# The Existence of Solution of Diffusion Equation with the General Conformable Derivative

Authors: Shanshan Li; Shuqin Zhang; Ruiying Liu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, the general conformable fractional derivative is used in the classical diffusion equations, and the corresponding maximum principle is obtained. By the maximum principle, this paper proves the uniqueness of the solution and the continuous dependence on source function and initial-boundary conditions of the solution. Furthermore, by employing the variable separation method, this paper obtains some existence results and the asymptotic behavior of the classical solution.


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## Body

## 1. Introduction

It has been several decades since the fractional calculus has attracted extensive attention in both pure and applied fields [1–8]. Fractional calculus has many definitions, such as the Riemann-Liouville type, Caputo type, Hadamard type, Erdélyi-Kober type, and Riesz type. Most of the above fractional derivatives are defined by their corresponding fractional integrals. Compared with some integral-order partial differential equations such as [9–19], fractional derivatives have hereditary and nonlocal property so that they are much more suitable for describing long-memory processes than the classical integer-order derivatives.However, the traditional fractional calculus have complex expressions which causes many difficulties in applying them in engineering calculation, physical application, numerical modeling, etc. In addition, the traditional fractional calculus loses some basic but important properties, such as product rule and chain rule. Khalil et al. [20] proposed a new local fractional derivative, called conformable derivative, and proved the product rule and the fractional mean value theorem. Abdeljawad supplemented the Taylor power series representation, the fractional chain rule, the Gronwall inequality, and the fractional Laplace transform in [21]. Zhao et al. [22, 23] further extended the definition to the general conformable fractional derivative (GCFD). Zhang et al. defined the conformable variable order derivative in [24]. Owing to its well-behaved properties and the close connection with the classical derivatives, conformable derivative generated great research interest [25–29].In [30, 31], the author considered a generalized time-fractional diffusion equation by replacing the first-order time derivative to the Caputo fractional derivative. Recently, some literatures about the fractional diffusion equations with various definitions of fractional calculus occur. Al-Refai et al. [32, 33] considered a nonlinear fractional diffusion equation with the Riemann-Liouville time-fractional derivative and obtained their corresponding maximum principle. Borikhanov et al. [34] considered a nonlinear time-fractional diffusion equation with the Atangana-Baleanu derivative. Chen et al. [35] used the finite element method to approximate the solutions to a time-fractional advection-diffusion equations with the Caputo variable order derivative. Zhang et al. [36] obtained the sharp blow-up and global existence of solutions, a time-fractional diffusion system with the Riemann-Liouville derivative. In this paper, we consider the same equation as in [30] with the GCFD on domain D×0,T, D⊂Rn, i.e.,
(1)Dtα,ψut=−Au+Fx,t,0<α≤1,with the nonhomogeneous initial-boundary conditions
(2)ut=0=u0x,x∈D¯,u∂D=vx,t,x,t∈∂D×0,T,where Fx,t, u0x, and vx,t are continuous functions, Dα,ψ is a GCFD of order α, ∂D is the boundary of D, and −A is a linear elliptic operator
(3)−Au≔pxΔu+∇p,∇u−qxu,x∈D¯,where Δ is the Laplace operator, ∇ is the gradient operator, p∈C1D¯, q∈CD¯, px>0, qx≥0, and the domain of definition of operator A is
