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

Authors: Zhuoyan Gao; JinRong Wang
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

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.


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## Body

## 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, [1–11] 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 time-fractional order diffusion equation (see [15, Chapter 6])(1)DL0+,tαux,t=λ2∂2ux,t∂x2,x∈R,t>0,λ>0,DL0+,tα-1ux,0=fx,limt→±∞⁡ux,t=0,and existence of solution to nonlinear problem(2)DL0+,tαux,t=λ2∂2ux,t∂x2+qx,t,ux,t,x∈R,t>0,λ>0,DL0+,tα-1ux,0=fx,limt→±∞ux,t=0,where f is a continuous function on R and q will be assumed to satisfy certain conditions and the symbol DL0+,tα denotes the Riemann-Liouville time-fractional derivatives of the order α∈0,1 (see [15, Chapter 6, p.349, (6.1.12)])(3)DL0+,tαux,t=∂∂tα+11Γ1-α∫0tux,st-sαds,x∈R,t>0,and α and α denote the integral and fractional parts of α and Γ· is the Euler-Gamma function.


## 2. Preliminaries

The two parameter Mittag-Leffler functionsEα,βz are defined Eα,βz=∑i=0∞zi/Γiα+β, z∈R, and α, β are positive real numbers. Next, Eαz=Eα,1z.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(4)ux,t=∫-∞+∞Gαx-ξ,tfξdξ,where fractional Green function Gαx,t involving Wright function ϕα,β;z is given by (5)Gαx,t=12λtα/2-1ϕ-α2,α2;-xλt-α/2=1π∫0∞tα-1Eα,α-λ2y2tαcosyxdy=12π∫-∞+∞eiyxtα-1Eα,α-λ2y2tαdy,ϕα,β;z=∑k=0∞zkk!Γαk+β=12πi∫Has-βes+zs-αds,provided that the integral in the right-hand side of (15) is convergent, where Ha denotes the Hankel path of integration in the complex s-plane.Note that [17, Lemma  2] for any λ>0 and t>0, Eα,α-tαλ≤1/Γα. Then, we have the following estimation.Lemma 2.
For anyt>0, x∈R, Gαx,t≤tα/Γα, and(6)∫-∞+∞Gαx-ξ,tfξdξ≤MftαπΓα,where(7)∫-∞+∞fξdξ≔Mf.Obviously, we have the following remarks.Remark 3.
Obviously, (7) can be fulfilled. For example, fξ=e-ξ-θ2, θ∈a,b. Moreover, we can obtain (8)Gαx,t≤tαΓαδx,δx=12π∫-∞+∞eiyxdy.Remark 4.
Ifα=1, then (1) becomes a classical heat conduction equation(9)∂∂tux,t=λ2∂2ux,t∂x2,x∈R,t>0,ux,0=fx,and its solution is given by(10)ux,t=∫-∞+∞G1x-ξ,tfξdξ,where(11)G1x,t=12λπt-1/2e-x2/4λ2t,ϕ-12,12;-xλt-1/2=e-x2/4λ2t.
Iffx≤M, then (4) satisfies the following inequality: (12)ux,t≤M,where we use the fact that(13)∫-∞+∞G1x-ξ,tdξ=∫-∞+∞12λπt-1/2e-x-ξ2/4λ2tdξ=∫-∞+∞1πe-ζ2dζ=1,ζ=ξ-x2λt.Note that [18, Lemma  3], for all t>0, tα-1Eα,α-ztα≤m~α,z/tα+1, where m~α,λ=∫0∞r1/αe-r1/αdr/sin⁡παπαz2. For more asymptotic expansions on Mittag-Leffler functions, one can refer to [19, Lemmas  2.2, 2.3, 2.4]. Next, we give asymptotic property of Gαx,t.Lemma 5.
Letmα,λ=∫0∞r1/αe-r1/αdr/sin⁡παπαλ4. Then(14)Gαx,t≤mα,λπt1+α∫0tcos⁡yxy4dy.(15)ux,t=∫-∞+∞Gαx-ξ,tfξdξ≤mα,λπt1+α∫-∞+∞∫0tcos⁡yx-ξy4dyfξdξ.


