STUDY OF SOLUTION FOR A PARABOLIC INTEGRODIFFERENTIAL EQUATION WITH THE SECOND KIND INTEGRAL CONDITION DEHILIS SOFIANE, BOUZIANI ABDELFATAH AND OUSSAEIF TAKI-EDDINE∗

In this paper, we establish sufficient conditions for the existence, uniqueness and numerical solution for a parabolic integrodifferential equation with the second kind integral condition. The existence, uniqueness of a strong solution for the linear problem based on a priori estimate ”energy inequality” and transformation of the linear problem to linear first-order ordinary differential equation with second member. Then by using a priori estimate and applying an iterative process based on results obtained for the linear problem, we prove the existence, uniqueness of the weak generalized solution of the integrodifferential problem. Also we have developed an efficient numerical scheme, which uses temporary problems with standard boundary conditions. A suitable combination of the auxiliary solutions defines an approximate solution to the original nonlocal problem, the algebraic matrices obtained after the full discretization are tridiagonal, then the solution is obtained by using the Thomas algorithm. Some numerical results are reported to show the efficiency and accuracy of the scheme.


Introduction
The topic of integro-differential equations which are combination of differential and integral has attracted many scientists and researchers due to their applications in many areas; see, for example, [16,17] .Many mathematical formulation of physical phenomena contain integro-differential equations, and these equations may arise in fluid dynamics, biological models, and chemical kinetics; for more details, see [20,40] .
Integro-differential equations are usually difficult to solve analytically, so it is required to obtain an efficient approximate solution.
Nowadays various nonlocal problems for partial differential equations have been actively studied and one can find a lot of papers dealing with them (see [13]- [29] , [12]- [21] and references therein).Afterwards, the nonlocal problems for integro-differential equation with integral conditions was studied by many authors, see A. Merad and A. Bouziani [23] , [26].Motivated by this we study a parabolic integrodifferential equation with nonlocal second kind integral condition.

Preliminaries and functional spaces
In the rectangular domain Ω = (0, 1) × (0, T ), with T < ∞,we consider the equation: with the initial data u = u(x, 0) = ϕ (x) , x ∈ (0, 1) , with the Second Kind Integral Conditions where f, ϕ, K 0 , K 1 ansd g are known functions.Note that a is bounded function where |a (t − s)| < a 0 , a 0 is a positive constant.
And the function g verify the following inequality We shall assume that the function ϕ satisfies a compatibility conditions with (2.3) and (2.4) , i.e., Some problems of modern physics and technology can be described in terms of partial differential equations with nonlocal conditions.The integral term of our problem (that is, t 0 a (t − s) g (s, u) ds appears,because in some fields such as the heat transfer, nuclear reactor dynamics and thermoelasticity, we need to reflect the effects of the memory of the system in model, but describing such a system as a function at a given space and time ignores the effect of past history.Therefore, the way of remedy this difficulty is including an integral term in the basic partial differential equation that leads to a Partial integro-differential equations( PIDE) [39].The study of the problem (2.1)-(2.2)withsome special types of boundary conditions of the form u x (0, t) = α(t) and 1 0 u (x, t) dx = E(t) motivated by the works of Dabas and Bahuguna [15],and Guezane-Lakoud et al. [18].
It is well known that the classical methods used widely to prove solvability of initial-boundary problems break down when applied to nonlocal problems.Nowadays some methods have been advanced for overcoming difficulties arising from nonlocal conditions.These methods are different and the choice of a concrete one depends on a form of a nonlocal condition.In this article, we focus on spatial nonlocal integral conditions like [30], of which we give three examples: Condition (2.5) is a nonlocal first kind condition, (2.6) and (2.7) are second kind nonlocal conditions.The kind of a nonlocal integral condition depends on the presence or lack of a term containing a trace of the required solution or its derivative outside the integral [30] .Problems with nonlocal conditions of the forms (2.5) and (2.7) are investigated in [30], [11],and [36].We pay attention on the second one, (2.6) which has not been studied so far with this class of integro-differential problems.
This paper is organized as follows.In Section 3, we establish the uniquness of solution by using a priori estimate method or the energy-integral method.In Sect 4, we first establish the existence of solutions of the linear problem by using the density of the range of the operator generated by the abstract formulation of the stated linear problem; secondly reformulating the integro-differential problem to a semi-linear problem, and after that we prove the slovability of semi-linear problem by using a priori estimate and applying an iterative process based on results obtained for the linear problem (see [34]), we prove the existence, uniqueness of the weak generalized solution of the integrodifferential problem.Section 5 is devoted to the construction of approximate solutions of problem (2.1)-(2.4),we discretize the problem by backward Euler in time and finite differences in space.The main numerical difficulty become visible after the discretization,the presence of an integral operator in the boundary conditions gives rise to rows/collumns, which are full.To avoid the problems with special solvers for algebraic systems, we design a very easy numerical algorithm, based on superposition principle, this technique lead to a linear systems have a tridiagonal coefficient matrix, so they can be solved very efficiently by fast Gauss elimination (which is also known as the Thomas algorithm).
Finally, in section 6 we presents two numerical examples to illustrate the performance and efficiency of the proposed algorithm.

