Article
Mathematics, Interdisciplinary Applications
Amanullah, Muhammad Yousaf, Salman Zeb, Mohammad Akram, Sardar Muhammad Hussain, Jong-Suk Ro
Summary: This paper investigates the use of the Hermite wavelet method (HWM) for solving 12th and 13th order boundary value problems (BVPs) of ordinary differential equations (ODEs). The proposed algorithm converts the ODEs into a system of algebraic equations, which are then solved to obtain the approximate solution. Test problems were considered and compared with exact solutions and other numerical methods in the literature, showing the HWM-based algorithm to be more accurate than the homotopy perturbation method (HPM) and the differential transform method (DTM).
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2023)
Article
Mathematics
Josep Vicent Cordoba, Marius Josep Alfonso
Summary: This work presents techniques for solving Initial Value Problems in a System of Ordinary Differential Equations (ODE). These techniques are applicable to adaptive step-size numerical methods, such as the Runge-Kutta-Fehlberg algorithm. By introducing an objective function and resizing local errors, catastrophic cancellations can be eliminated and integration stability can be ensured. These techniques are effective in solving physical problems, such as the numerical resolution of Lemaitre-Tolman-Bondi space-time solutions of Einstein Equations.
Article
Multidisciplinary Sciences
Kairat Usmanov, Batirkhan Kh Turmetov, Kulzina Zh Nazarova
Summary: In this paper, a multipoint boundary value problem for systems of integro-differential equations with involution is studied, and the parameterization method is used to solve the problem. By decomposing the problem into the Cauchy problem and a system of linear equations based on the parameterization method, necessary and sufficient conditions for the unique solvability of the problem are determined.
Article
Mathematics, Applied
Mohammad Esmael Samei, Lotfollah Karimi, Mohammed K. A. Kaabar
Summary: In this research, we analyze a multi-singular pointwise defined fractional q-integrodifferential equation under certain boundary conditions using the Riemann-Liouville q-integral and Caputo fractional q-derivatives. New existence results are obtained based on the alpha-admissible map and fixed point theorem for alpha-psi-contraction map. Finally, we provide an example with application and algorithms to illustrate the primary effects.
Article
Mechanics
Giovanni Formica, Franco Milicchio, Walter Lacarbonara
Summary: An enhanced pathfollowing strategy utilizing a modified Newton-Raphson solver accelerated by a low-cost routine through a Krylov subspace iterator has been proposed recently. The computation of the Jacobian of the Poincare map, serving as the iteration matrix, is a key aspect of this numerical strategy that governs the accuracy and robustness of the entire approach. In this study, a new technique is introduced to assemble the monodromy matrix by performing a central finite difference of the vector field associated with the ODE problems within the time-step integration, aiming to mitigate errors caused by numerical differentiation. Extensive numerical experiments on meaningful multi-dof dynamical systems demonstrate that the enhanced pathfollowing scheme achieves higher accuracy, robustness, and improved convergence properties compared to the standard approach. The C++ code implementing the proposed methodology, including both the classic and new approaches for computing the Jacobian matrix, is freely available at https://zenodo.org/record/ 7245478.
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS
(2023)
Article
Mathematics, Applied
Higinio Ramos, Gurjinder Singh
Summary: This article presents an optimized third-derivative hybrid block method for solving second-order two-point boundary value problems. The method is developed using a purely interpolation and collocation approach, with consideration of optimal point selection. It produces an approximate solution over the entire integration interval and demonstrates good performance.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Engineering, Multidisciplinary
Mubashir Qayyum, Qursam Fatima, Syed Tauseef Saeed, Ali Akgul, Wajaree Weera, Wedad R. Alharbi
Summary: In this paper, an extended residual power series method (RPSM) is proposed to solve ordinary differential equations with boundary conditions. The reliability of the proposed methodology is tested by comparing the results with other schemes for different boundary value problems (BVPs). The analysis shows that the extended approach can handle BVPs easily and provide a convergent series solution without the need for discretization, perturbation, or linearization. Therefore, RPSM is a better choice for scientists and researchers working in various fields of engineering and sciences.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Mathematics, Applied
Akira Imakura, Keiichi Morikuni, Akitoshi Takayasu
Summary: This paper proposes operator analogues of Sakurai-Sugiura-type complex moment-based eigensolvers for computing partial eigenpairs of differential eigenvalue problems (DEPs). By using higher-order complex moments, the computational costs of solving a large number of ordinary differential equations (ODEs) can be reduced without sacrificing accuracy. Numerical results show that the proposed methods are much faster and still maintain high accuracy compared to the operator analogue of FEAST. This study is important for promoting the solve-then-discretize paradigm for solving DEPs and improving solutions in real-world applications.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Yang Wang, Francesco Topputo
Summary: This study introduces a homotopy method based on the Theory of Functional Connections (TFC), which implicitly defines infinite homotopy paths and selects the most promising ones to solve zero-finding problems. A two-layer continuation algorithm is designed, retaining the simplicity of direct continuation while allowing flexible path switching.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
A. K. B. Chand, K. R. Tyada, M. A. Navascues
Summary: Fractal interpolation functions (FIFs) are functions that can capture the irregularity or smoothness of a function. This work proposes the use of cubic spline FIFs to solve a two-point boundary value problem involving a complicated non-smooth function. Moments are computed and used to construct the cubic fractal spline solution, capturing the non-smooth nature of the problem. The convergence of the proposed method is proven through truncation error analysis.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Correction
Mathematics, Applied
Paul W. Eloe, Johnny Henderson, Jeffrey T. Neugebauer
Summary: This paper corrects an error in a previously published article and presents a new theorem.
ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Alec J. Linot, Joshua W. Burby, Qi Tang, Prasanna Balaprakash, Michael D. Graham, Romit Maulik
Summary: Careful consideration is needed in capturing the dynamics of high wavenumbers in data-driven modeling of spatiotemporal phenomena, especially when shocks or chaotic dynamics are present. To address this challenge, a new architecture called stabilized neural ordinary differential equation (ODE) is proposed, which accurately captures shocks and chaotic dynamics. By combining the outputs of two neural networks, one learning the linear term and the other the nonlinear term, the proposed architecture learns the right-hand-side (RHS) of the ODE. Experimental results on the viscous Burgers equation and the Kuramoto-Sivashinsky equation demonstrate that stabilized neural ODEs outperform standard neural ODEs in short-time tracking, prediction of energy spectrum, robustness to noisy initial conditions, and long-time trajectory keeping on the attractor.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics
Angelamaria Cardone, Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster
Summary: The present paper illustrates the use of multivalue methods for the numerical solution of ordinary and fractional differential equations. Two-step and mixed collocation methods for ordinary differential equations, as well as two-step spline collocation methods for fractional differential equations, are discussed in detail. The paper reports on the construction, convergence, and stability analysis of these methods, and presents numerical experiments to demonstrate their efficiency.
Article
Mathematics, Applied
Eleonora Amoroso, Gabriele Bonanno, Giuseppina D'Agui, Salvatore De Caro, Salvatore Foti, Donal O'Regan, Antonio Testa
Summary: This paper focuses on the study of a second-order differential equation of type Sturm-Liouville with coefficients sign changing, and it obtains the existence of one positive solution by requiring a specific growth of the nonlinearity. In particular, this study is useful in analyzing the dynamical performance of a class of power converter.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Health Care Sciences & Services
Philipp Wendland, Colin Birkenbihl, Marc Gomez-Freixa, Meemansa Sood, Maik Kschischo, Holger Froehlich
Summary: Individual organizations face limitations in collecting representative disease population data, hindering the generalization ability of statistical models and scientific insights. Legal restrictions impede data sharing across organizations, and existing federated data access concepts are challenging to implement. This study proposes a multimodal AI approach called MultiNODEs, which generates realistic synthetic patient trajectories on a continuous time scale, enabling smooth interpolation and extrapolation of clinical studies.
NPJ DIGITAL MEDICINE
(2022)
Article
Mathematics, Applied
Winfried Auzinger, Thomas Kassebacher, Othmar Koch, Mechthild Thalhammer
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2017)
Article
Mathematics, Applied
Winfried Auzinger, Othmar Koch
APPLIED MATHEMATICS LETTERS
(2018)
Article
Mathematics, Applied
Chris Budd, Othmar Koch, Leila Taghizadeh, Ewa Weinmueller
Article
Computer Science, Interdisciplinary Applications
Winfried Auzinger, Iva Brezinova, Harald Hofstaetter, Othmar Koch, Michael Quell
COMPUTER PHYSICS COMMUNICATIONS
(2019)
Article
Mathematics, Applied
Winfried Auzinger, Harald Hofstaetter, Othmar Koch, Michael Quell, Mechthild Thalhammer
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2019)
Article
Mathematics, Applied
Harald Hofstaetter, Othmar Koch
NUMERISCHE MATHEMATIK
(2019)
Article
Mathematics, Applied
Othmar Koch
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
(2019)
Article
Computer Science, Software Engineering
Tobias Jawecki, Winfried Auzinger, Othmar Koch
BIT NUMERICAL MATHEMATICS
(2020)
Article
Mathematics, Applied
Winfried Auzinger, Harald Hofstaetter, Othmar Koch
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Pierluigi Amodio, Chris J. Budd, Othmar Koch, Vivi Rottschafer, Giuseppina Settanni, Ewa Weinmueller
PHYSICA D-NONLINEAR PHENOMENA
(2020)
Article
Mathematics, Applied
Winfried Auzinger, Harald Hofstaetter, Othmar Koch
APPLIED MATHEMATICS LETTERS
(2019)
Article
Mathematics, Applied
Winfried Auzinger, Harald Hofstaetter, Othmar Koch, Karolina Kropielnicka, Pranav Singh
APPLIED MATHEMATICS AND COMPUTATION
(2019)
Article
Optics
Ralf Wanzenboeck, Stefan Donsa, Harald Hofstaetter, Othmar Koch, Peter Schlagheck, Iva Brezinova
Summary: The study investigates the chaos phenomenon in the mean-field limit of a bosonic quantum many-body system, demonstrating that the system rapidly loses coherence with a rate determined by the Lyapunov exponent, which in turn affects the visibility of interference fringes.
Proceedings Paper
Computer Science, Interdisciplinary Applications
Winfried Auzinger, Alexander Grosz, Harald Hofstaetter, Othmar Koch
LARGE-SCALE SCIENTIFIC COMPUTING (LSSC 2019)
(2020)
Proceedings Paper
Computer Science, Theory & Methods
Harald Hofstaetter, Winfried Auzinger, Othmar Koch
COMPUTER ALGEBRA IN SCIENTIFIC COMPUTING (CASC 2019)
(2019)