    # 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<α <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.

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## 1. 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

, the non-local relationships in space and time are used to model a complex phenomena, such as in electroanalytical chemistry, viscoelasticity , porous environment, fluid flow, thermodynamic , diffusion transport, rheology , electromagnetism, signal processing , electrical network and others . 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 . 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 . 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 , Meerschaert , Shen and Liu and many others.

In this study, we are interested in a problem (FPDE) with boundary conditions of integrals type . 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 is approximated by a scheme called , 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..

### 1.1. 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 denote the gamma function. For any positive non-integer value the caputo derivative defined as follows:

###### Definition 1.

Let us denote by the space of continuous fonctions with compact support in and its bilinear form is given by

 ((u,w))=1∫0Imxu.Imxwdx \ \ \ \ \ \ \ \ (m∈N∗), (1)

where

 Imxu=x∫0(x−ξ)m−1(m−1)!u(ξ,t)dξ        (m∈N∗).

For , we have and The bilinear form is considered as scalar product on when is not complete.

###### Definition 2.

We denote  by

 Bm2(0,1)={L2(0,1)% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for m=0u/Imxu∈L2(0,1) \  \ \ \ for m∈N∗,

the completion of for the scalar product defined by .The associated norm to the scalar product is given by

 ∥u∥Bm2(0,1)=∥Imxu∥L2(0,1)=⎛⎜⎝T∫0(Imxu)2dx⎞⎟⎠12.
###### Lemma 3.

For all we obtain

 ∥u∥Bm2(0,1)≤(12)m∥u∥2L2(0,1). (2)
###### Definition 4.

Let be a Banach space with the norm , and let u : be an abstract functions, by we denote the norm of the element at a fixed t.

We denote by the set of all measurable abstract functions from into such that

 ∥u∥L2(0,T;X)=⎛⎜⎝T∫0∥u(.,t)∥Xdt⎞⎟⎠12<∞.
###### Lemma 5.

For all and arbitrary variables a,b we have the following inequality:

 |ab|≤ε2|a|2+12ε|b|2. (3)
###### Definition 6.

The left Caputo derivative for can be expressed as

###### Definition 7.

The integral of order of the function  is defined by:

 Iα0f(t)=1Γ(α)t∫0f(s)(t−s)1−αds; t>0.
###### Lemma 8.

For all real we have the inequality

 ∫10c0∂αt(Ixu)2dx≤2∫10(c0∂αtu)(Ixu)dx.
###### Lemma 9.

For all real we have the inequality

 ∫Q(c0∂αtu)(Ixu)dxdt≤∫Q(c0∂α2tIxu)2dxdt.

## 2. Statement of the problem

In the rectangular domain

 Q={(x,t)∈R2:00,

we consider the fractional differential equation:

 \poundsv=c0∂αtv+a(x,t)∂2v∂x2+b(x,t)∂v∂x+c(x,t)v=g(x,t), where 1<α<2, (4)

to the equation , we associate the initial conditions:

 (5)

and the purely integrals conditions

 (6)

where and are  known continuous functions.

Assumptions:

1) for all , we assume that:

 sup Qa(x,t)≤0,supQ∂a4(x,t)∂x4≥0,infQ∂3b(x,t)∂x3≤0,c(x,t)≥0,supQ∂c2(x,t)∂x2≥0, (7)

2) for all , we assume that:

 0

3 The functions and satisfy the following compatibility conditions:

 ∫10Φdx=μ(0),∫10xΦdx=E(0),∫10Ψdx=μ′(0),∫10xΨdx=E′(0). (9)

We transform a problem with nonhomegenous integral conditions to the equivalent problem with homogenous integral conditions, by introducing a new unknown function defined by

 v(x,t)=˜u(x,t)+U(x,t), (10)

where

 U(x,t)=2(2−3x)μ(t)+6(2x−1)E(t). (11)

Now we study a new problem with homegenous integral conditions

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩\pounds˜u=c0∂αt˜u+a(x,t)∂2˜u∂x2+b(x,t)∂˜u∂x+c(x,t)˜u=h(x,t),ℓv=˜u(x,0)=φ(x),x∈(0,1),qv=˜u(x,0)∂t=ψ(x),x∈(0,1),∫10˜u(x,t)dx=0,t∈(0,T),∫10x˜u(x,t)dx=0,t∈(0,T), (12)

where

 h(x,t) = g(x,t)−\poundsU(x,t), φ(x) = Φ(x)−ℓU, ψ(x) = Ψ(x)−qU

and

 ∫10φ(x)dx=0,∫10xφ(x)dx=0,∫10ψ(x)=0,∫10xψ(x)=0.