## 3. 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.
LetFx,t be jointly continuous on R×0,+∞. The solution of Cauchy problem for inhomogeneous partial differential equations of the type(16)DL0,tαyx,t=λ2∂2yx,t∂x2+Fx,t,x∈R,t>0,DL0,tα-1yx,0=fx,is(17)yx,t=∫-∞+∞Gαx-ξ,tfξdξ+∫0t∫-∞+∞Gαx-ξ,t-τFξ,τdξdτ.Proof.
Using the superposition principle, the following Cauchy problem (16) can be decomposed into two Cauchy problems:(18)DL0,tαy1x,t=λ2∂2y1x,t∂x2,x∈R,t>0,DL0,tα-1y1x,0=fx,(19)DL0,tαy2x,t=λ2∂2y2x,t∂x2+Fx,t,x∈R,t>0,DL0,tα-1y2x,0=0.
By Lemma1, the solution of (18) is (20)y1x,t=∫-∞+∞Gαx-ξ,tfξdξ.
By virtue of homogeneous theorem the solution of (19) can be written as(21)y2x,t=∫0tϖx,t;τdτ,where ϖ=ϖx,t;τ is the solution of(22)DL0,tαϖx,t=λ2∂2ϖx,t∂x2,x∈R,t>τ,DL0,tα-1ϖx,τ=Fx,τ.
By virtue of homogeneous theorem, Lemma1, and (21) we obtain yx,t=y1x,t+y2x,t which is the desired result.Letϵ>0 and φ~t:R→0,+∞ be a nonnegative and increasing function.Consider (1) and the following two inequalities:(23)DL0+,tαyx,t-λ2∂2yx,t∂x2≤ε,x∈R,t>0,λ>0,0<α≤1,(24)DL0+,tαyx,t-λ2∂2yx,t∂x2≤εφ~t,x∈R,t>0,λ>0,0<α≤1.Now we are ready to introduce the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability concepts for (1).Definition 7.
Equation (1) is Hyers-Ulam stable if there exists a number c>0 such that for each solution y of inequality (23) there exists a solution u of (1) with (25)yx,t-ux,t≤cε,x∈R,t>0.Remark 8.
A functionyx,t is a solution of inequality (23) if and only if there is Fx,t such that(i)
Fx,t≤ε, x∈R, t>0;(ii)
DL0+,tαyx,t=λ2∂2yx,t/∂x2+Fx,t.Remark 9.
Ifyx,t is a solution of inequality (23), then y is a solution of the following integral inequality:(26)yx,t-∫-∞+∞Gαx-ξ,tfξdξ≤ε∫0t∫-∞+∞Gαx-ξ,t-τdξdτ.By Remark8 and (17), we get (27)yx,t-∫-∞+∞Gαx-ξ,tfξdξ≤∫0t∫-∞+∞Gαx-ξ,t-τFξ,τdξdτ≤ε∫0t∫-∞+∞t-ταπΓαFξ,τdξdτ.Letφx,t:R×0,+∞→0,+∞ be a nonnegative function.Definition 10.
Equation (1) is generalized Hyers-Ulam-Rassias stable if there exists a number c>0 such that for each solution y of inequality (24) there exists a solution u of (1) with (28)yx,t-ux,t≤cφx,t,x∈R,t>0.Remark 11.
A functionyx,t is a solution of inequality (24) if and only if there is Fx,t such that(i)
Fx,t≤φ~t, x∈R, t>0;(ii)
DL0+,tαyx,t=λ2∂2yx,t/∂x2+Fx,t.Remark 12.
Ifyx,t is a solution of inequality (24), then y is a solution of the following integral inequality:(29)yx,t-∫-∞+∞Gαx-ξ,tfξdξ≤∫0t∫-∞+∞Gαx-ξ,t-τφ~ξdξdτ≤ε∫0t∫-∞+∞Mt-ταπΓαφ~ξdξdτ.