An energy estimate and uniqueness of solution
The method used here is one of the most efficient functional analysis methods and important techniques for solving partial differential equations with integral conditions, which has been successfully used in investigating the existence, uniqueness, and continuous dependence of the solutions of PDE's, the so-called a priori estimate method or the energy-integral method.This method is essentially based on the construction of multiplicators for each specific given problem, which provides the a priori estimate from which it is possible to establish the solvability of the posed problem.More precisely, the proof is based on an energy inequality and the density of the range of the operator generated by the abstract formulation of the stated problem, so to investigated the posed problem, we introduce the needed function spaces.In this paper, we prove the existence and the uniqueness for solution of the problem (2.1) − (2.4) as a solution of the operator equation Where L = (L, ), with domain of difinition E consisting of functions u ∈ L 2 (0, T, L 2 (0, 1)) := L 2 (Ω) such that u x ∈ L 2 (Ω) and u satisfies condition (2.3) and (2.4) ; the operator L is considered from E to F, where E is the Banach space consisting of all functions u(x, t) having a finite norm and F is the Hilbert space consisting of all elements = (f, ϕ) for which the norm is finite.
Theorem 3.1.If ε > 0, where ε << 1 2 .Then for any function u ∈ E and we have the inequality where c is a positive constant independent of u.
By using the Cauchy inequality with ε; we have To obtain the estimate, we need the inequalities which easily follow from the equalities Also by (2.3) and (2.4), we obtain So, by using Holder inequality, we have where the constant Then, we get Remains apply the inequality the Cauchy inequality with ε to the end terms of the right-hand side part of (3.4) and using (3.7) and (3.8) , we get Then, we obtain Using Lemma 1 of Gronwall in [32] , we have By integrating the inequality (3.10) over (0, T ) , we obtain the desired inequality, where c = (T d)

Existence of solution of the integrodifferential problem
This section is consecrated to the proof of the existence of the solution on the data of the problem (2.1) − (2.4).we can reformulating the integro-differential problem to a semi-linear problem by putting Where exists a positive constant δ such that Therefore to study the existence of solution of previous problem (2.1)−(2.4), is enough to study the following semi-linear problem: with the initial data with the Second Kind Integral Conditions Let us consider the following auxiliary problem with homogeneous equation If u is a solution of problem (4.1) − (4.4) and w is a solution of problem (4.5) − (4.8), then y = u − w satisfies y = y(x, 0) = 0, (4.10) Where G (x, t, y) = H (x, t, y + w) , As the function H, the function G satisfies the condition (C * ) , that is there exists a positive constant δ such that To show the existence of solutions of the problem (4.5) − (4.8), it is enough to transform the problem to the linear first-order ordinary differential equation with second member.
For that we integrate the equation (4.5) over [0, 1] and using (4.7) − (4.8), we get then, we obtain So, we can prove that there existe a function ψ verify that Clearly, that the solution of (4.5) by using (4.6) is given by Therefore, the existence of solution is guaranteed.
According to this results, we deduce that problem (4.5) − (4.8) admits a unique solution.Therefore it remains to solve and prove that the problem (4.9) − (4.12) has a unique weak solution.
Let us construct an iteration sequence in the following way: Starting with y (0) = 0, the sequence y (n) n∈N is defined as follows: given the element y (n−1) , then for n = 1, 2, ... solve the problem: ) x (0, t) = 0, (4.17) Clearly, for fixed n, each problem (4.15) − (4.18) has a unique solution y ), then we have the new problem where where M is a positive constant given by Proof.Multiplying the equation (4.19) by Z (n) and integrating over Ω τ , where Ω τ = (0, 1) × (0, τ ), we get Integrating by parts the second term of the left-hand side in (4.24) and taking into account conditions (4.20) , (4.21) and (4.22), we obtain 1 2 Using the Cauchy inequality to the right-hand side of (4.25), we get Using Lemma of Gronwall, we obtain On the other hand., by virtue of condition (C * * ) , we have dt.
The right hand side here is independent of τ ; hence, replacing the left hand side by the upper bound with respect to τ , we obtain dt.
Now by integrating over (0, T ), we get dt.
So, we obtain dt Finally, we find where From the criteria of convergence of series, we see that the series , then it follows that the sequence (y (n) ) n∈N defined by converges to an element y ∈ L 2 0, T ; H 1 (0, 1) .
Remains to precise the concept of the solution we are considering.Let v = v(x, t) be any function from We shall compute the integral Ω Gvdxdt, for this we assume v x (0, t) = v x (1, t) = 0.By using conditions on y, we have Then we put Definition 4.1.For every v ∈ C 1 (Ω), a function y ∈ L 2 (0, T ; H 1 (0, 1)) is called a weak solution of problem (4.9) − (4.12) if (4.30) holds under the conditions of y.Now, we must show that the limit function y is a solution of the problem under study.To do this, we will show that y verifies (4.30) as mentioned in definition 1.So, we consider the weak formulation of problem (4.9) − (4.12) : From (4.30) , we have However, we apply Holder inequality, we get then the problem (4.9) − (4.12) admits a weak solution in L 2 0, T ; H 1 (0, 1) .
It remains to prove that problem (4.9) − (4.12) admits a unique solution.