Again we introduce new function defined by

 u(x,t)=˜u(x,t)−ψ(x)t−φ(x), (13)

therefore the problem can be given as follow

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩\poundsu=c0∂αtu+a(x,t)∂2u∂x2+b(x,t)∂u∂x+c(x,t)u=f(x,t),ℓu=u(x,0)=0,x∈(0,1),qu=u(x,0)∂t=0,x∈(0,1),∫10u(x,t)dx=0,t∈(0,T),∫10xu(x,t)dx=0,t∈(0,T). (14)

Thus, instead of seeking a solution of the problem , we establish the existence and uniqueness of solution of the problem and solution will simply be given by:

 v(x,t)=˜u(x,t)+U(x,t). (15)

## 3. Inequality of energy and its consequences

The solution of the problem can be considered as a solution of the problem in operational form:

 Lu=\tciFourier,

where is considered from to , where is a Banach space of functions , whose norm:

 ∥u∥B=(∫Q(c0∂α2t(Ixu))2dxdt+∫Q(Ixu)2dxdt)12 (16)

is finite, and is a Hilbert space consisting of all the elements whose norm is given by:

 ∥∥\tciFourier∥∥F=(∫Q\ f 2dxdt)12. (17)

Now we let be the domain of  the opérator for the set of all functions such as that: and satisfies the integral conditions in problem Then,

###### Theorem 10.

Under assumptions -

, the condition satisfied then we have the estimate

 ∥u∥B≤C∥Lu∥F, (18)

where is a positive constant and independent of where .

###### Proof.

Multiplying the fractional differential equation in the problem by and integrating it on we obtain

 −2∫Q(c0∂αtu)I2xudxdt−2∫Qa(x,t)∂2u∂x2I2xudxdt (19) −2∫Qb(x,t)∂u∂xI2xudxdt−2∫Qc(x,t)uI2xudxdt = −2∫QfI2xudxdt.

Integrating by parts of four integrals in the left side of , we get

 −2∫Q(c0∂αtu)I2xudxdt=2∫Q(c0∂αtIxu)(Ixu)dxdt, (20)
 −2∫Qa(x,t)∂2u∂x2I2xudxdt = 4∫Q∂2a∂x2(Ixu)2dx−2∫Qau2dxdt (21) −∫Q∂a4∂x4(I2xu)2dx,
 −2∫Qb(x,t)∂u∂xI2xudx=∫Q∂3b∂x3(I2xu)2dx−3∫Q∂b∂x(Ixu)2dx, (22)
 −2∫Qc(x,t)uI2xudx=−∫Q∂2c∂x2(I2xu)2dx+2∫Qc(Ixu)2dx (23)

Substituting in , we have

 2∫Q(c0∂αtIxu)(Ixu)dx+4∫Q∂2a∂x2(Ixu)2dx−2∫Qau2dx (24) −∫Q∂a4∂x4(I2xu)2dx+∫Q∂3b∂x3(I2xu)2dx−3∫Q∂b∂x(Ixu)2dx −∫Q∂2c∂x2(I2xu)2dx+2∫Qc(Ixu)2dx = −2∫QfI2xudx.

By the elementary inequalities in lemmas (8), (9) respectively and assumptions give

 2∫Q(c0∂α2t(Ixu))2dxdt+∫Q(4∂2a∂x2−4supau (25) −12∂a4∂x4+12inf∂3b∂x3−3∂b∂x −12sup∂2c∂x2+2c)(Ixu)2dxdt ≤ −2∫QfI2xudxdt.

The estimate of the right side of gives:

 ∫Q(c0∂α2t(Ixu))2dxdt+∫Q(4∂2a∂x2−4supau (26) −12∂a4∂x4dxdt+12inf∂3b∂x3−3∂b∂x −2sup∂2c∂x2+2c−12ε)(Ixu)2dxdt ≤ ε∫Qf2dxdt.

So, by using the assumptions we find

 2∫Q(c0∂α2t(Ixu))2dxdt+M∫Q(Ixu)2dxdt (27) ≤ ε∫Qf 2dxdt

Finally, we obtain a priori estimate

 ∥u∥B≤C∥Lu∥F, (28)

where

 C=(εmin(2,M))12.

