A Nonlinear Fractional Problem with a Second Kind Integral Condition for Time-Fractional Partial Differential Equation

Abstract

The aim of this research is to demonstrate the existence and the uniqueness of the weak solution for a semilinear fractional parabolic problem with the special case of the second integral boundary condition. For this aim, we split the proof into two parts; to study the main linear problem part, we used the variable separation method, and concerning the semilinear problem part, we apply an iterative method and a priori estimate for the study of the weak solution.

1. Introduction

Fractional calculus is a mathematical analysis branch that explores the various different possibilities of describing the differentiation operator power of real or complex numbers. Fractional differential equations are extraordinary differential equations [1]. Many natural phenomena and modern problems of physics, mechanics, biology and technology, chemistry, engineering, etc. can be modeled by fractional partial differential equations (FPDEs). For details, see ([2–12]) and the references therein.

The first who drew attention to these problems with an integral one-space variables condition is Cannon [13], which gold of the study of heat conduction in a bar heated thin is demonstrated by using the potential method, and the importance of the problems with integral conditions has been pointed out by Samarskii [14]. The basic physical meaning of integral conditions (total energy, average temperature, the total mass of impurities, total flux, moments, etc.) has served as the main reason for the growing interest in this kind of problem.

In modern physics and technology, many problems are found using nonlocal conditions for partial differential equations, which are defined using integral conditions. So, the first type of integral condition is given by the following: where is a given function.

Or second type can be used when it is impossible to directly measure the quantity sought on the border. Its total value or its average is known. We find a lot of research studied this kind of integral like [15–17].

Many researchers have widely studied the solvability “existence and uniqueness” of solutions of fractional and integer partial differential equation problem by many method; see, for example, [18–23].

There are a few works that study the nonlinear fractional partial differential equation with initial and boundary conditions, not to mention the nonlocal problem like [24].

In this work, we developed the study of fractional problems to partial differential equations on the one hand and beyond this in the nonlinear direction, which simulates heat diffusion in the complex phenomena. We also relied on the modeling of Neumann’s condition, i.e., heat flow by an integral condition, which is closer to reality in this situation.

Motivated by this, in our article, we treat for the existence and the uniqueness of the weak solution of the mixed semilinear problem for a fractional partial differential equation with an integral condition of the second type, where we start by studying the nonlocal linear problem by the variable separation method, to demonstrate the existence and the uniqueness of the weak solution of the semilinear problem; we then apply an iterative method based on the results obtained by the linear problem.

2. Preliminaries

Definition 1 (see [25, 26]). For any , the Caputo and Riemann Liouville derivatives are defined, respectively, as follows: (1)The left Caputo derivatives:(2)The left Riemann-Liouville derivatives:(3)The right Caputo derivatives:(4)The right Riemann-Liouville derivatives:

Proposition 2 (see [25, 26]). For , we have If , then we have

Definition 3 (see [27]). For any real and finite interval of the real axis , we define the seminorm and norm We then define as the closure of with respect to the norm .

Definition 4 (see [27]). For any real , we define the seminorm and norm We then define as the closure of with respect to the norm .

Definition 5. For any real , we define the seminorm and norm

Lemma 6 (see [27, 28]). For any real if and , then

Lemma 7 (see [27, 28]). For on a

Lemma 8 see ([27, 28]). For , the seminorms and are equivalent. Then, we pose

Lemma 9 see ([27, 28]). For any real , the space with respect to the norm (12) is complete.

3. Formulation of the Nonlinear Problem (18)

Let .

We would like to deal with the following nonlinear problem: where and .