## 4. Hyers-Ulam Stability

In this section, we present the stability results.Theorem 13.
Assume that there existscλ,α>0 such that(30)supx,t∈R×0,+∞⁡∫0t∫-∞+∞Gαx-ξ,t-τdξdτ=c<+∞.Then (1) is Hyers-Ulam stable on finite time interval 0,T with respect to ε and c.Proof.
Letyx,t be a solution of inequality (23) and ux,t the solution of Cauchy problem (1), and its expression is (31)ux,t=∫-∞+∞Gαx-ξ,tfξdξ. Keeping in mind (26), we have (32)yx,t-∫-∞+∞Gαx-ξ,tfξdξ≤ε∫0t∫-∞+∞Gαx-ξ,t-τdξdτ. By (30) we obtain (33)yx,t-ux,t=yx,t-∫-∞+∞Gαx-ξ,tfξdξ≤ε∫0t∫-∞+∞Gαx-ξ,t-τdξdτ=cε,which implies that (1) is Hyers-Ulam stable with respect to ε and c. The proof is completed.Example 14.
Setε>0, α=1, x∈0,l, and t∈0,T. Then the following heat conduction equation(34)∂∂tux,t=λ2∂2ux,t∂x2,x∈0,l,t∈0,T, is Ulam-Hyers stable with respect to ε and (35)cλ,1=supx,t∈0,l×0,T⁡∫0t∫-∞+∞G1x-ξ,t-τdξdτ=T, where we use (13) in Remark 4.Theorem 15.
Assume that there existsm>0 such that(36)∫-∞+∞φ~ξdξdτ≤m,for  eachx∈R,t>0.Then (1) is generalized Hyers-Ulam-Rassias stable with respect to φ(t) and c.Proof.
Letyx,t be a solution of inequality (24) and ux,t the solution of Cauchy problem (1), and its expression is (37)ux,t=∫-∞+∞Gαx-ξ,tfξdξ.Keeping in mind (29), we have (38)yx,t-∫-∞+∞Gαx-ξ,tfξdξ≤∫0t∫-∞+∞Gαx-ξ,t-τφ~τdξdτ.Hence, we have (39)yx,t-ux,t≤∫0t∫-∞+∞t-ταπΓαφ~τdξdτ≤mtα+1πα+1Γα≔cφx,t,which implies that (1) is Hyers-Ulam-Rassias stable with respect to φx,t≔tα+1 and c≔m/πα+1Γα. The proof is completed.Remark 16.
Condition (36) can be changed to (40)∫0t∫-∞+∞Gαx-ξ,t-τφ~τdξdτ≤cφx,t,for  eachx∈R,t>0.which is reasonable due to the fact that (see [15, (6.3.7)])(41)FxLtGαx,t=tαtα+λ2x2Fxφ~x,x∈R,t>0,where the symbols Fx and Lt denote the Fourier transform and Laplace transform, respectively.
Note that, forφ~t=tv, v≠0, one has (see [15, (1.3.55)]) (42)Fxφ~x=2v+1πΓv+1/2Γ-v/2x-v-1=-2sin⁡vπ2Γv+1x-v-1.Then we can choose φx,t=x-v-1tα/tα+λ2x2 and c=2Γv+1.Example 17.
Setα=1 and φ~t=t2. Then the following heat conduction equation(43)∂∂tux,t=λ2∂2ux,t∂x2,x∈R,t>0,is Hyers-Ulam-Rassias stable with respect to φx,t=x-3t2/t2+λ2x2 and c=2, where we use (13) in Remark 4 again.


## 5. Existence of Solution

LetB be the Banach space of all bounded continuous function ux,t on R with fixed t equipped with the norm u·,tB=supx∈R⁡ux,t. Consider the Banach space B~=C0,t0,B of all bounded continuous function defined on 0,T equipped with the norm u·,tB~=supt∈0,T⁡u·,t. Next, we introduce Banach space B~α=u:t1-αu·,t∈C0,t0,B equipped with the norm uB~α=supt∈0,T⁡t1-αu·,t.We introduce the following assumptions:[H1]:qx,t,ux,t is jointly continuous on R×0,T×R with qx,t,0=0.[H2]: For anyM>0, there exists a constant L>0 such that(44)q·,t,u·,t-q·,t,v·,tB≤Lu·,t-v·,tBfor all t∈0,T and all u,v∈B with u·,tB≤M and v·,tB≤M.Define(45)Υ=u∈B~α:u·,·-∫-∞+∞Gα·-ξ,·fξdξB~α≤fB,t∈0,tm,where tm=min⁡Γα+1/2L,Γα+1/2L1/α.Theorem 18.
Assume that [H1] and [H2] hold. Then (2) has unique solution in Υ and uB~α≤1+1/ΓαfB.Proof.
DefineQ:Υ→B~α given by(46)Qux,t=∫-∞+∞Gαx-ξ,tfξdξ+∫0t∫-∞+∞Gαx-ξ,t-τqξ,τ,uξ,τdξdτ.
Note that(47)∫-∞+∞Gα·-ξ,tfξdξB≤∫-∞+∞tαΓαδ·-ξfξBdξ=tαΓαfξB. Multiplying t1-α to the above inequality and taking the supremum, we have(48)∫-∞+∞Gα·-ξ,·fξdξB~α≤1ΓαfB~. Observing the definition of Υ and triangular inequality, one has (49)uB~α≤1+1ΓαfB.
Next, we show thatQ:Υ→Υ. By [H2] with qx,t,0=0, (50)Qu·,t-∫-∞+∞Gα·-ξ,tfξdξB≤∫0tt-τα-1Γαq·,τ,u·,τBdτ≤LΓα∫0tt-τα-1dτu·,ξB=LtαΓα+1u·,ξB.Multiplying t1-α to the above inequality and taking the supremum, we have (51)Qu-∫-∞+∞Gα·-ξ,·fξdξB~α≤LtΓα+1u·,·B~≤12u·,·B~<u·,·B~.
Now we check thatQ is contraction. Observe that (52)Qu·,t-Qv·,tB≤∫0tLt-τα-1Γαu·,τ-v·,τBdτ≤LtαΓα+1u·,t-v·,tB.Multiplying t1-α to the above inequality and taking the supremum, we have (53)Qu-QvB~α≤LtαΓα+1u·,·-v·,·B~α≤12u·,·-v·,·B~α.By contraction mapping principle, Q has a unique fixed point in Υ which is the solution of (2).



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