Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

open access articleMotivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations

A Parametric Method Optimised for the Solution of the (2+1)-Dimensional Nonlinear Schrödinger Equation

open access articleWe investigate the numerical solution of the nonlinear Schrödinger equation in two spatial dimensions and one temporal dimension. We develop a parametric Runge–Kutta method with four of their coefficients considered as free parameters, and we provide the full process of constructing the method and the explicit formulas of all other coefficients. Consequently, we produce an adaptable method with four degrees of freedom, which permit further optimisation. In fact, with this methodology, we produce a family of methods, each of which can be tailored to a specific problem. We then optimise the new parametric method to obtain an optimal Runge–Kutta method that performs efficiently for the nonlinear Schrödinger equation. We perform a stability analysis, and utilise an exact dark soliton solution to measure the global error and mass error of the new method with and without the use of finite difference schemes for the spatial semi-discretisation. We also compare the efficiency of the new method and other numerical integrators, in terms of accuracy versus computational cost, revealing the superiority of the new method. The proposed methodology is general and can be applied to a variety of problems, without being limited to linear problems or problems with oscillatory/periodic solutions

Identifying Parkinson’s Disease Through the Classification of Audio Recording Data

Developments in artificial intelligence can be leveraged to support the diagnosis of degenerative disorders, such as epilepsy and Parkinson’s disease. This study aims to provide a software solution, focused initially towards Parkinson’s disease, which can positively impact medical practice surrounding degenerative diagnoses. Through the use of a dataset containing numerical data representing acoustic features extracted from an audio recording of an individual, it is determined if a neural approach can provide an improvement over previous results in the area. This is achieved through the implementation of a feedforward neural network and a layer recurrent neural network. By comparison with the state-of-the-art, a Bayesian approach providing a classification accuracy benchmark of 87.1%, it is found that the implemented neural networks are capable of average accuracy of 96%, highlighting improved accuracy for the classification process. The solution is capable of supporting the diagnosis of Parkinson’s disease in an advisory capacity and is envisioned to inform the process of referral through general practice

A Neural Network for Interpolating Light-Sources

This study combines two novel deterministic methods with a Convolutional Neural Network to develop a machine learning method that is aware of directionality of light in images. The first method detects shadows in terrestrial images by using a sliding-window algorithm that extracts specific hue and value features in an image. The second method interpolates light-sources by utilising a line-algorithm, which detects the direction of light sources in the image. Both of these methods are single-image solutions and employ deterministic methods to calculate the values from the image alone, without the need for illumination-models. They extract real-time geometry from the light source in an image, rather than mapping an illumination-model onto the image, which are the only models used today. Finally, those outputs are used to train a Convolutional Neural Network. This displays greater accuracy than previous methods for shadow detection and can predict light source-direction and thus orientation accurately, which is a considerable innovation for an unsupervised CNN. It is significantly faster than the deterministic methods. We also present a reference dataset for the problem of shadow and light direction detection. © 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

