Many studies on fractional calculus and fractional differential equations, involving different operators such as Riemann-Liouville operators, Erdlyi-Kober operators, Weyl-Riesz operators, Caputo operators, and Grünwald-Letnikov operators, have appeared during the past three decades with its applications in other field. Moreover, the existence and uniqueness of holomorphic solutions for nonlinear fractional differential equations such as Cauchy problems and diffusion problems in complex domain are established and posed [8–15].

The present article deals with a nonhomogeneous fractional differential equation. The nonhomogeneous fractional differential equations involving the Bessel differential equation which appears frequently in practical problems and applications. These equations have proved useful in many branches of physics and engineering. They have been used in problems of treating the boundary value problems exhibiting cylindrical symmetries.

In [1], Srivastava and Owa gave definitions for fractional operators (derivative and integral) in the complex 𝑧-plane ℂ as follows.

Definition 1.1. The fractional derivative of order 𝛼 is defined, for a function 𝑓(𝑧) by 𝐷^𝛼_𝑧𝑓(𝑧)1∶=Γ()𝑑1−𝛼𝑑𝑧^𝑧₀𝑓(𝜁)(𝑧−𝜁)^𝛼𝑑𝜁;0≤𝛼<1, (1.1) where the function 𝑓(𝑧) is analytic in simply-connected region of the complex 𝑧-plane ℂ containing the origin, and the multiplicity of (𝑧−𝜁)^−𝛼 is removed by requiring log(𝑧−𝜁) to be real when (𝑧−𝜁)>0.

Definition 1.2. The fractional integral of order 𝛼 is defined, for a function 𝑓(𝑧), by 𝐼^𝛼_𝑧𝑓(𝑧)1∶=Γ(𝛼)^𝑧₀𝑓(𝜁)(𝑧−𝜁)^𝛼−1𝑑𝜁;𝛼>0, (1.2) where the function 𝑓(𝑧) is analytic in simply-connected region of the complex 𝑧-plane (ℂ) containing the origin, and the multiplicity of (𝑧−𝜁)^𝛼−1 is removed by requiring log(𝑧−𝜁) to be real when (𝑧−𝜁)>0.

Remark 1.3. From Definitions 1.1 and 1.2, we have 𝐷⁰_𝑧𝑓(𝑧)=𝑓(0), lim_𝛼→0𝐼^𝛼_𝑧𝑓(𝑧)=𝑓(𝑧) and lim_𝛼→0𝐷_𝑧^1−𝛼𝑓(𝑧)=𝑓^′(𝑧). Moreover, 𝐷^𝛼_𝑧𝑧^𝜇=Γ𝜇+1Γ𝑧𝜇−𝛼+1^𝜇−𝛼𝐼,𝜇>−1;0≤𝛼<1,_𝛼𝑧𝑧^𝜇=Γ𝜇+1Γ𝑧𝜇+𝛼+1^𝜇+𝛼,𝜇>−1;𝛼>0. (1.3) Further properties of these operators can be found in [10, 11, 13].

2. Nonhomogeneous Fractional Differential Equation

In this section, we will express the solution for the nonhomogeneous problem 𝑧^𝛼𝐷^𝛼_𝑧𝑢(𝑧)+𝑧𝑢^′(𝑧)+𝑧²−𝑎²𝑢(𝑧)=^∞_𝑛=0𝑎_𝑛𝑧^𝑛+𝛼,1<𝛼≤2, (2.1) where the radius of convergence of power series is at least 𝜌, for all 𝑧∈𝑈 and 𝑢∶𝑈→ℂ. Note that for 𝛼=2, (2.1) reduces to the Bessel differential equation.

Theorem 2.1. Consider the problem (2.1) with 𝑎²<𝛼,1<𝛼≤2, and there exists a constant 𝛿>0 satisfying the condition 𝑎_𝑛+2≤Γ(𝑛+𝛼+1)+𝑛+𝛼−𝑎²Γ(𝑛+1)()𝑐𝛿Γ𝑛+1^𝑛, (2.2) for all sufficiently large integers 𝑛, where 𝑐_𝑛=𝑎₀Γ(𝛼+1)−𝑎²𝑎,𝑛=0_𝑛−𝑐_𝑛−2Γ(𝑛+𝛼+1)/Γ(𝑛+1)+(𝑛+𝛼)−𝑎²,𝑛≥1, (2.3) where 𝑐₋₁=0. Then every solution 𝑢∶𝑈→ℂ of the fractional differential equation (2.1) can be expressed by 𝑢(𝑧)=𝑢_ℎ(𝑧)+^∞_𝑛=0𝑐_𝑛𝑧^𝑛+𝛼, (2.4) where 𝑢_ℎ(𝑧) is a solution of the homogeneous 𝑧^𝛼𝐷^𝛼_𝑧𝑢(𝑧)+𝑧𝑢^′(𝑧)+𝑧²−𝑎²𝑢(𝑧)=0. (2.5)

