Solving the Fractional Schr¨odinger Equation with Singular Initial Data in the Extended Colombeau Algebra of Generalized Functions

. Tis manuscript aims to highlight the existence and uniqueness results for the following Schr¨odinger problem in the extended Colombeau algebra of generalized functions.


Introduction
In the theory of distributions, it is well known that the multiplication of two distributions is not always well defned; 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 diferential algebra in which we can inject the space of distributions D ′ . In [2], Colombeau was able to answer to this question by replacing continuous functions with indefnitely diferentiable functions. Te thing that has allowed us to investigate nonlinear diferential 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 defned 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 defnitions to the level of equivalence classes. Te derivative operator is one of the most essential operators, and its description in this context allows us to investigate diferential equations in terms of distributions. Unfortunately, according to the defnition, it does not pose a difculty 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. Te 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][5][6][7][8]. Te study of fractional order integral and derivative operators over real and complex domains, as well as their applications, is the subject of fractional calculus. Te features of fractional derivatives make them ideal for modeling complicated systems. Te 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][10][11][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. Te 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.
Te 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.

Preliminaries
To defne the full algebra of Colombeau, for r ∈ N * we defne We denote by where Te full Colombeau algebra is defned by To solve ODE and DED with integer and fractional derivatives with initial data distributions, we need to recall the defnition of the extension of the fractional derivative in the Colombeau algebra.
We denote the set of all extended moderate functions by and the set of all extended negligible functions by 2 International Journal of Diferential Equations Te extended Colombeau algebra of generalized functions is the factor set.
Now, we give the defnitions of the fractional calculus theory.
Defnition 1 (see [17,18]). Te fractional integral is defned as follows: Te fractional derivative of order β > 0 in the Caputo sense is defned as follows: Te fractional derivative and the fractional integral are injected into the extension of the Colombeau algebra G e which is given by the formulas in the following proposition: Proposition 1 (see [14]).
We end this section by recalling the association relation on the Colombeau algebra G s . It identifes elements of u ∈ G s if they coincide in the weak limit.
Defnition 2 (see [19]). Let u 1 , u 2 ∈ G s (R n ) such that u 1,ϵ and u 2,ϵ are their representatives, respectively. We say that u 1 and u 2 are associated in G s (R n ), and we write u 1 ≈ u 2 , if for every μ ∈ D(R n ).

Main Results
Te nonlinear Schrödinger equation with singular potential and initial data is considered.
We shall use the regularization for the initial and Dirac measure. where

Existence and Uniqueness Results in the Colombeau Algebra
Theorem 1. Let equation (12) have the regularized equation: where v ϵ and u 0,ϵ are regularized of v and u 0 , respectively.

International Journal of Diferential Equations
Ten, the problem (12) has a unique solution in G s (R + × R n ). (15) (see [13]) is given by.

Proof. Te integral solution of equation
where κ n (t, x) is the heat kernel. Ten, By Gronwall inequality, Ten, there exists N > 0 such that For the frst derivative to Ten, By Gronwall inequality, By the previous step, there exist N > 0 such that 4 International Journal of Diferential Equations For the second derivative, for y j , j ∈ 1, n { }, we get Using Gronwall inequality, we obtain the following equation: International Journal of Diferential Equations By the previous step, there exist N > 0 such that Let us prove the uniqueness. Suppose that there exist two solutions u 1 , u 2 to problem (12), then we obtain the following equation: where N ϵ (t, x) ∈ ℵ s (R + × R n ), N 0,ϵ (x) ∈ ℵ s (R n ). Ten, and thus By Gronwall inequality, Ten, Ten, the problem (12) has a unique solution in G s (R + × R n ).

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 6 International Journal of Diferential Equations we prove the moderateness and the negligibility for entire and fractional derivatives to the spatial variable x.
Theorem . Let the following problem be the regularized equation of (12) such that v ϵ and u 0,ϵ are regularizations of v and u 0 , respectively. Ten, the problem (12) has a unique solution in G e (R + × R n ).
Proof. We prove only the fractional part since the entire part is already proved in Teorem 1. Te fractional derivative, D β , 0 < β < 1, is considered. Without loss of generality, the same holds for m − 1 < β < m, m ∈ N. Te fractional derivative is taken to the spatial variable to equation (12). We have Ten, By Gronwall inequality, we get From Teorem 1 and hypothesis (13), we obtain Ten, there exist N > 0 such that International Journal of Diferential Equations 7 It follows moderateness for the fractional derivatives in the space G e (R + × R n ). For uniqueness, let 0 < β < 1, and apply D β to (29) we get: So, By Gronwall inequality,