Construction of approximate solutions
In order to solve the problem (2.1) − (2.4) , first we divide the time interval [0, T ] into N ∈ N equidistant subintervals (t j−1 , t j ) for t j = jτ , where τ = T N We introduce the following notation after replacing the derivative ∂u ∂t by backward finite difference approximations u j − u j−1 τ and the integral by rectangular rule.Then problem (2.1) − (2.4) reduced to the solutions of recurrent system of ODE problems at each successive time point t j for j = 1, ..., N find, successively for j = 1, ..., N ; functions u j : (0, 1) → R such that: The main numerical difficulty become visible after the full discretization of these nonlocal problem , the presence of an integral BC in the problem gives rise to rows, which are full(see Algorithm 1).

Algorithm 1 :(A1).
For The space discretization we use the finite differences scheme .
we divide the space interval [0, 1] into M ∈ N equidistant subintervals of equal lengths h = 1 M second-order difference is used to approximate the second order spatial derivative : where u i,j = u(x i , t j ), and employing central-differences to approximat the first order spatial derivative in the boundary condition : we construct a difference scheme for the problem (5.1)-(5.4): after some rearrangement, the Equation (5.5) becomes : where r = τ h 2 .we approximate the integral in (5.6)-(5.7)numerically by the trapezoidal numerical integration rule: which is the same second-order of accuracy in space as the methods used for spatial derivative .Equation (5.9) presents M + 1 linear equations in M + 3 unknowns u −1 , u 0 , ..., u M +1 .Eliminating of the "fictitious" value u −1,j beteween (5.8) i=0 and (5.9) gives : Similarly, eliminating u M +1,j beteween (5.8) i=M and (5.10) gives : Combining (5.11), (5.9), with (5.12) yields an (M + 1) × (M + 1) linear system of equations whose coefficient matrix A j has the form: where a 00 , a 01 , ..., a 0M and a M 0 , a M 1 , ..., a M M are the coefficients in (5.11) and (5.12), respectively.We will denote the right-side of the system by b ) + τ f i,j , i = 0, ..., M .We write the system in the matrix form : which have to be solved successively with increasing time step j = 1, .., N .The main numerical problem is the special character of the algebraic matrix obtained, tridiagonal except that their first and last rows are full,this needs a special solver to get a result.But there exist a simple way how to avoid this complication, we explain it in algorithm 2.