Proof. We assume that 𝑢∶𝑈→ℂ is a function given in the form (2.4), and we define that 𝑢_𝑝(𝑧)=𝑢(𝑧)−𝑢_ℎ(𝑧)=∑^∞_𝑛=0𝑐_𝑛𝑧^𝑛+𝛼. Then, it follows from (2.2) and (2.3) that lim_𝑛→∞𝑐_𝑛+2𝑐_𝑛=lim_𝑛→∞1Γ(𝑛+𝛼+1)+𝑛+𝛼−𝑎²Γ(()𝑎𝑛+1)/Γ𝑛+1^𝑛+2−𝑐_𝑛𝑐_𝑛=lim_𝑛→∞1Γ(𝑛+𝛼+1)+𝑛+𝛼−𝑎²Γ()()𝑎𝑛+1/Γ𝑛+1𝑐^𝑛+2_𝑛≤1−1𝛿,𝛿>0. (2.6)
That is, the power series for 𝑢_𝑝(𝑧) converges for all 𝑧∈𝑈. Hence, we see that the domain of 𝑢(𝑧) is well defined. We now prove that the function 𝑢_𝑝(𝑧) satisfies the nonhomogeneous equation (2.1). Indeed, it follows from (2.3) that𝑧^𝛼𝐷^𝛼_𝑧𝑢_𝑝(𝑧)+𝑧𝑢^′(𝑧)+𝑧²−𝑎²𝑢_𝑝(𝑧)=^∞_𝑛=0Γ(𝑛+𝛼+1)Γ()𝑐𝑛+1^𝑛𝑧^𝑛+𝛼+^∞_𝑛=1(𝑛+𝛼)𝑐_𝑛𝑧^𝑛+𝛼+^∞_𝑛=0𝑐_𝑛𝑧^𝑛+𝛼+2−^∞_𝑛=0𝑐_𝑛𝑎²𝑧^𝑛+𝛼=^∞_𝑛=0Γ(𝑛+𝛼+1)Γ()𝑐𝑛+1^𝑛𝑧^𝑛+𝛼+^∞_𝑛=1(𝑛+𝛼)𝑐_𝑛𝑧^𝑛+𝛼+^∞_𝑛=2𝑐_𝑛−2𝑧^𝑛+𝛼−^∞_𝑛=0𝑐_𝑛𝑎²𝑧^𝑛+𝛼=Γ(𝛼+1)−𝑎²𝑐₀𝑧^𝛼+Γ(𝛼+2)+(𝛼+1)−𝑎²𝑐₁𝑧^𝛼+1+^∞_𝑛=2Γ(𝛼+𝑛+1)Γ()+()𝑛+1𝛼+𝑛−𝑎²𝑐_𝑛+𝑐_𝑛−2𝑧^𝛼+𝑛∶=^∞_𝑛=0𝑎_𝑛𝑧^𝑛+𝛼. (2.7)
Hence 𝑢_𝑝(𝑧) is a particular solution of the nonhomogeneous equation (2.1). On the other hand, since every solution to (2.1) can be expressed as a sum of a solution 𝑢_ℎ(𝑧) of the homogeneous equation and a particular solution 𝑢_𝑝(𝑧) of the nonhomogeneous equation, every solution of (2.1) is certainly of the form (2.4).

3. Ulam Stability

A classical problem in the theory of functional equations is that if a function 𝑓 approximately satisfies functional equation ℰ when does, there exists an exact solution of ℰ which 𝑓 approximates. In 1960, Ulam [16] imposed the question of the stability of Cauchy equation, and, in 1941, Hyers solved it [17]. In 1978, Rassias [18] provided a generalization of Hyers, theorem by proving the existence of unique linear mappings near approximate additive mappings. The problem has been considered for many different types of spaces (see [19–21]). Recently, Li and Hua [22] have discussed and proved the Hyers-Ulam stability of spacial type of finite polynomial equation, and Bidkham et al. [23] have introduced the Hyers-Ulam stability of generalized finite polynomial equation.

In this section, we consider the Hyers-Ulam stability for fractional differential equation (2.1). Let ℋ be the space of all analytic functions on 𝑈.

Definition 3.1. Let 𝑝 be a real number. We say that ^∞_𝑛=0𝑎_𝑛𝑧^𝑛+𝛼=𝑓(𝑧) (3.1) has the generalized Hyers-Ulam stability if there exists a constant 𝐾>0 with the following property: for every 𝜖>0,𝑤∈𝑈=𝑈⋃𝜕𝑈, if ^∞_𝑛=0𝑎_𝑛𝑤^𝑛+𝛼≤𝜖^∞_𝑛=0𝑎_𝑛^𝑝2^𝑛, (3.2) then there exists some 𝑧∈𝑈 that satisfies (2.1) such that 𝑧^𝑚−𝑤^𝑚≤𝜖𝐾,𝑧,𝑤∈𝑈,𝑚∈ℕ. (3.3)

We need the following results.