(4)MA=f,f∈C2D∩C1D¯,fsatisfiestheboundaryconditionandAf∈L2D.The GCFD is defined in the following sense (see [22]):
(5)Dα,ψft=limε→0ft+εψt,α−ftε,t>0,and the GCFD at 0 is defined as Dα,ψf0=limt→0+Dα,ψft, where ψt,α is a continuous real function depending on t and the fractional order α and satisfying the below conditions
(6)ψt,α≠0,ψt,1=1,and the relationship between the function ψt,α and the order α should be one-to-one. By the definition (5), we know that the GCFD is an extension of the classical derivative α=1orψt,α=1 and the conformable derivative ψt,α=t1−α defined in [20, 21]. Compared with the definition of conformable derivative in [20, 21] and the Definition 2.5 in [24], the GCFD in relation (5) becomes the conformable variable order derivative defined in [24] if ψt,α=t1−α and the order α is a time-dependent function αt, that is
(7)Dαtft=limε→0ft+εt1−αt−ftε,t>0.If the limit (5) exists, it is called that f is α-differentiable. Furthermore, if f is differentiable, then by direct calculation of definition, we can obtain that f is α-differentiable and
(8)Dα,ψft=f′tψt,α.In this paper, we introduce the fractional Taylor power series expansion (see Lemma5) and prove the theorem of term-by-term integration and differentiation (see Lemma 7 and 8) with the general conformable fractional calculus. Then by using the above and some other properties of the general conformable factional calculus (see Section 2), we obtain the maximum principle (see Theorems 9 and 10) for the classical diffusion Equation (1) with the GCFD and get some existence (see Theorems 17 and 18) and uniqueness (see Theorem 12) results of the classical solution of (1). Finally, we get the asymptotic behavior of the classical solution (see Theorem 19). The problems (1) and (2) have a solution implies that u is α-differentiable and Dtα,ψut is continuous on 0,T. We define that ux,t is called a classical solution of problems (1) and (2), if ux,t∈CD¯×0,T∩Cx2D∩Ct10,T and satisfies Equation (1) and the initial-boundary condition (2).


## 2. Preliminaries

In this section, we will recall some properties of the GCFD and its fractional integral calculus.Definition 1 (see [22]).
The integral of a functionf:0,t⟶R of order α is defined by
(9)Iα,ψft=∫0tfτψτ,αdτ,t>0,where the integral is the Riemann integral.Remark 2.
To further study the properties of general conformable fractional calculus, we assume thatψt,α>0 and 1/ψ·,·:0,T×0,1⟶0,+∞ is Riemann integrable and this assumption is valid in the following sections.Lemma 3 (see [20–22]).
Iff,g is α-differentiable at t, then
(1)
Dα,ψc=0,wherecisanyrealconstant,
(2)
LinearityDα,ψaft+bgt=aDα,ψft+bDα,ψgt,a,b∈R,
(3)
ProductruleDα,ψftgt=ftDα,ψgt+gtDα,ψft,Dα,ψft/gt=gtDα,ψft−ftDα,ψgt/g2t,
(4)
ChainruleIffisα−differentiableatgandg≠0,then
(10)Dα,ψfgt=Dα,ψfgt·Dα,ψgt·1ψgt,α
Iff is differentiable at g, then Dα,ψfgt=f′gtDα,ψgt.Remark 4.
Especially, by the definition (5), the relation (8) and Lemma 3 (4), we have
(11)Dα,ψ∫0t1ψτ,αdτ=1,Dα,ψe∫0t1ψτ,αdτ=e∫0t1ψτ,αdτ,which coincides with the classical derivatives ψt,α=1 and the conformable fractional derivatives ψt,α=t1−α defined in [20, 21].Lemma 5 (Fractional Taylor power series expansions).
Assume thatf is an infinite α-differentiable function at a neighbourhood of t0. Then f has the fractional power series expansion at point t0:
(12)ft=∑k=0+∞Dα,ψfkt0k!∫t0t1ψτ,αdτk,t0<t<t0+R,R>0,where Dα,ψfkt0 means the application of the GCFD k times.Proof.
The proof is similar to that of Theorem17 in [21]. And this result coincides with the classical derivatives and the conformable derivatives. Especially, the exponential function e∫t0t1/ψτ,αdτ has the factional Taylor power series expansion at point t0:
(13)e∫t0t1ψτ,αdτ=∑k=0+∞1k!∫t0t1ψτ,αdτk.Lemma 6 (see [22]).
Letf be a continuous differentiable function on 0,T, then
(14)Iα,ψDα,ψft=ft−f0,Dα,ψIα,ψft=ft.Lemma 7.