Algorithm 2:(A2).
To get rid of the nonlocal BC,we make use of a slightly modified idea of [37], for any given j we introduce three auxiliary problems.The first one with an unknown function v i is given as: and the initial condition The second one with the unknown z reads as: The third one with the unknown w reads as Let us note that the temporary problems are standard problems.
Let α j and β j be any real number, the principle of linear superposition gives that ω j := v j + α j z + β j w is the solution to the following BVP and the initial condition We have to pick up the appropriate value of the free parameter α j and β j for which the function ω j be a solution to problem (5.1)-(5.4).We are looking for an α j and β j such that then ω j will be a solution to problem (5.1)−(5.3)if and only if the pair (α j , β j ) is a solution of the following system of equations we have to check if the determinant of system (5.18) is different from zero.
if D = 0 then we easily deduce : Proof.One can see that the solution of the second auxiliary problem is : and the solution of the third auxiliary problem is : The variational formulations of temporary problems are: we set Φ = z into (5.21) and we get analogously for w (w, Φ) + τ ( dw dx , dΦ dx ) = τ Φ(1), for any Φ ∈ H 1 (0, 1) (5.22) we set Φ = w into (5.22) and we get Lebegue domineted theorem says : w 2 = 0 cauchy inequality says from the definition of a limit we easily arrive at for ε = 1/2 there exists a τ 0 such that: for any 0 < τ < τ 0 we haveD 1/2 .
For the space discretization we use the same scheme in algorithm 1 for a better comparison.We construct a difference scheme for the first auxiliary problem (5.14) : after some rearrangement, the Equation (5.23) becomes : where r = τ h 2 .There are M + 1 linear equations in M + 3 unknowns v −1,j , v 0,j , ..., v M +1;j Eliminating of the "fictitious" value v −1,j beteween (5.23) i=0 and (5.24) gives : Eliminating v M +1,j beteween (5.23) i=M and (5.25) gives : Combining (5.27), (5.26), with (5.27) yields an (M + 1) × (M + 1) linear system of equations, we write the system in the matrix form : where Then at each time level, the difference scheme can be written as systems of M + 1 tridiagonal linear algebraic equations, which is solved by Thomas' algorithm.After that computing the value of α j and β j from equation (5.20) .The integrals are approximated by the composite trapezoidal rule: then the approximative solution of (2.1)-(2.4) is obtained by: u i,j = v i,j + α j z i + β j w i , i = 0..., M, j = 1, ..., N.

Numerical experiment
To test the above algorithms , we use two examples as follows: Example 1.Consider (2.1) − (2.4) in Ω = (0, 1) × (0, 1) , with It is easy to check that the exact solution of this test problem is It is easy to check that the exact solution of this test problem is  Our numerical experiment are performed using Matlab and we used an Intel Core i3 with 2.1 GHz .Table 1 and table 4 gives some numerical results and exact values at some points at the time t = 0.5.Table 2 and table 5 gives the absolute errors of the numerical solutions at some points at the time t = 0.5, and this is also shown in figure 1 and figure 2. Table 3 and table 6 gives the maximum errors of the numerical solutions.
The maximum error is defined as follows: The results obtained using algorithm 1 and algorithm 2 have the same accuracy .It is also noted that the algorithm 2 will require less CPU time than algorithm 1 (see table 2 and table 5).From table 3 and table 6,we may see the errors decrease about by a factor of 4 as the spatial mesh size is reduced by a factor of 2 and the time mesh size is reduced by a factor of 4.

Conclusion
It is important to note that, for non-local problems, there is not yet a general theory analogous to that of classical problems.This is due to the relative novelty of this topic on the one hand and to the complexity of the questions it raises on the other hand.Each problem then requires a specific treatment, which highlights the topicality of the subject tackled in this article.Especially, when combined a parabolic integrodifferential equation with the second kind integral condition.So in this paper, we establish sufficient conditions for the existence, uniqueness and numerical solution for a parabolic integrodifferential equation with the second kind integral condition.For the theoretical studies we use the energy inequality and fixed point theorem methods.Also we construct a new numerical scheme to solve parabolic integrodifferential equation with the second kind integral condition, which has the following advantage: The coefficient matrices of the scheme is tridiagonal,to solve the linear system of equations by Thomas algorithm the cost is about 8M − 7 (M the order of The coefficient matrices ) , will save remarkable CPU time.

Figure 1 .
Figure 1.The errors of the numerical solutions at t=0.5 for example1.

uFigure 2 .
Figure 2. The errors of the numerical solutions at t=0.5 for example2.

Table 1 .
Some numerical results at t = 0.5 with τ =

Table 2 .
The absolute errors of some numerical solutions at

Table 3 .
The maximum errors of the numerical solutions for Example 1.

Table 4 .
Some numerical results at t = 0.5 with τ =

Table 5 .
The absolute errors of some numerical solutions at t = 0.5 with τ =

Table 6 .
The maximum errors of the numerical solutions for Example 2.