On Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

In this paper, we are interested in the study of a problem with fractional derivatives having boundary conditions of integral types. The problem represents a Caputo type advection-diffusion equation where the fractional order derivative with respect to time with $1<\alpha<2$. The method of the energy inequalities is used to prove the existence and the uniqueness of solutions of the problem. The finite difference method is also introduced to study the problem numerically in order to find an approximate solution of the considered problem. Some numerical examples are presented to show satisfactory results.


Introduction
Fractional Partial Differential Equations (FPDE) are considered as generalizations of partial differential equations having an arbitrary order and play essential role in engineering, physics and applied mathematics. Due to the properties of Fractional Differential Equations (FDE), the non-local relationships in space and time are used to model a complex phenomena, such as in electroanalytical chemistry, viscoelasticity [10,21], porous environment, fluid flow, thermodynamic [11,34,35], diffusion transport, rheology [5,7,15,26,31,33], electromagnetism, signal processing [20,21,30], electrical network [20] and others [9,13,26,27]. Several problems have been studied in modern physics and technology by using the partial differential equations (PDEs) where the nonlocal conditions were described by integrals, further these integral conditions are of great interest due to their applications in population dynamics, models of blood circulation, chemical engineering thermoelasticity [34]. At the same time, the existence and uniqueness of the solutions for these type of problems have been studied by several researchers, see for example [2,12,16,27,28,29]. Some results have been obtained by construction of variational formulation and depends on the choice of spaces along their norms, Lax-Milgram theorem, Poincaré theorem, fixed point theory. For the numerical studies of (EDPF) with classical boundary nonlocal conditions, we can cite the works of A. Alikhanov [3,5,6,7], Meerschaert [15], Shen and Liu [26] and many others.
In this study, we are interested in a problem (FPDE) with boundary conditions of integrals For the theoretical study, we use the energy inequalities method to prove the existence and the uniqueness. However the numerical study is based on the finite difference method to obtain an approximate numerical solution of the proposed problem. We use a uniform discretization of space and time and the fractional operator in the Caputo sense having order α (1 < α < 2) is approximated by a scheme called L2 [26], similarly the integer-order differential operators are also approximated by central and advanced numerical schemes. For the stability and convergence of obtained numerical scheme, the conditionally stable method is used and we prove the convergence. Numerical tests are carried out in order to illustrate satisfactory results from the point of view that the values of the approximate solution that is very close to the exact solution. In the process of numerical and graphical results we applied MATLAB software..

Notions and preleminaries.
In this section we recall some early results that we need, such as, the definition of Caputo derivative to explain the problem that we shall study in this work: let Γ (.) denote the gamma function. For any positive non-integer value 1 < α < 2, the caputo derivative defined as follows: [12]) . Let us denote by C 0 (0, 1) the space of continuous fonctions with compact support in (0, 1) , and its bilinear form is given by where (1) is considered as scalar product on C 0 (0, 1) when is not complete.
Definition 2. (See [12]) . We denote by the completion of C 0 (0, 1) for the scalar product defined by (1).The associated norm to the scalar product is given by Lemma 3. (See [8]) . For all m ∈ N * , we obtain Definition 4. (See [12]) . Let X be a Banach space with the norm u X , and let u :(0, T ) → X be an abstract functions, by u (., t) X we denote the norm of the element u (., t) ∈ X at a fixed t.

Statement of the problem
In the rectangular domain we consider the fractional differential equation: to the equation (4), we associate the initial conditions: and the purely integrals conditions where Φ, Ψ, µ, E, a, b, c and g are known continuous functions.

Assumptions:
1) for all (x, t) ∈ Q, we assume that: 2) for all (x, t) ∈ Q, we assume that: 3) The functions Φ(x) and Ψ(x) satisfy the following compatibility conditions: We transform a problem (4) -(6) with nonhomegenous integral conditions to the equivalent problem with homogenous integral conditions, by introducing a new unknown function u defined by where Now we study a new problem with homegenous integral conditions where Again we introduce new function u defined by therefore the problem (12) can be given as follow Thus, instead of seeking a solution v of the problem (4) − (6), we establish the existence and uniqueness of solution u of the problem (14) and solution v will simply be given by:

Inequality of energy and its consequences
The solution of the problem (14) can be considered as a solution of the problem in operational form: is finite, and F is a Hilbert space consisting of all the elements F = (f, 0, 0) whose norm is given by: Now we let D(L) be the domain of the opérator L for the set of all functions u such as that: ∂x 2 ∈ L 2 (Q) and u satisfies the integral conditions in problem (14) . Then, Theorem 10. Under assumptions (7)-(8), the condition satisfied then we have the estimate where C is a positive constant and independent of u where u ∈ D(L).
Proof. Multiplying the fractional differential equation in the problem (14) by M u = −2 2 x u and integrating it on Q we obtain Integrating by parts of four integrals in the left side of (19), we get By the elementary inequalities in lemmas (8), (9) respectively and assumptions (7) The estimate of the right side of (25) gives: So, by using the assumptions (7) − (8) we find Finally, we obtain a priori estimate where (14) is unique if it exists, and depends continuously on F = (f, 0, 0).