Let a sequence of functionsfit, i=1,2,⋯ satisfying the following conditions:
(1)
for any givenα>0, there exists fractional integrals Iα,ψfit, i=1,2,⋯
(2)
both the series∑i=1+∞fit and ∑i=1+∞Iα,ψfit are uniformly convergent for any t∈0,T then the function ∑i=1+∞fit is α-integrable on 0,T and
(15)Iα,ψ∑i=1+∞fit=∑i=1+∞Iα,ψfit.Proof.
Due to the uniformly convergence of∑i=1+∞fit and ∑i=1+∞Iα,ψfit, then for any ε>0, there exists a positive integer N such that for any n>N and t∈0,T,
(16)∑i=1+∞fit−∑i=1nfit<ε2∫0T1/ψτ,αdτ,∑i=n+∞Iα,ψfit<ε2.
Therefore, by (16) and the linearity of the operator Iα,ψ, we obtain
(17)Iα,ψ∑i=1+∞fit−∑i=1+∞Iα,ψfit≤Iα,ψ∑i=1+∞fit−∑i=1nfit+∑i=n+1+∞Iα,ψfit≤∫0t∑i=1+∞fiτ−∑i=1nfiτψτ,αdτ+∑i=n+1+∞Iα,ψfit<ε.i.e., the function ∑i=1+∞fit is α-integrable and Iα,ψ∑i=1+∞fit=∑i=1+∞Iα,ψfit, then the proof is completed.Lemma 8.
Let a continuous differentiable sequence of functionsfit, i=1,2,⋯ satisfying the following conditions:
(1)
for any givenα>0, there exists fractional integrals Dα,ψfit,i=1,2,⋯
(2)
both the series∑i=1+∞fit and ∑i=1+∞Dα,ψfit are uniformly convergent for any t∈0,T. Moreover, ∑i=1+∞Dα,ψfit∈L0,T, then the function ∑i=1+∞fit is α-differentiable on 0,T and
(18)Dα,ψ∑i=1+∞fit=∑i=1+∞Dα,ψfit.Proof.
Since∑i=1+∞fit and ∑i=1+∞Dα,ψfit are uniformly convergent for any t∈0,T, then by Lemma 6 and Lemma 7.
(19)Iα,ψ∑i=1+∞Dα,ψfit=∑i=1+∞Iα,ψDα,ψfit=∑i=1+∞fit−∑i=1+∞fi0.
The left side of the above equality isα-differentiable; then by Lemma 3(1), ∑i=1+∞fit is α-differentiable and
(20)Dα,ψ∑i=1+∞fit=Dα,ψ∑i=1+∞fit−∑i=1+∞fi0=∑i=1+∞Dα,ψfit.


## 3. The Uniqueness of Solution

Theorem 9.
Letu be a classical solution of Equation (1) in the domain D×0,T and Fx,t≤0, then
(21)ux,t≤max0,maxx,t∈Sux,t,x,t∈D¯×0,T,where S≔D¯×0∪∂D×0,T.Proof.
The proof follows by Lemma3 (4) and setting the same auxiliary function wx,t≔ux,t+ε/2·T−t/T as [30]. Due to u is a classical solution of the problems (1) and (2), then u is differentiable with respect to t. Therefore, we have Dtα,ψut0=ψt0,αutx0,t0=0, where x0,t0 is the maximum point of u over the domain D¯×0,T.Theorem 10.
Let u be a classical solution of Equation (1) in the domain D×0,T and Fx,t≥0, x,t∈D×0,T, then
(22)ux,t≥minx,t∈Sux,t,x,t∈D¯×0,T.Theorem 11.
Letu be a classical solution of problems (1) and (2) and F∈CD¯×0,T with the norm FCD¯×0,T=supD¯×0,TFx,t≔M1.
Denote byM2=u0CD¯=supD¯u0x, M3=vC∂D×0,T=sup∂D×0,Tvx,t, then the following estimate of the solution norm holds true:
(23)uCD¯×0,T≤2CM1+maxM2,M3,where C≔∫0T1/ψτ,αdτ.Proof.
