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We discuss stability of time-fractional order heat conduction equations and prove the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability of time-fractional order heat conduction equations via fractional Green function involving Wright function. In addition, an interesting existence result for solution is given.

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, [

Recently, Hegyi and Jung [

In this paper, we study the stability of Nigmatullin’s time-fractional order diffusion equation (see [

The two parameter Mittag-Leffler functions

The solvability of (

Equation (

Note that [

For any

Obviously, we have the following remarks.

Obviously, (

If

If

Note that [

Let

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 (

Let

Using the superposition principle, the following Cauchy problem (

By Lemma

By virtue of homogeneous theorem the solution of (

By virtue of homogeneous theorem, Lemma

Let

Consider (

Now we are ready to introduce the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability concepts for (

Equation (

A function

If

By Remark

Let

Equation (

A function

If

In this section, we present the stability results.

Assume that there exists

Let

Set

Assume that there exists

Let

Condition (

Note that, for

Set

Let

We introduce the following assumptions:

[H1]:

[H2]: For any

Define

Assume that [H1] and [H2] hold. Then (

Define

Note that

Next, we show that

Now we check that

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Unite Foundation of Guizhou Province (