Solving the Fractional Schrödinger Equation with Singular Initial Data in the Extended Colombeau Algebra of Generalized Functions
Abstract
This manuscript aims to highlight the existence and uniqueness results for the following Schrödinger problem in the extended Colombeau algebra of generalized functions. where δ is the Dirac distribution. The proofs of our main results are based on the Gronwall inequality and regularization method. We conclude our article by establishing the association concept of solutions.
1. Introduction
In the theory of distributions, it is well known that the multiplication of two distributions is not always well defined; for more details, see [1]. However, until 1980, this operation was not used in a rigorous way. Following, Schwarz’s renowned impossibilities relating to the build of a commutative, associative, and differential algebra in which we can inject the space of distributions . In [2], Colombeau was able to answer to this question by replacing continuous functions with indefinitely differentiable functions. The thing that has allowed us to investigate nonlinear differential equations formed from natural occurrences with parameters in the sense of distributions in a rational manner; for that, see [1]. We are talking about an algebra that is defined as the quotient of the algebra of moderate functions by the algebra of negligible functions. So, it should come as no surprise that the following presentations’ main topic will be to raise componentwise definitions to the level of equivalence classes. The derivative operator is one of the most essential operators, and its description in this context allows us to investigate differential equations in terms of distributions. Unfortunately, according to the definition, it does not pose a difficulty for generalized functions, which is not the case for the fractional derivative. In [3], Mirjana presents a method for dealing with fractional derivatives including singularities based on Colombeau’s idea of algebras of generalized functions. The Colombeau algebra of generalized functions is extended to fractional derivatives by the same author. It is used to solve ODEs and PDEs with entire and fractional derivatives in terms of temporal and spacial variables; see [4–8]. The study of fractional order integral and derivative operators over real and complex domains, as well as their applications, is the subject of fractional calculus. The features of fractional derivatives make them ideal for modeling complicated systems. The ordinary derivative is a local operator, whereas fractional operators are nonlocal. Fractional derivatives exhibit nonlocal dynamics as a result of this, i.e. the process dynamics have some memory; for more information, see [9–12]. When the nonlinear term is a L∞loc-function that does not satisfy the Lipschitz condition, the nonlinear Schrödinger equation with singular potential and initial data is studied in [13]. Recently, in [14], Benmerrous et al. established the existence and uniqueness of solutions for the Schrödinger equation with singular potential and initial data in the Colombeau algebra. The reader can consult articles as well [15, 16] and the references therein for more details on the fractional Schrödinger equation.
Motivated by the above works precisely by [14], we study the existence and uniqueness results for the Schrödinger equation with singular initial data in the extended Colombeau algebra of generalized functions. Also, we give the association of solutions.
The present article is organized as follows: In Section 2, we recall some fundamental properties of the generalized function theory. Section 3 is consecrated for the proofs of the existence and uniqueness of solutions for the Schrödinger problem (12) in the Colombeau algebra and the extended Colombeau algebra. We conclude this article by proving the association of solutions.
2. Preliminaries
To solve ODE and DED with integer and fractional derivatives with initial data distributions, we need to recall the definition of the extension of the fractional derivative in the Colombeau algebra.
Now, we give the definitions of the fractional calculus theory.
Definition 1. (see [17, 18]). The fractional integral is defined as follows:
The fractional derivative of order β > 0 in the Caputo sense is defined as follows:
The fractional derivative and the fractional integral are injected into the extension of the Colombeau algebra which is given by the formulas in the following proposition:
Proposition 1. (see [14]). Let , then
We end this section by recalling the association relation on the Colombeau algebra . It identifies elements of if they coincide in the weak limit.
Definition 2. (see [19]). Let such that and are their representatives, respectively. We say that and are associated in , and we write , if for every .
3. Main Results
3.1. Existence and Uniqueness Results in the Colombeau Algebra
Theorem 1. Let equation (12) have the regularized equation:
Then, the problem (12) has a unique solution in .
Proof. The integral solution of equation (15) (see [13]) is given by.
Then,
By Gronwall inequality,
Then, there exists N > 0 such that
For the first derivative to xi, i ∈ {1, n}, we obtain
By Gronwall inequality,
By the previous step, there exist N > 0 such that
For the second derivative, for yj, j ∈ {1, n}, we get
So,
Using Gronwall inequality, we obtain the following equation:
By the previous step, there exist N > 0 such that
Let us prove the uniqueness. Suppose that there exist two solutions to problem (12), then we obtain the following equation:
By Gronwall inequality,
Then,
Then, the problem (12) has a unique solution in .
3.2. Existence and Uniqueness Results in the Extension of the Colombeau Algebra
In a framework of extended algebra of generalized functions, we prove the existence uniqueness result for nonlinear Schrödinger equations with singular potential and initial data and an equation driven by the fractional derivative of the delta distribution. It means that we prove the moderateness and the negligibility for entire and fractional derivatives to the spatial variable x.
Theorem 2. Let the following problem
Proof. We prove only the fractional part since the entire part is already proved in Theorem 1. The fractional derivative, , is considered. Without loss of generality, the same holds for . The fractional derivative is taken to the spatial variable to equation (12). We have
Then,
By Gronwall inequality, we get
From Theorem 1 and hypothesis (13), we obtain
Then, there exist N > 0 such that
It follows moderateness for the fractional derivatives in the space . For uniqueness, let 0 < β < 1, and apply to (29) we get:
So,
By Gronwall inequality,
Using Theorem 1, we obtain
3.3. Association of Solutions
Proposition 2. The generalized solution u of the problem (12) is associated with w1 + w2.
Proof. Let w1,ε be the classical solution to the problem.
Then,
The integral solution is
So,
Thanks to the Gronwall lemma, we get
4. Conclusion
In this article, we studied the existence and uniqueness results for the Schrödinger equation with distribution type initial conditions in the extended Colombeau algebra of generalized functions. The existence theorem is proved by using some regularizations of the proposed problem and Gronwall inequality.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Open Research
Data Availability
The data generated or analyzed to support the findings of this study are included within the article.