Define the auxiliary functiong:
(24)gx,t=ux,t−∫0tM1ψτ,αdτ,x,t∈D¯×0,T.
By the continuity ofψ and Remark 4, then g is a classical solution of the following problem:
(25)Dtα,ψgt=−Ag+F1x,t,x,t∈D×0,T,gt=0=u0,x∈D¯,g∂D=vx,t−∫0tM1ψτ,αdτ≔v1x,t,x,t∈∂D×0,T,where F1x,t=Fx,t−M1−qx∫0tM1/ψτ,αdτ≤0. According to the conditions, we have
(26)v1C∂D×0,T≤M3+CM1.
Then by Theorem9, the classical solution g satisfies the estimate,
(27)gx,t≤maxM2,M3+CM1,x,t∈D¯×0,T,which means that
(28)ux,t≤2CM1+maxM2,M3.
Similarly, consider of the auxiliary functiong1 and using the Theorem 10
(29)g1x,t=ux,t+∫0tM1ψτ,αdτ,x,t∈D¯×0,T,then we have
(30)g1x,t≥max−M2,−M3−CM1,x,t∈D¯×0,T,i.e., ux,t≥−2CM1−maxM2,M3. Combined with (28), we obtain
(31)uCD¯×0,+∞≤2CM1+maxM2,M3.Theorem 12.
The problems (1) and (2) have at most one classical solution and the solution continuously depends on the data given in the problem in the sense that if
(32)F1−F2CD¯×0,T≤ε1,u01−u02CD¯≤ε2,v1−v2C∂D×0,T≤ε3,then for the corresponding classical solution u1 and u2, the estimate holds:
(33)u1−u2CD¯×0,T≤2Cε1+maxε2,ε3.Proof.
Letu1, u2 be the classical solution of the following problem, respectively,
(34)Dtα,ψut=−Au+Fix,t,x,t∈D×0,T,ut=0=u0it,x∈D¯,u∂D=vix,t,x,t∈∂D×0,T,where i=1,2. Then u=u1−u2 is the classical solution of the corresponding problem
(35)Dtα,ψut=−Au+F1x,t−F2x,t,x,t∈D×0,T,ut=0=u01t−u02t,x∈D¯,u∂D=v1x,t−v2x,t,x,t∈∂D×0,T.
By (23), the estimate (33) holds. The uniqueness of the classical solution of (1) and (2) is a direct consequence of Theorem 9, Theorem 10 and the estimate (33).


## 4. The Existence of Solution

In this section, we will consider some existence results of solution of problems (1) and (2) with homogenous boundary conditions
(36)ut=0=u0x,x∈D¯,u∂D=0,x,t∈∂D×0,T.Firstly, we consider the homogenous equation, i.e.,Fx,t≡0. In the following, we seek the solution of Equation (1) by using the variable separation method in the form
(37)ux,t=XxTt,x,t∈D×0,T,which satisfies the boundary condition. Substituting expression (37) in Equation (1) and separating the variables, we obtain the following sense:
(38)Dα,ψTtTt=−AXxXx=−λ,where λ is a constant independent on t and x. Therefore, addition to the boundary condition, we have
(39)Dα,ψTt+λTt=0,and the eigenvalues problem for the operator A
(40)AXx=λXx,x,t∈D×0,T,X∂D=0,x,t∈∂D×0,T.The properties ofA is well known in [37] that A is a positive and self-adjoint linear operator. Moreover, (40) has a counted number of positive eigenvalues 0<λ1≤λ2≤⋯, λk⟶+∞k⟶+∞ with finite multiplicity, and any function f∈MA can be represented through its Fourier series:
(41)fx=∑i=1+∞f,XiXix,where Xi∈MA is the corresponding eigenfunction of the eigenvalue λi. In fact, in the following, we choose Xii=1+∞ to be real and orthonormal.For Equation (39) and its inhomogenous equation