###### Corollary 11.

A strong solution of problem is unique if it exists, and depends continuously on

###### Corollary 12.

The range of the operator is closed in and

## 4. 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 is dense in

###### Theorem 13.

Let us suppose that the assumptions and integral conditions are filled, and for and for all , we have

 ∫Q\poundsu.ωdxdt=0, (29)

then almost everywhere in

###### Proof.

We can rewrite the equation as follows

 ∫Q(c0∂αtuω)dxdt = −∫Qa(x,t)∂2u∂x2ωdxdt−∫Qb(x,t)∂u∂xωdxdt (30) −∫Qc(x,t)uωdxdt,

Further, we express the function in terms of as follows :

 ω=−2I2xu (31)

Substituting by its representation in integrating by parts, and taking into account the conditions , we obtain:

 2∫Q(c0∂αtIxu)Ixudxdt=−4∫Q∂2a∂x2(Ixu)2dxdt+2∫Qau2dxdt+∫Q∂4a∂x4(Ixu)2dxdt
 −∫Q∂3b∂x3(Ixu)2dxdt+3∫Q∂b∂x(Ixu)2dxdt+∫Q∂2c∂x2(Ixu)2dxdt−2∫Qc(Ixu)2dxdt,

on using under assumptions and conditions , we obtain

 2∫Q(c0∂αtIxu)Ixudxdt=−∫Q(4∂2a∂x2+4supau +12∂4a∂x4−12inf∂3b∂x3+3∂b∂x+2sup∂2c∂x2−2c)(Ixu)2dxdt,

 2∫Q(c0∂αtIxu)Ixudxdt≤−(12ε+M)∫Q(Ixu)2dxdt.

By lemmas ( and ( we obtain

 2∫Q(c0∂α2t(Ixu))2dxdt≤−(12ε+M)∫Q(Ixu)2dxdt.

Then

 (Ixu)2=0 (32)

and we obtain

 u=0.

So in wich gives in

## 5. Finite Difference Method

### 5.1. Discretization of the problem

Now, we consider a uniform subdivision of intervals and as follows

 xi=ih; i=0,...,N and tk=kht; k=0,...,M.

Then, denote by the approximate solution of at points , and the operator is defined by

 L=a∂2∂x2+b∂∂x+c, L(.)ki=aki∂2(.)∂x2+bki∂(.)∂x+cki (33)

where

 aki=a(xi,tk),bki=b(xi,tk),cki=c(xi,tk).

From the Taylor devlopment of function at the point we have

 (34)

Substituting in the operateur expressed in gives

 Lvk+1i=(ak+1ih2+bk+1ih)vk+1i+1+(ck+1i−2ak+1ih2−bk+1ih)vk+1i+ak+1ih2vk+1i−1. (35)

The discretization of Caputo derivative fractional operator with defined by

 (c0∂αtv)k+1i≃γk∑j=0(v%k−j−1i−2v k−ji+v k−j+1i)dj\ . (36)
 where{dj=(j+1)2−α−j2−αd0=1;k=1,...,M \  ,γ=h− αtΓ(3−α).

Writing fractional differential equation in points , we find

 γk∑j = 0(v k−j−1i−2v k−ji+v % k−j+1i)dj+Lvk+1i=g k+1i, i=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯1,N−1 (37)

then

 Fk+1ivk+1i−1+Ak+1ivk+1i+Bk+1ivk+1i +1−2γdkvki+γdkvk−1i+γk−1∑j=1Sjdj+γ(v−1i−2v0i+v1i)dk=gk+1i (38)

where

 Ak+1i = γ+ck+1i−2ak+1ih2−bk+1ih,Bk+1i=ak+1ih2+bk+1ih, Fk+1i = ak+1ih2,Sj=v k−j−1i−2v k−ji+v k−j+1i.

In order to eliminate , we use initial condition , and we find

 (∂v∂t)ni≃vni−vn−1iht

therefore

Substituting in we obtain

 Fk+1ivk+1i−1+Ak+1ivk+1i+Bk+1ivk+1i+1−2γdkvki+γdkvk−1i+γk−1∑j=1Sjdj=dkγv0i+dkγhtΨi−dkγv1i+gk+1i. (40)

For , the relation gives