Hyers-Ulam Stability and Existence of Solutions for Nigmatullin ’ s Fractional Diffusion Equation

Motivated by a famous question of Ulam concerning the stability of group homomorphisms, Hyers and Rassias introduced the concepts of Hyers-Ulam and Hyers-Ulam-Rassias stability, respectively, in the case of the Cauchy functional equation in Banach spaces, which received a great influence in the development of the generalized Hyers-Ulam-Rassias stability of all kinds of functional equations. There are many interesting results on this topic in the case of functional equations, ordinary differential equations, partial differential equations, and impulsive differential equations; see, for example, [1–11] and the recent survey [12, 13]. Recently, Hegyi and Jung [14] presented the generalized Hyers-Ulam-Rassias stability of the classical Laplace’s equation Δu = 0 in the class of spherically symmetric functions via harmonic functions method. Meanwhile, the same topic of fractional evolution equations via functional analysis methods has attracted attention of researchers. However, to the best of our knowledge, stability of fractional partial differential equations via direct analysis methods has not been discussed yet. In this paper, we study the stability of Nigmatullin’s timefractional order diffusion equation (see [15, Chapter 6]) D0+,tu (x, t) = λ2 ∂ 2u (x, t) ∂x2 , x ∈ R, t > 0, λ > 0, LD 0+,tu (x, 0) = f (x) , lim t→±∞u (x, t) = 0, (1)


Introduction
Motivated by a famous question of Ulam concerning the stability of group homomorphisms, Hyers and Rassias introduced the concepts of Hyers-Ulam and Hyers-Ulam-Rassias stability, respectively, in the case of the Cauchy functional equation in Banach spaces, which received a great influence in the development of the generalized Hyers-Ulam-Rassias stability of all kinds of functional equations.There are many interesting results on this topic in the case of functional equations, ordinary differential equations, partial differential equations, and impulsive differential equations; see, for example, [1][2][3][4][5][6][7][8][9][10][11] and the recent survey [12,13].
Recently, Hegyi and Jung [14] presented the generalized Hyers-Ulam-Rassias stability of the classical Laplace's equation Δ = 0 in the class of spherically symmetric functions via harmonic functions method.Meanwhile, the same topic of fractional evolution equations via functional analysis methods has attracted attention of researchers.However, to the best of our knowledge, stability of fractional partial differential equations via direct analysis methods has not been discussed yet.
In this paper, we study the stability of Nigmatullin's timefractional order diffusion equation (see [15, and existence of solution to nonlinear problem where  is a continuous function on R and  will be assumed to satisfy certain conditions and the symbol    0+, denotes the Riemann-Liouville time-fractional derivatives of the order  ∈ (0, 1] (see [15,Chapter 6, p.349, (6.1.12)]) and [] and {} denote the integral and fractional parts of  and Γ(⋅) is the Euler-Gamma function.
The solvability of (1) has been reported in [15, Chapter 6.2.1].Here we collect the following result.
Lemma 1 (see [15,Corollary 6.1] or [16, (4.19)]).Equation ( 1) is solvable, and its solution has the form where fractional Green function   (, ) involving Wright function (, ; ) is given by provided that the integral in the right-hand side of ( 15) is convergent, where   denotes the Hankel path of integration in the complex -plane.

Fractional Duhamel's Principle, Stability Concepts, and Remarks
The standard Duhamel principle adopts the idea form ODEs in studying Cauchy problem for inhomogeneous partial differential equations by linking Cauchy problem for corresponding homogeneous equation.In this section, we establish a fractional Duhamel principle which helps us to study Ulam's stability of ( 1) and existence of solution to (2).Lemma 6.Let (, ) be jointly continuous on R × (0, +∞).

The solution of Cauchy problem for inhomogeneous partial differential equations of the type
is Proof.Using the superposition principle, the following Cauchy problem ( 16) can be decomposed into two Cauchy problems: By Lemma 1, the solution of ( 18) is By virtue of homogeneous theorem the solution of ( 19) can be written as where  = (, ; ) is the solution of By virtue of homogeneous theorem, Lemma 1, and (21) we obtain (, ) =  1 (, ) +  2 (, ) which is the desired result.

Hyers-Ulam Stability
In this section, we present the stability results.(30) Then ( 1) is Hyers-Ulam stable on finite time interval [0, ] with respect to  and .
Proof.Let (, ) be a solution of inequality (23) and (, ) the solution of Cauchy problem (1), and its expression is where we use (13) in Remark 4.
Proof.Let (, ) be a solution of inequality (24) and (, ) the solution of Cauchy problem (1), and its expression is Keeping in mind (29), we have Hence, we have which implies that (1) is Hyers-Ulam-Rassias stable with respect to (, ) fl  +1 and  fl /( + 1)Γ().The proof is completed.(40) which is reasonable due to the fact that (see [15, (6.3.7)]) where the symbols F  and L  denote the Fourier transform and Laplace transform, respectively.