(42)Dα,ψTt+λTt=ft,we consider the following first-order linear differential equation for λ=λi:
(43)Ti′t+λiψt,αTit=ftψt,α.It is well known that the solution of Equation (43) is given by
(44)Tit=e−∫0tλiψτ,αdτci+∫0te∫0τλiψs,αdsfτψτ,αdτ,where ci are some constants. Obviously, Tit∈C0,T∩C10,T. Moreover, Tit in (44) satisfies the following equation:
(45)ψτ,αTi′t+λiTit=ft.Hence, according to the relation (8) we have
(46)Dα,ψTit+λiTit=ψτ,αTi′t+λiTit=ft,which means the function in (44) is the solution of Equation (42). Correspondingly, the solution of Equation (39) for λ=λi is
(47)Tit=cie−∫0tλiψτ,αdτ.Naturally, each function(48)TitXix=cie−∫0tλiψτ,αdτXix,and its finite sum
(49)ukx,t=∑i=1kTitXix=∑i=1kcie−∫0tλiψτ,αdτXix,satisfy Equation (1) and the boundary condition.In addition to the initial condition, we define the formal solution of problems (1) and (36) as the following sense:
(50)ux,t=∑i=1+∞TitXix=∑i=1+∞u0,Xie−∫0tλiψτ,αdτXix.Next, we consider the case of inhomogenous equation. Similarly to the case of homogenous condition, the formal solution of inhomogenous equation also has the expression (50) with appropriate Tit. For any i, we multiply Equation (1) by Xi, integrating over D. Then by Lemma 8 and using the self-adjoint property of the operator A, combined by the orthogonality of Xii=1+∞, we obtain
(51)Dα,ψTit=∫DDtα,ψut·Xixdx=∫DDtα,ψ∑i=1+∞TitXix·Xixdx=∫D∑i=1+∞Dtα,ψTitXix·Xixdx=−Au,Xi+F,Xi=−λiu,Xi+F,Xi=−λiTi+F,Xi.That is(52)Dα,ψTit+λiTit=Fx,t,Xix.According to the relation (44), the solution of Equation (52) is
(53)Tit=e−∫0tλiψτ,αdτci+∫0te∫0τλiψs,αdsFx,τ,Xixψτ,αdτ,where ci are some constants. Consequently, combined with the initial condition, we define the formal solution of problems (1) and (36) as the following sense:
(54)ux,t=∑i=1+∞TitXix=∑i=1+∞u0,Xie−∫0tλiψτ,αdτXix+∑i=1+∞e−∫0tλiψτ,αdτ∫0te∫0τλiψs,αdsFx,τ,XixXixψτ,αdτ.Remark 13.
For Equation (52), we try applying the operator Iα,ψ to its both sides, then by Definition 1 and Lemma 6, we have
(55)Tit=Ti0−λi∫0tTiτψτ,αdτ+∫0tFx,τ,Xixψτ,αdτ.
Next, we apply the method of successive approximations to solve this integral equation by setting(56)Ti0t=Ti0,(57)Timt=Ti0t−λi∫0tTim−1τψτ,αdτ+∫0tFx,τ,Xixψτ,αdτ,m∈N+.
Taking (56) into account, then
(58)Ti1t=Ti0−λiTi0∫0t1ψτ,αdτ+∫0tFx,τ,Xixψτ,αdτ.
Similarly, using (56)–(58), we have
(59)Ti2t=Ti0−λiTi0∫0t1ψτ,αdτ+λi2Ti0∫0t∫0s1ψτ,α·1ψs,αdsdτ+∫0tFx,τ,Xixψτ,α1−λi∫τt1ψs,αdsdτ=∑k=02−λikTi0k!∫0t1ψτ,αdτk+∫0tFx,τ,Xixψτ,α∑k=01−λikk!∫τt1ψτ,αdskdτ.
Continuing this process, we obtain the following expression forTimt:
(60)Timt=∑k=0m−λikTi0k!∫0t1ψτ,αdτk+∫0tFx,τ,Xixψτ,α∑k=0m−1−λikk!∫τt1ψτ,αdskdτ.
