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

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.

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][26][27][28][29].
In [30,31], the author considered a generalized timefractional 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 × ð0, TÞ, D ⊂ R n , i.e., with the nonhomogeneous initial-boundary conditions where Fðx, tÞ, u 0 ðxÞ, and vðx, tÞ are continuous functions, D α,ψ is a GCFD of order α, ∂D is the boundary of D, and −A is a linear elliptic operator where Δ is the Laplace operator, ∇ is the gradient operator, p ∈ C 1 ð DÞ, q ∈ Cð DÞ, pðxÞ > 0, qðxÞ ≥ 0, and the domain of definition of operator A is The GCFD is defined in the following sense (see [22]): and the GCFD at 0 is defined as ðD α,ψ f Þð0Þ = lim where ψðt, αÞ is a continuous real function depending on t and the fractional order α and satisfying the below conditions and the relationship between the function ψðt, αÞ 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 ðα = 1 or ψðt, αÞ = 1Þ and the conformable derivative ðψðt, αÞ = t 1−α Þ 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 ψðt, αÞ = t 1−α and the order α is a time-dependent function αðtÞ, that is 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 In this paper, we introduce the fractional Taylor power series expansion (see Lemma 5) 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 ðD α,ψ t uÞðtÞ is continuous on ð0, TÞ. We define that uðx, tÞ is called a classical solution of problems (1) and (2), if uðx, tÞ ∈ Cð D × ½0, TÞ ∩ C 2 x ðDÞ ∩ C 1 t ð0, TÞ and satisfies Equation (1) and the initial-boundary condition (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 function f : ð0, tÞ ⟶ R of order α is defined by where the integral is the Riemann integral.

Lemma 5 (Fractional Taylor power series expansions).
Assume that f is an infinite α-differentiable function at a neighbourhood of t 0 . Then f has the fractional power series expansion at point t 0 : where ðD α,ψ f Þ ðkÞ ðt 0 Þ means the application of the GCFD k times.
Proof. The proof is similar to that of Theorem 17 in [21]. And this result coincides with the classical derivatives and the conformable derivatives. Especially, the exponential function has the factional Taylor power series expansion at point t 0 : Lemma 6 (see [22]). Let f be a continuous differentiable function on ð0, TÞ, then Lemma 7. Let a sequence of functions f i ðtÞ, i = 1, 2, ⋯ satisfying the following conditions: (1) for any given αð>0Þ, there exists fractional integrals Proof. Due to the uniformly convergence of ∑ +∞ i=1 f i ðtÞ and ∑ +∞ i=1 ðI α,ψ f i ÞðtÞ, then for any ε > 0, there exists a positive integer N such that for any n > N and t ∈ ð0, TÞ, Therefore, by (16) and the linearity of the operator I α,ψ , we obtain i.e., the function (1) for any given αð>0Þ, there exists fractional integrals Proof. Since ∑ +∞ i=1 f i ðtÞ and ∑ +∞ i=1 ðD α,ψ f i ÞðtÞ are uniformly convergent for any t ∈ ½0, T, then by Lemma 6 and Lemma 7.
Proof. Define the auxiliary function g: By the continuity of ψ and Remark 4, then g is a classical solution of the following problem: where Then by Theorem 9, the classical solution g satisfies the estimate, which means that Similarly, consider of the auxiliary function g 1 and using the Theorem 10 then we have i.e., uðx, tÞ≥−2CM 1 − max fM 2 , M 3 g: Combined with (28), we obtain 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 then for the corresponding classical solution u 1 and u 2 , the estimate holds: Proof. Let u 1 , u 2 be the classical solution of the following problem, respectively, where i = 1, 2. Then u = u 1 − u 2 is the classical solution of the corresponding problem 4 Journal of Function Spaces 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).

The Existence of Solution
In this section, we will consider some existence results of solution of problems (1) and (2) with homogenous boundary conditions Firstly, we consider the homogenous equation, i.e., Fðx, tÞ ≡ 0. In the following, we seek the solution of Equation (1) by using the variable separation method in the form which satisfies the boundary condition. Substituting expression (37) in Equation (1) and separating the variables, we obtain the following sense: where λ is a constant independent on t and x. Therefore, addition to the boundary condition, we have and the eigenvalues problem for the operator A The properties of A 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 < λ 1 ≤ λ 2 ≤⋯, λ k ⟶+∞ðk⟶+∞Þ with finite multiplicity, and any function f ∈ M A can be represented through its Fourier series: where X i ∈ M A is the corresponding eigenfunction of the eigenvalue λ i . In fact, in the following, we choose fX i g +∞ i=1 to be real and orthonormal.

