Solving the Fractional Schrödinger Equation with Singular Initial Data in the Extended Colombeau Algebra of Generalized Functions

El Mfadel, Ali; Melliani, Said; Taqbibt, Abdellah; Elomari, M’hamed

doi:https://doi.org/10.1155/2023/3493912

International Journal of Differential Equations

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 3493912 | https://doi.org/10.1155/2023/3493912

Solving the Fractional Schrödinger Equation with Singular Initial Data in the Extended Colombeau Algebra of Generalized Functions

Ali El Mfadel,^1,2Said Melliani,¹Abdellah Taqbibt,¹and M’hamed Elomari¹

Academic Editor: Andrea Scapellato

Received21 Jul 2022

Revised12 Oct 2022

Accepted07 Nov 2022

Published02 May 2023

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 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 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 define the full algebra of Colombeau, for we define

We denote bywhere

The full Colombeau algebra is defined by

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.

We denote the set of all extended moderate functions byand the set of all extended negligible functions by

The extended Colombeau algebra of generalized functions is the factor set.

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 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 , thenWe 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

The nonlinear Schrödinger equation with singular potential and initial data is considered.

We shall use the regularization for the initial and Dirac measure.where .

3.1. Existence and Uniqueness Results in the Colombeau Algebra

Theorem 1. Let equation (12) have the regularized equation:where and are regularized of and , respectively.

Then, the problem (12) has a unique solution in .

Proof. The integral solution of equation (15) (see [13]) is given by.where is the heat kernel.
Then,By Gronwall inequality,Then, there exists such thatFor the first derivative to , we obtainThen,By Gronwall inequality,By the previous step, there exist such thatFor the second derivative, for , we getSo,Using Gronwall inequality, we obtain the following equation:By the previous step, there exist such thatLet us prove the uniqueness. Suppose that there exist two solutions to problem (12), then we obtain the following equation:where . Then,and thusBy 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 .

Theorem 2. Let the following problembe the regularized equation of (12) such that and are regularizations of and , respectively. Then, the problem (12) has a unique solution in .

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 haveThen,By Gronwall inequality, we getFrom Theorem 1 and hypothesis (13), we obtainThen, there exist such thatIt 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 obtainwhich completes the proof.

3.3. Association of Solutions

Let be the solution of the problem.and be a solution of the problem.

Proposition 2. The generalized solution of the problem (12) is associated with .

Proof. Let be the classical solution to the problem.and be the classical solution to the problem.Then,The integral solution isSo,Thanks to the Gronwall lemma, we getby passing to the limit, and we obtainwhich completes the proof.

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.

Data Availability

The data generated or analyzed to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2023 Ali El Mfadel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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