Taking the limit asm⟶+∞, we obtain
(61)Tit=∑k=0+∞−λikTi0k!∫0t1ψτ,αdτk+∫0tFx,τ,Xixψτ,α∑k=0+∞−λikk!∫τt1ψτ,αdskdτ.
By Lemma5, the expression (61) is the fractional Taylor power series expansion of (53).Definition 14.
A formal solution of the problems (1) and ((36)) is called the Fourier series in the form (50) F≡0 and (54) F≠0.Definition 15.
Let the sequences ofu0k∈CD¯ and vk∈C∂D×0,T, Fk∈CD¯×0,T, k=1,2,⋯ such that for k⟶+∞, u0k⟶u0, vk⟶v, Fk⟹F in L2D and t∈0,T. And for any k=1,2,⋯, there exists a classical solution uk of the following problem
(62)Dtα,ψukt=−Auk+Fkx,t,ukt=0=u0kx,x∈D¯,uk∂D=vkx,t,x,t∈∂D×0,T.
Suppose there exists a functionu∈CD¯×0,T such that uk⟶u as k⟶+∞, then the function u is called a generalized solution of the problems (1) and (2).Remark 16.
By Theorem12, the convergence of u0k, vk, Fk and the completeness of CD¯×0,T,·CD¯×0,T, we know that there always exists a function u∈CD¯×0,T which is the limit of the sequence uk. In fact, under the assumption u0,F∈MA, the formal solution is also the generalized solution of problems (1) and (36). That is the following theorem.Theorem 17.
Suppose thatu0,F∈MA, then the formal solution (54) is the generalized solution of the problems (1) and (36).Proof.
For everyk=1,2,⋯,
(63)ukx,t=∑i=1ku0,Xie−∫0tλiψτ,αdτXix+∑i=1ke−∫0tλiψτ,αdτ∫0te∫0τλiψs,αdsFx,τ,XixXixψτ,αdτ,is the classical solution of the problems (1) and (36) with the initial condition
(64)u0kx=∑i=1ku0,XiXix.
Note thatu0∈MA, u0k⟶u0 as k⟶+∞ in D¯×0,T and e−∫0tλi/ψτ,αdτ≤1; therefore, ∑i=1ku0,Xie−∫0tλi/ψτ,αdτXix uniformly converges on D¯×0,T.
In view ofF∈MA, then Fx,t=∑i=1+∞Fx,t,XiXix uniformly converges on D¯×0,T. Therefore,
(65)∑i=1ke−∫0tλiψτ,αdτ∫0te∫0τλiψs,αdsFx,τ,XixXixψτ,αdτ≤∫0t1ψτ,α·∑i=1kFx,τ,XiXixdτ⟶∫0t1ψτ,α·∑i=1+∞Fx,τ,XiXixdτask⟶+∞.
Consequently, (54) is the generalized solution of the problems (1) and (36).Theorem 18.
Suppose that u0,F∈MA, Au0,A2u0∈MAand for any t∈0,T, AF,A2F∈MA, then the formal solution (54) is the classical solution of the problems (1) and (36).Proof.
By Theorem17, the formal solution (54) is the generalized solution; it remains to prove that (54) is at least twice differentiable with respect to the spatial variable x and differentiable with respect to the time variable t.
Differentiating term-by-term (54) with respect to x, we construct a series
(66)∑i=1+∞u0,Xie−∫0tλiψτ,αdτ∇Xix+∑i=1+∞e−∫0tλiψτ,αdτ∫0te∫0τλiψs,αdsFx,τ,Xix∇Xixψτ,αdτ.