For Equation (39) and its inhomogenous equation
we consider the following first-order linear differential equation for λ = λ i : It is well known that the solution of Equation (43) is given by where c i are some constants. Obviously, T i ðtÞ ∈ C½0, T ∩ C 1 ð0, TÞ. Moreover, T i ðtÞ in (44) satisfies the following equation: Hence, according to the relation (8) we have which means the function in (44) is the solution of Equation (42). Correspondingly, the solution of Equation (39) for λ = λ i is Naturally, each function and its finite sum 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: 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 T i ðtÞ. For any i, we multiply Equation (1) by X i , integrating over D. Then by Lemma 8 and using the 5 Journal of Function Spaces self-adjoint property of the operator A, combined by the orthogonality of fX i g +∞ i=1 , we obtain That is According to the relation (44), the solution of Equation (52) is where c i are some constants. Consequently, combined with the initial condition, we define the formal solution of problems (1) and (36) as the following sense: Remark 13. For Equation (52), we try applying the operator I α,ψ to its both sides, then by Definition 1 and Lemma 6, we have Next, we apply the method of successive approximations to solve this integral equation by setting Taking (56) into account, then Similarly, using (56)-(58), we have Continuing this process, we obtain the following expression for T ðmÞ i ðtÞ: Taking the limit as m ⟶ +∞, we obtain By Lemma 5, the expression (61) is the fractional Taylor power series expansion of (53).
Definition 15. Let the sequences of u 0k ∈ Cð DÞ and v k ∈ C ð∂D × ½0, TÞ, F k ∈ Cð D × ½0, TÞ, k = 1, 2, ⋯ such that for k ⟶ +∞, u 0k ⟶ u 0 , v k ⟶ v, F k ⟹ F in L 2 ðDÞ and t ∈ ½0, T. And for any k = 1, 2, ⋯, there exists a classical solution u k of the following problem Journal of Function Spaces Suppose there exists a function u ∈ Cð D × ½0, TÞ such that u k ⟶ u as k ⟶ +∞, then the function u is called a generalized solution of the problems (1) and (2).
Remark 16. By Theorem 12, the convergence of u 0k , v k , F k and the completeness of Cð D × ½0, T, k·k Cð D×½0,TÞ Þ, we know that there always exists a function u ∈ Cð D × ½0, TÞ which is the limit of the sequence u k . In fact, under the assumption u 0 , F ∈ M A , the formal solution is also the generalized solution of problems (1) and (36). That is the following theorem. Proof. For every k = 1, 2, ⋯, is the classical solution of the problems (1) and (36) with the initial condition Consequently, (54) is the generalized solution of the problems (1) and (36).  (1) and (36).
Proof. By Theorem 17, 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 Now we shall prove that the above series (66) is uniformly convergent. Since u 0 ∈ M A , F ∈ M A , then Aðu 0 Þ, AðFÞ ∈ L 2 ðDÞ. Moreover, Aðu 0 Þ, AðFÞ ∈ M A , so Aðu 0 Þ and AðFÞ can be expanded into uniformly convergent series for any t ∈ ð0, TÞ. Then by the Parseval equality, Considering that the series ∑ +∞ i=1 ðj∇X i j 2 /λ 2 i Þ uniformly convergent (see [38]), then applying the Cauchy inequality we have which implies that the series ∑ +∞ i=1 jðFðx, tÞ, X i ðxÞÞ∇X i ðxÞj is uniformly convergent on D for any t ∈ ð0, TÞ. In view that fX i g +∞ i=1 is a complete orthonormal system in L 2 ðDÞ, using 7 Journal of Function Spaces the above facts, the estimates (68)-(70) and applying the Cauchy inequality, we get where C = Ð T 0 ð1/ψðτ, αÞÞdτ. Consequently, the series (66) uniformly converges to ∇u and the generalized solution (54) belongs to C 1 x ðDÞ. Applying the similarly method, we can show the generalized solution (54) belongs to C 2 x ðDÞ as well and the relation holds true. In fact, since Aðu 0 Þ ∈ M A , AðFÞ ∈ M A , then A 2 ðu 0 Þ ∈ L 2 ðDÞ, A 2 ðFÞ ∈ L 2 ðDÞ. Moreover, due to A 2 ðu 0 Þ, A 2 ðFÞ ∈ M A , so A 2 ðu 0 Þ and A 2 ðFÞ can be expanded into uniformly convergent series for any t ∈ ð0, TÞ. Note that the series ∑ +∞ i=1 ðjΔX i j 2 /λ 3 i Þ is uniformly convergent (see [38]), then by the Cauchy inequality, we have Due to lim i→+∞ λ i = +∞, there exists a constant l such that λ 3 i < lλ 4 i , i = 1, 2, ⋯, where l 1 is a constant depending on l. Then (73) implies that the series ∑ +∞ i=1 jðFðx, tÞ, X i ðxÞÞΔX i ðxÞj is uniformly convergent on D for any t ∈ ð0, TÞ. Using the above facts and applying the Parseval equality and Cauchy inequality, we get Consequently, the relation (72) holds true and the generalized solution (54) belongs to C 2 x ðDÞ. It remains to prove that (54) belongs to C 1 t ð0, TÞ. We take a derivative term by term with respect to t (54) and construct the series 8 Journal of Function Spaces