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Fractal Fract
Special Issues
Frontiers in Fractional Schrödinger Equation
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Frontiers in Fractional Schrödinger Equation

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 December 2021) | Viewed by 7888

Special Issue Editor

Prof. Dr. Muhammad Bhatti

E-Mail Website
Guest Editor
Department of Physics and Astronomy, University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
Interests: fractional Schrödinger Equation and its application to quantum systems; computational physics and mathematical physics; fractional-order circuits and systems

Special Issue Information

Dear Colleagues,

The fields of fractional-order quantum mechanics and systems combine concepts from fractional calculus into their modeling and design. These projects, focused on non-integer order differentiation and integration of mathematical operations, are discovered across many fields of science and engineering. Concentrating on their incorporation into quantum mechanical and electronic circuits, these ideas are being discovered to design oscillators, circuits, and control systems.

Application of B-poly basis set is used to solve fractional-order differential equations (FDEs) that modeled Schrödinger equation and electric circuits using fractional-order polynomials (α as the fractional-order of polys). The method accomplishes the desired solution in terms of continuous finite number of generalized fractional B-polys in an interval [a, b].

The focus of this research is to advance research on topics relating to the theory, project design, and application of fractional-order quantum systems and circuits. Topics of research include:

  • Fractional order Forced Harmonic Oscillator (FHO);
  • Fractional-order Schrödinger and Dirac equations and prediction of compacted hydrogenic orbitals (so called hydrino particles which may lead to dark matter);
  • Fractional-order quantum mechanical theory, commutators;
  • Exploration of atomic and molecular structure using fractional-order theory;
  • Applications of fractional-order circuit models for energy storage elements, supper capacitors and batteries.

Prof. Dr. Muhammad Bhatti
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (3 papers)

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Research

26 pages, 2885 KiB  
Article
Chirped Periodic and Solitary Waves for Improved Perturbed Nonlinear Schrödinger Equation with Cubic Quadratic Nonlinearity
by Aly R. Seadawy, Syed T. R. Rizvi and Saad Althobaiti
Fractal Fract. 2021, 5(4), 234; https://doi.org/10.3390/fractalfract5040234 - 19 Nov 2021
Cited by 10 | Viewed by 1730
Abstract
In this paper, we study the improved perturbed nonlinear Schrödinger equation with cubic quadratic nonlinearity (IPNLSE-CQN) to describe the propagation properties of nonlinear periodic waves (PW) in fiber optics. We obtain the chirped periodic waves (CPW) with some Jacobi elliptic functions (JEF) and [...] Read more.
In this paper, we study the improved perturbed nonlinear Schrödinger equation with cubic quadratic nonlinearity (IPNLSE-CQN) to describe the propagation properties of nonlinear periodic waves (PW) in fiber optics. We obtain the chirped periodic waves (CPW) with some Jacobi elliptic functions (JEF) and also obtain some solitary waves (SW) such as dark, bright, hyperbolic, singular and periodic solitons. The nonlinear chirp associated with each of these optical solitons was observed to be dependent on the pulse intensity. The graphical behavior of these waves will also be displayed. Full article
(This article belongs to the Special Issue Frontiers in Fractional Schrödinger Equation)
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19 pages, 3173 KiB  
Article
Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases
by Muhammad I. Bhatti and Md. Habibur Rahman
Fractal Fract. 2021, 5(4), 208; https://doi.org/10.3390/fractalfract5040208 - 11 Nov 2021
Cited by 2 | Viewed by 2606
Abstract
A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, [...] Read more.
A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, minimization of error in the coefficients was found by employing the Galerkin method. Before the Galerkin method was applied, the linear FPDE was transformed into an operational matrix equation that was inverted to provide the values of the unknown coefficients in the approximate solution. A valid multidimensional solution was determined when an appropriate number of basis sets and fractional-order of B-polys were chosen. In addition, initial conditions were applied to the operational matrix to seek proper solutions in multidimensions. The technique was applied to four examples of linear FPDEs and the agreements between exact and approximate solutions were found to be excellent. The current technique can be expanded to find multidimensional fractional partial differential equations in other areas, such as physics and engineering fields. Full article
(This article belongs to the Special Issue Frontiers in Fractional Schrödinger Equation)
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15 pages, 2420 KiB  
Article
Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases
by Muhammad I. Bhatti, Md. Habibur Rahman and Nicholas Dimakis
Fractal Fract. 2021, 5(3), 106; https://doi.org/10.3390/fractalfract5030106 - 31 Aug 2021
Cited by 7 | Viewed by 2767
Abstract
A multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys). To approximate the anticipated solution of the NPD equation, a linear product of variable coefficients ai(t) and [...] Read more.
A multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys). To approximate the anticipated solution of the NPD equation, a linear product of variable coefficients ai(t) and Bi(x) B-polys has been employed. Additionally, the variable quantities in the anticipated solution are determined using the Galerkin method for minimizing errors. Before the minimization process is to take place, the NPDE is converted into an operational matrix equation which, when inverted, yields values of the undefined coefficients in the expected solution. The nonlinear terms of the NPDE are combined in the operational matrix equation using the initial guess and iterated until converged values of coefficients are obtained. A valid converged solution of NPDE is established when an appropriate degree of B-poly basis is employed, and the initial conditions are imposed on the operational matrix before the inverse is invoked. However, the accuracy of the solution depends on the number of B-polys of a certain degree expressed in multidimensional variables. Four examples of NPDE have been worked out to show the efficacy and accuracy of the two-dimensional B-poly technique. The estimated solutions of the examples are compared with the known exact solutions and an excellent agreement is found between them. In calculating the solutions of the NPD equations, the currently employed technique provides a higher-order precision compared to the finite difference method. The present technique could be readily extended to solving complex partial differential equations in multivariable problems. Full article
(This article belongs to the Special Issue Frontiers in Fractional Schrödinger Equation)
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