Now we shall prove that the above series (66) is uniformly convergent. Since u0∈MA, F∈MA, then Au0,AF∈L2D. Moreover, Au0,AF∈MA, so Au0 and AF can be expanded into uniformly convergent series
(67)Au0=∑i=1+∞Au0,XiXi,AF=∑i=1+∞AF,XiXi,for any t∈0,T. Then by the Parseval equality,
(68)Au022=∑i=1+∞Au0,Xi2=∑i=1+∞u0,AXi2=∑i=1+∞λi2u0,Xi2<+∞,(69)AF22=∑i=1+∞AF,Xi2=∑i=1+∞F,AXi2=∑i=1+∞λi2F,Xi2<+∞.
Considering that the series∑i=1+∞∇Xi2/λi2 uniformly convergent (see [38]), then applying the Cauchy inequality we have
(70)∑i=1+∞Fx,t,Xix∇Xix≤∑i=1+∞λiFx,t,Xix·∇Xixλi≤∑i=1+∞λi2Fx,t,Xix21/2·∑i=1+∞∇Xix2λi21/2≤AF2∑i=1+∞∇Xix2λi21/2,which implies that the series ∑i=1+∞Fx,t,Xix∇Xix is uniformly convergent on D for any t∈0,T. In view that Xii=1+∞ is a complete orthonormal system in L2D, using the above facts, the estimates (68)–(70) and applying the Cauchy inequality, we get
(71)∑i=1+∞u0,Xie−∫0tλiψτ,αdτ∇Xix+∑i=1+∞e−∫0tλiψτ,αdτ∫0te∫0τλiψs,αdsFx,τ,Xix∇Xixψτ,αdτ≤∑i=1+∞u0,Xi∇Xix+∑i=1+∞∫0tFx,τ,Xix∇Xixψτ,αdτ≤∑i=1+∞λiu0,Xi·∇Xixλi+∫0t1ψτ,α∑i=1+∞λiFx,τ,Xix·∇Xixλidτ≤∑i=1+∞λi2u0,Xi21/2·∑i=1+∞∇Xix2λi21/2+∫0t1ψτ,α∑i=1+∞λi2F,Xi21/2·∑i=1+∞∇Xix2λi21/2dτ≤Au02+CAF2∑i=1+∞∇Xix2λi21/2,where C=∫0T1/ψτ,αdτ. Consequently, the series (66) uniformly converges to ∇u and the generalized solution (54) belongs to Cx1D.
Applying the similarly method, we can show the generalized solution (54) belongs to Cx2D as well and the relation
(72)Δu=∑i=1+∞u0,Xie−∫0tλiψτ,αdτΔXix+∑i=1+∞e−∫0tλiψτ,αdτ∫0te∫0τλiψs,αdsFx,τ,XixΔXixψτ,αdτholds true. In fact, since Au0∈MA, AF∈MA, then A2u0∈L2D, A2F∈L2D. Moreover, due to A2u0,A2F∈MA, so A2u0 and A2F can be expanded into uniformly convergent series for any t∈0,T. Note that the series ∑i=1+∞ΔXi2/λi3 is uniformly convergent (see [38]), then by the Cauchy inequality, we have
(73)∑i=1+∞Fx,t,XixΔXix≤∑i=1+∞λi32Fx,t,Xix·ΔXixλi3/2≤l1∑i=1+∞λi4Fx,t,Xix21/2·∑i=1+∞ΔXix2λi31/2≤l1A2F2∑i=1+∞ΔXix2λi31/2.
Due tolimi→+∞λi=+∞, there exists a constant l such that λi3<lλi4, i=1,2,⋯, where l1 is a constant depending on l. Then (73) implies that the series ∑i=1+∞Fx,t,XixΔXix is uniformly convergent on D for any t∈0,T. Using the above facts and applying the Parseval equality and Cauchy inequality, we get
(74)∑i=1+∞u0,Xie−∫0tλiψτ,αdτΔXix+∑i=1+∞e−∫0tλiψτ,αdτ∫0te∫0τλiψs,αdsFx,τ,XixΔXixψτ,αdτ≤∑i=1+∞u0,XiΔXix+∑i=1+∞∫0tFx,τ,XixΔXixψτ,αdτ≤∑i=1+∞λi32u0,Xi·ΔXixλi3/2+∫0t1ψτ,α∑i=1+∞λi32Fx,τ,Xix·ΔXixλi3/2dτ≤l1∑i=1+∞λi4u0,Xi21/2·∑i=1+∞ΔXix2λi31/2+∫0tl1ψτ,α∑i=1+∞λi4F,Xi21/2·∑i=1+∞ΔXix2λi31/2dτ≤l1A2u02+Cl1A2F2∑i=1+∞ΔXix2λi31/2.