Numerical solution of differential equations using special methods

In the present thesis we examine systems of first-order ordinary differential equations with oscillating solutions which are integrated numerically. For their solution we use explicit Runge-Kutta methods that integrate exactly a set of special functions. The thesis consists of two main parts. In the first part we work on the properties of phase-lag and dissipation. At first we construct a family of methods with fifth algebraic order and constant or variable coefficients with maximized or infinite order of phase-lag respectively. We test these methods along with a group of classical methods in five known problems with oscillating solutions. Afterwards we construct a method with fifth algebraic order, infinite order of phase-lag and infinite order of dissipation and compare it with other methods during the integration of three known orbital problems. In the second part we study the radial one-dimensional time-independent Schrodinger equation and construct two families of exponentially-fitted methods for its integration. The first family consists of two methods with fifth algebraic order and first and second exponential order, while the second consists of three methods with sixth algebraic order and first, second and third exponential order. After the analysis of the local truncation error of all methods we stress the critical role of the maximum exponent of energy in the error. We note the increase of the efficiency of the exponentially-fitted methods in comparison to the classical ones and especially when using high values of energy.Στην παρούσα διδακτορική διατριβή εξετάζουμε συστήματα πρωτοβάθμιων συνήθων διαφορικών εξισώσεων με λύση ταλαντωτικής μορφής, τα οποία ολοκληρώνουμε αριθμητικά. Για την επίλυσή τους χρησιμοποιούμε άμεσες μεθόδους Runge-Kutta που ολοκληρώνουν ακριβώς ένα σύνολο από ειδικές συναρτήσεις. Η διατριβή αποτελείται από δύο βασικά μέρη. Στο πρώτο μέρος ασχολούμαστε με τις ιδιότητες της υστέρησης φάσης και της απώλειας. Αρχικά παράγουμε μια οικογένεια μεθόδων πέμπτης αλγεβρικής τάξης με σταθερούς ή μεταβλητούς συντελεστές που έχουν αντίστοιχα μεγιστοποιημένη ή άπειρη τάξη υστέρησης φάσης. Τις μεθόδους αυτές μαζί με μια ομάδα κλασικών μεθόδων τις δοκιμάζουμε σε πέντε γνωστά προβλήματα με ταλαντωτική λύση. Στη συνέχεια κατασκευάζουμε μια μέθοδο με πέμπτη αλγεβρική τάξη, άπειρη τάξη υστέρησης φάσης και άπειρη τάξη απώλειας την οποία συγκρίνουμε με άλλες μεθόδους κατά την ολοκλήρωση τριών γνωστών τροχιακών προβλημάτων. Στο δεύτερο μέρος της διατριβής μελετούμε την ανεξάρτητη του χρόνου μονοδιάστατη ακτινική εξίσωση Schrodinger και κατασκευάζουμε δύο οικογένειες εκθετικά προσαρμοσμένων μεθόδων για την ολοκλήρωσή της. Η πρώτη οικογένεια αποτελείται από δύο μεθόδους πέμπτης αλγεβρικής τάξης και πρώτης και δεύτερης εκθετικής, ενώ η δεύτερη οικογένεια αποτελείται από τρεις μεθόδους έκτης αλγεβρικής τάξης και πρώτης, δεύτερης και τρίτης εκθετικής. Μετά από ανάλυση του τοπικού σφάλματος αποκοπής όλων των μεθόδων επισημαίνουμε τον κρίσιμο ρόλο που παίζει ο μέγιστος εκθέτης της ενέργειας στο σφάλμα. Παρατηρούμε την αύξηση της αποδοτικότητας των εκθετικά προσαρμοσμένων μεθόδων σε σχέση με τις κλασικές και ειδικά για μεγάλες τιμές της ενέργειας

Explicit Almost P-Stable Runge-Kutta-Nyström Methods for the Numerical Solution of the Two-Body Problem

The Publisher's final version can be found by following the DOI link.In this paper, three families of explicit Runge–Kutta–Nyström methods with three stages and third algebraic order are presented. Each family consists of one method with constant coefficients and one corresponding optimized “almost” P-stable method with variable coefficients, zero phase-lag and zero amplification error. The firstmethod with constant coefficients is new, while the second and third have been constructed by Chawla and Sharma. The newmethod with constant coefficients, constructed in this paper has larger interval of stability than the two methods of Chawla and Sharma. Furthermore, the optimized methods possess an infinite interval of periodicity, excluding some discrete values, while being explicit, which is a very desired combination. The preservation of the algebraic order is examined, local truncation error and stability/periodicity analysis are performed and the efficiency of the new methods is measured via the integration of the two-body problem

A 6(4) optimized embedded Runge–Kutta–Nyström pair for the numerical solution of periodic problems

The Publisher's final version can be found by following the DOI link.In this paper an optimization of the non-FSAL embedded RKN 6(4) pair with six stages of Moawwad El-Mikkawy, El-Desouky Rahmo is presented. The new method is derived after applying phase-fitting and amplification-fitting and has variable coefficients. The preservation of the algebraic order is verified and the principal term of the local truncation error is evaluated. Furthermore, periodicity analysis is performed, which reveals that the new method is ‘‘almost’’ P-stable. The efficiency of the new method is measured via the integration of several initial value problems

A variable step-size implementation of the hybrid Nyström method for integrating Hamiltonian and stiff differential systems

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.The approximate solution to second-order Hamiltonian and stiff differential systems is obtained using an efficient hybrid Nyström method (HNM) in this manuscript. The development of the method considers three hybrid points that are selected by optimizing the local truncation errors of the main formulas. The properties of the proposed HNM are studied. An embedding-like procedure is explored and run in variable step-size mode to improve the accuracy of the HNM. The numerical integration of some second-order Hamiltonian and stiff model problems, such as the well-known Vander Pol, Fermi-Pasta-Ulam, and Duffing problems, demonstrate the improved impact of our devised error estimation and control strategy. Finally, it is essential to note that the proposed technique is efficient in terms of computational cost and maximum global errors