In the remainder of this section, we assume to be Lipschitz; i.e., there is a constant as

4. Study of the Associated Linear Problem

4.1. Position of the Associated Linear Problem (20)

In , with we examine the following associated linear problem: which can be written in this operational form:

With the initial condition

Dirichlet boundary condition and the integral condition of the second kind

4.2. Resolution of Problem (20) by the Variable Separation Method

Let the following associated homogeneous linear problem:

We can certainly show that problem (25) admits a unique solution, for that we put

Substituting (26) into (25), we obtain

Therefore, we get for

We begin by showing a Sturm-Liouville problem:

With , it has a solution which is given by

Or A and B are two real arbitrary.

Using the Dirichlet condition, we find

It follows that

Now, we use the integral condition, and we get

Hence, we can assert that the eigenvalue is given by the following equation:

Our next goal is to determine the explicit form of . According to the superposition theorem, we pose

Replacing (35) by (20) leads to which implies

As then hence

We can then solve in a simple way the fractional ordinary differential problem (37)−(40) using the Laplace transform method; so it comes then

Therefore,

Finally, we obtain

Hence, we obtain the solution of (25) in the following explicit form:

5. Solvability of the Weak Solution of the Nonlinear Problem (18)

The key point to discuss in this section is to study the existence and uniqueness of the problem’s weak solution (18). The key idea is to apply an iterative method and estimate a priori.

In this way, we define an auxiliary problem with a homogeneous equation:

If is a solution of (18) and is a solution to (46), then satisfies or . From now on, we make the assumption: the function is Lipschitz, that is, there is a positive constant as

In order to get the desired results, it is necessary to propose the concept of the studied solution.

Let arbitrary function of as Multiply 9 by and integrate it on ; we find By the use of integration by parts and the conditions on , we get Then, (54) establishes the following formula: or

Definition 10. We say the solution to problems (47)–(50) is weak; any function such as (33), (49), and (50) are achieved.

Now, we build a recurring sequence defined as follows: from we can define , where the first element is given by ; then, for , the following problem is solved:

For any fixed, thanks to the study of (20) which we gave the solution explicitly using the variable separation method; then, problem (57) has a unique solution .

Now, suppose ; so we get a new problem: or

Lemma 11. We assume that (51) be satisfied. So we obtain for problem (58) the following a priori estimate: or

Proof. We multiply by this equation and we integrate it on ; we get Performing integration by part and using lemma 2, 3, 4, and 5 by the same way in the articles [15–24], we obtain Using Cauchy with inequality, we get Exploiting the fact that we conclude from (51) that Eliminating the first term from (64), we get which enables us to deduce

Our next concern will be the convergence of the sequence , so we are thus looking for the convergence of the series .

For this, we use the criterion of convergence of series which gives us

Then,

As and , we have

Note that we have actually proved that the sequence defined by is convergent to an element . Having disposed of this preliminary step, the task is now to show that is a solution of problems (47)–(50).

In order to get this result, it will be necessary to check that satisfied From (57), we get

As is linear, we get

Applying the Cauchy Schwartz inequality on , we get where is Poincare’s inequality constant. On the other hand, as

So

Letting we find

From (80) and going to the limit in (76), we obtain

Theorem 12. If the condition (51) is satisfied and So problems (47)–(50) admit a weak solution which belongs to .

It remains to be proven that problems (47)−(50) admit a unique solution.

Theorem 13. According to (51), the solution of problems (47)−(50) is unique.

Proof. Suppose that and in are two solutions of (47)−(50), then satisfied and Or Similar analysis to that in the proof of Lemma 11, it shows that Or is the same constant of Lemma 11. As so according to (85), it comes that We conclude that in is the desired conclusion.

6. Conclusion

The study of fractional nonlinear parabolic problems is of great theoretical interest, because of new modeling with fractional derivatives which plays an important role in the description in the part of modeling in mathematics. Also, the use of the condition of Neumann in terms of integral condition which means the flux of the solution is an average gives a very good model of boundary condition. Then, there remains the numerical part as perspective in the fractional problem especially when there is a nonlinear term and integral condition of the second type.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research was supported by the Taif University Researchers Supporting Project number (TURSP-2020/77), Taif, Saudi Arabia.