Corollary 12.
The range of the operator L is closed in F and R(L) = R(L).

Existence of solutions
In thei section, we prove the uniqueness of solution, if there is a solution. However, we have not demonstrated it yet. To do it, we will just prove that R(L) is dense in F. Theorem 13. Let us suppose that the assumptions (7) − (8)and integral conditions (6) are filled, and for ω ∈ L 2 (Q) and for all u ∈ D(L), we have then ω almost everywhere in Q.
Proof. We can rewrite the equation (29) as follows Further, we express the function ω in terms of u as follows : Substituting ω by its representation (31) in (30) , integrating by parts, and taking into account the conditions (6), we obtain: on using under assumptions (7) − (8) and conditions (9), we obtain and this leads that By lemmas (2), (3) and (4) we obtain Then and we obtain u = 0.
So u = 0 in Ω wich gives ω = 0 in L 2 (Q). Then, denote by v k i the approximate solution of v (x i , t k ) at points (x i , t k ), and the operator L is defined by

Finite Difference Method
Substituting (34) in the operateur L k i expressed in (33) gives The discretization of Caputo derivative fractional operator c where Writing fractional differential equation (4) in points (ih, (k + 1) h t ), we find In order to eliminate v −1 i , we use initial condition (5), and we find ∂v ∂t Substituting (39) in (38) , we obtain For k = 0, the relation (40) gives By conditions (6) , and trapezoid method we obtain, For i = 1,

Matrix's form
We denote by is square matrix and defined by Taking account (41) , (42) , and (43) , we obtain the matrix system where To solve the system (44) we can apply one of direct methods.

General case.
It is readily checked that, for k ≥ 1 Lemma 14. If k ≥ 1; the numerical scheme (40) is equivalent to Proof. From the scheme (40), we have Using the conditions (6) , and by trapezoid method we obtain: For i = 1,

Matrix's form
We take expression (48) for i = 2, N − 2 and equations (??), (49) to formulate the matrix systems: where P k+1 = l k+1 i,j N −1,N −1 is square matrix defined by l k+1 In order to prove system (50) has a unique solution we denote ρ as an eigenvalue of the matrix P k , and X = (x 1 , x 2 , ..., x N −1 ) T is an nonzero eigenvector corresponding to ρ. Then, we choose i such as |x i | = max{|x j | : j = 1; ...; N − 1} then Substituting the values of l i,j into (51) , and taking into account that F k i , a k i are negative and xj xi ≤ 1 we get, for i = 1, From the above we conclude for i = 1, , ρ > 0, then all eigenvalues of matrix P k+1 are strictly positive, therefore P k+1 is invertible.

Stability. Since, we have
then we let u k+1 i be the approximate solution of (48) , and e k+1 i , the error at point ( for k = 0 we apply (41) we get Therefore the method is stable.
Lemma 15. For k ≥ 1 the scheme (47) is stable and we have Proof. We use the mathematical induction.
We assume E j ≤ c j E 0 , and C max = max c j ; where c j 0, j = 1, k from (48) we get Therefore the method is stable.

Convergence.
Let v(x i ; t k+1 ) as the exact solution and v k+1 i is the approximate solution of scheme (37), we put v( substitution into (37) and using (34), (55) leads to Taking k l = k = Max we have We assume : j l ≤ O(h + h t ); j = 1, k from (56) we get we have Therefore, the method is convergent.

Applications
In this section, we give some numerical investigation tests.
The analytical solution is given by v(x, t) = t 3 2 e x .
For h = 0.01, α = 1.5 The Fig.12 ,13 and 14 show where the space step is fixed at h = 0.01 and the time step h t decreases towards zero (h t = 0.001, h t = 0.0001, h t = 0.00001), the approximate solution u 1 tends to the exact solution v 1 , in the case where h t = 0.00001 we see that the two curves of u 1 and v 1 are almost identical. Table 10 shows the error norm E k ∞ for defferent value of α defined by We see in the table 10, for the space step h = 0.1, and for the defferent values of α, the error norm tends to zeros when the time step h t takes values close to zeros, with convergence order O(h + h t ).

Conclusion
In this paper, we study a problem with fractional derivatives with boundary conditions of integral types. The study concerns a Caputo-type advection-diffusion equation where the fractional order derivative α with respect to time with 1 < α < 2. The existence and uniqueness are proven by the method of energy inequalities. The numerical study of this problem based on the finite difference method. Applications on certain examples clearly show that the numerical results obtained are very satisfactory, where we see the approximate solution u tends to the exact solution v for the defferent value of α.