Lemma 3.2 (see [24]). If 𝑓∶𝐷→𝑋 (𝑋 is a complex Banach space) is holomorphic, then ‖𝑓‖ is a subharmonic of 𝑧∈𝐷⊂ℂ. It follows that ‖𝑓‖ can have no maximum in 𝐷 unless ‖𝑓‖ is of constant value throughout 𝐷.

Theorem 3.3. Let 𝑓(𝑧)∶=𝑧^𝛼𝐷^𝛼_𝑧𝑢(𝑧)+𝑧𝑢^′(𝑧)+(𝑧²−𝑎²)𝑢(𝑧) be holomorphic in the unit disk 𝑈 and then (2.1) has the generalized Hyers-Ulam stability.

Proof. In virtue of Theorem 2.1, (2.2) has a holomorphic solution in the 𝑈. According to Lemma 3.2, we have 𝑓(𝑧)<1,𝑧∈𝑈. (3.4) Let 𝜖>0 and 𝑤∈𝑈 be such that ^∞_𝑛=0𝑎_𝑛𝑤^𝑛+𝛼≤𝜖^∞_𝑛=0𝑎_𝑛^𝑝2^𝑛. (3.5) We will show that there exists a constant 𝐾 independent of 𝜖 such that 𝑤^𝑚−𝑢^𝑚≤𝜖𝐾,𝑤∈𝑈,𝑢∈𝑈 (3.6) and satisfies (3.1). We put the function 𝑓(𝑤)=−1𝜆𝑎_𝑚^∞_{𝑛=0,𝑛≠𝑚}𝑎_𝑛𝑤^𝑛+𝛼,𝑎_𝑚≠0,0<𝜆<1, (3.7) thus, for 𝑤∈𝜕𝑈, we obtain 𝑤^𝑚−𝑢^𝑚=𝑤^𝑚−𝜆𝑓(𝑤)+𝜆𝑓(𝑤)−𝑢^𝑚≤𝑤^𝑚−𝜆𝑓(𝑤)+𝜆𝑓(𝑤)−𝑢^𝑚<𝑤^𝑚−𝜆𝑓(𝑤)+𝜆𝑤^𝑚−𝑢^𝑚=𝑤^𝑚+1𝑎_𝑚^∞_{𝑛=0,𝑛≠𝑚}𝑎_𝑛𝑤^𝑛+𝛼+𝜆𝑤^𝑚−𝑢^𝑚=1𝑎_𝑚^∞_𝑛=0𝑎_𝑛𝑤^𝑛+𝛼+𝜆𝑤^𝑚−𝑢^𝑚. (3.8)
Without loss of the generality, we consider |𝑎_𝑚|=max_𝑛≥1(|𝑎_𝑛|) yielding 𝑤^𝑚−𝑢^𝑚≤1𝑎_𝑚(1−𝜆)^∞_𝑛=0𝑎_𝑛𝑤^𝑛+𝛼≤𝜖𝑎_𝑚(1−𝜆)^∞_𝑛=0𝑎_𝑛^𝑝2^𝑛≤𝜖𝑎_𝑚^𝑝−1(1−𝜆)^∞_𝑛=012^𝑛≤2𝜖𝑎_𝑚^𝑝−1(1−𝜆)∶=𝐾𝜖. (3.9) This completes the proof.

Theorem 3.4. If ^∞_{𝑛=0,𝑛≠1}𝑎_𝑛≤𝑎₁,𝑎₁≠0, (3.10) then (2.1) has a unique solution in the unit disk.

Proof. By setting 𝑓(𝑧)=−1𝑎₁^∞_{𝑛=0,𝑛≠1}𝑎_𝑛𝑧^𝑛+𝛼, (3.11) for |𝑧|≤1, we pose 𝑓(𝑧)=−1𝑎₁^∞_{𝑛=0,𝑛≠1}𝑎_𝑛𝑧^𝑛+𝛼=1𝑎₁^∞_{𝑛=0,𝑛≠1}𝑎_𝑛𝑧^𝑛+𝛼≤1𝑎₁^∞_{𝑛=0,𝑛≠1}𝑎_𝑛<1. (3.12)
Since |𝑓(𝑧)|<1 for |𝑧|=1, hence for |𝑓(𝑧)|=|−𝑧| and by Rouche’s theorem, we observe that 𝑓(𝑧)−𝑧 has exactly one zero in 𝑈, which yields that 𝑓 has a unique fixed point in 𝑈.

4. Conclusion

From above, we conclude that fractional differential equations of Bessel type have holomorphic solutions in the unit disk. The uniqueness imposed by employing the Rouche’s theorem. Furthermore, this solution satisfied the generalized Ulam stability for infinite series of fractional power.