Consequently, the relation (72) holds true and the generalized solution (54) belongs to Cx2D. It remains to prove that (54) belongs to Ct10,T. We take a derivative term by term with respect to t (54) and construct the series
(75)−∑i=1+∞λiu0,Xiψt,αe−∫0tλiψτ,αdτXix−∑i=1+∞λie−∫0tλi/ψτ,αdτψt,α∫0te∫0τλiψs,αdsFx,τ,XixXixψτ,αdτ+∑i=1+∞Fx,τ,XixXixψt,α.
Now lett∈ε,T, 0<ε<T. Note that 1/ψt,α is the Riemann integrable on ε,T; there exists M>0 such that 1/ψt,α≤M. Next we shall to prove that the above series (75) is uniformly convergent on any interval ε,T, 0<ε<T. By the uniformly convergence of ∑i=1+∞Xi2/λi (see [38]), applying the Parseval equality and the Cauchy inequality, we get
(76)∑i=1+∞λiu0,Xiψt,αe−∫0tλiψτ,αdτXix+∑i=1+∞Fx,τ,XixXixψt,α+∑i=1+∞λie−∫0tλi/ψτ,αdτψt,α∫0te∫0τλiψs,αdsFx,τ,XixXixψτ,αdτ≤∑i=1+∞Mλiu0,XiXix+∑i=1+∞MFx,τ,XixXix+∑i=1+∞∫0tMλi∣Fx,τ,XixXix∣ψτ,αdτ≤∑i=1+∞Mλi3/2u0,Xi·Xixλi1/2+∑i=1+∞Mλi1/2F,Xi·Xixλi1/2+∫0tMψτ,α∑i=1+∞λi3/2Fx,τ,Xix·Xixλi1/2dτ≤M1∑i=1+∞λi4u0,Xi21/2·∑i=1+∞Xix2λi1/2+M2∑i=1+∞λi2F,Xi21/2·∑i=1+∞Xix2λi1/2+∫0tM1ψτ,α∑i=1+∞λi4Fx,τ,Xi21/2·∑i=1+∞Xix2λi1/2dτ≤M1A2u02+M2AF2+CM1A2F2·∑i=1+∞Xix2λi1/2.
Once again, due tolimi→+∞λi=+∞, there exists a constant l such that λi3<lλi4, λi<lλi2i=1,2,⋯, where M1,M2 are some constants depending on l and M. Then, the series (75) uniformly convergent on any interval ε,T, 0<ε<T. Therefore, the generalized solution (54) belongs to Ct10,T and the proof is complete.Theorem 19.
The classical solution (50) of (1) and (36) with Fx,t≡0 decay to zero with the rate e−∫0tλ1/ψs,αds.Proof.
In fact, sinceu0∈MA, then ∑i=1+∞u0,XiXix=∣u0∣<+∞. Then for any t>0 and x∈D¯, we have
(77)∑i=1+∞u0,Xie−∫0tλiψτ,αdτXix≤e−∫0tλ1ψs,αds∑i=1+∞u0,Xie−∫0tλi−λ1ψτ,αdτXix≤∣u0∣e−∫0tλ1ψs,αds⟶0ast⟶+∞.


## 5. Conclusions

The GCFD is an extension of the classical derivatives and the conformable derivatives. In this paper, we introduce its fractional Taylor power series expansions and prove the theorem of term-by-term integration and differentiation. Utilizing its properties, we obtain the uniqueness and existence of the classical solution of the classical diffusion equations with GCFD by using variable separation method.



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*Source: 3965269-2020-05-11.xml*