Review
Computer Science, Artificial Intelligence
Nurettin Dogan, Selami Bayeg, Raziye Mert, Oemer Akin
Summary: In this work, the authors introduce singularly perturbed fuzzy initial value problems (SPFIVPs) and present an algorithm for solving them using the extension principle proposed by Zadeh. They also provide some results on the behavior of the α-cuts of the solutions. To demonstrate the robustness of the algorithm, they fuzzify some examples from the literature and apply the algorithm.
EXPERT SYSTEMS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Zhongdi Cen, Jian Huang, Aimin Xu
Summary: This paper considers a system of singularly perturbed initial value problems with weak constrained conditions on the coefficients. The system is transformed into a system of first-order singularly perturbed problems with integral terms, and a hybrid difference method is utilized to approximate the transformed system. A posteriori error analysis for the discretization scheme on an arbitrary mesh is presented, and a solution-adaptive algorithm based on a posteriori error estimation is devised. Numerical experiments show the uniform convergence behavior of second-order for the scheme.
Article
Mathematics, Applied
Gung-Min Gie, Chang-Yeol Jung, Hoyeon Lee
Summary: In this study, new semi-analytic time differencing methods were implemented for singularly perturbed non-linear initial value problems. By introducing correctors to improve the classical integrating factor and exponential time differencing methods, better approximations for stiff problems were achieved. Numerical simulations confirmed the effectiveness of the new enriched schemes compared to classical methods without correctors.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
S. Malek
Summary: This paper examines a family of linear partial differential equations that are singularly perturbed in a complex parameter c and in complex time t at the origin. The solutions of these equations are built up by means of Laplace transforms and Fourier inverse integral, and the asymptotic expansions with respect to t and epsilon are investigated.
RESULTS IN MATHEMATICS
(2022)
Article
Mathematics
Robert Vrabel
Summary: In this paper, the paper establishes sufficient conditions for the uniform convergence of solutions to a reduced problem for a singularly perturbed differential equation. The analysis is done for four different types of boundary conditions using the notion of (I-q)-stability and the method of lower and upper solutions. Additionally, the structure of the solutions of the reduced problem is analyzed using the Peano phenomenon.
Article
Mathematics
Hidetoshi Tahara
Summary: In this paper, the initial value problem for singularly perturbed nonlinear holomorphic differential equations with a small parameter is studied in a complex domain. The asymptotic behavior of solutions with respect to the parameter is investigated. It is shown that a formal power series solution can be constructed, and this solution is proven to be summable in a suitable direction. This guarantees the existence of a holomorphic solution that has a Gevrey type asymptotic expansion.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Multidisciplinary Sciences
Gemechis File Duressa
Summary: The article presents a novel numerical method for solving singularly perturbed boundary value problems with negative shift parameter, providing accurate solutions in the inner region of the boundary layer at a second order rate of convergence.
Article
Mathematics, Applied
Carlo Marcati, Maxim Rakhuba, Johan E. M. Ulander
Summary: This paper investigates the rank bounds on the quantized tensor train (QTT) compressed approximation for singularly perturbed reaction diffusion boundary value problems. It shows that the QTT-compressed solutions converge rapidly to the exact solution, regardless of the scale of the singular perturbation parameter. The paper also addresses the known stability issues of the QTT-based solution by adapting a preconditioning strategy.
Article
Mathematics, Applied
Devendra Kumar, Parvin Kumari
Summary: A numerical scheme is proposed for problems with small diffusion and convection parameters where the convection and source terms have a jump discontinuity. Theoretical error bounds for the singular and regular components of the solution are obtained through rigorous analysis, showing uniform convergence regardless of the sizes of the parameters epsilon(1) and epsilon(2). Two test problems are used to validate the theoretical results.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Hao Zhang, Na Wang
Summary: This paper discusses a class of nonlinear singular perturbation problems with Robin boundary values in critical cases. By using the boundary layer function method and successive approximation theory, the corresponding asymptotic expansions of small parameters are constructed, and the existence of uniformly efficient smooth solutions is proved. Meanwhile, we give a concrete example to prove the validity of our results.
ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Robert Vrabel
Summary: In this brief note, we study the asymptotic behavior of solutions for non-resonant, singularly perturbed linear Neumann boundary value problems and propose an approach based on the analysis of an integral equation associated with this problem, with an indication of possible extension to more complex cases.
Article
Mathematics, Applied
Robert Vrabel
Summary: In this paper, sufficient conditions for the uniform convergence of the solutions of singularly perturbed Neumann boundary value problems for second-order differential equations to the solution of their reduced problems are established, using the method of lower and upper solutions and the notion of (I-q)-stability.
Article
Mathematics
Samuel Jelbart, Kristian Uldall Kristiansen, Martin Wechselberger
Summary: We study the transition of smooth systems to piecewise-smooth systems with a boundary-focus bifurcation as epsilon -> 0, and identify different bifurcation structures. We uncover the evolution characteristics of cycles associated with BF bifurcations in the smooth system, and prove the existence of a family of stable limit cycles.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Yanhui Lv, Jin Zhang
Summary: This paper considers the two-parameter singularly perturbed problems and proposes a balanced norm to capture each exponential layer. The uniform convergence and optimal order of the finite element method on a Shishkin mesh are proved using this norm.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Interdisciplinary Applications
Jin Zhang, Xiaowei Liu
Summary: In this article, we analyze the convergence of a weak Galerkin method on Bakhvalov-type mesh. By carefully defining the parameters and interpolant, the method is proved to have optimal convergence order.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Mathematics, Applied
Meenakshi Shivhare, Pramod Chakravarthy Podila, Higinio Ramos, Jesus Vigo-Aguiar
Summary: In this paper, we studied a time-dependent singularly perturbed differential-difference equation with small shifts in the field of neuroscience. We approximated the terms containing delay and advance parameters using Taylor's series expansion. The continuous problem was semi-discretized using the Crank-Nicolson finite difference method in the time direction and quadratic B-spline collocation method in the space direction. The method was proven to have second-order uniform convergence in both space and time directions, and theoretical estimates were carried out to support the obtained numerical experiments.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Musa Ahmed Demba, Higinio Ramos, Poom Kumam, Wiboonsak Watthayu, Norazak Senu, Idris Ahmed
Summary: In this study, a trigonometrically adapted 6(4) explicit Runge-Kutta-Nystrom pair with six stages is formulated, which can integrate the usual test equation exactly. The local truncation error and the periodicity interval of the new method are calculated, demonstrating its maintained algebraic order and almost P-stability.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Sunil Kumar, Kuldeep, Higinio Ramos, Joginder Singh
Summary: An efficient hybrid numerical method is developed for solving coupled systems of singularly perturbed linear parabolic problems, with first order convergence in time and third order convergence in space. The additive scheme helps decouple the vector approximate solution components for increased computational efficiency. Numerical results confirm theoretical convergence and demonstrate the efficiency of the proposed strategy.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Chemistry, Multidisciplinary
Higinio Ramos, Mufutau Ajani Rufai
Summary: This study introduces a new one-step method with three intermediate points to solve stiff differential systems. The method utilizes interpolation and collocation techniques and adopts an embedded strategy to improve performance. Finally, the proposed technique is evaluated through numerical solutions of various initial-value problems, showcasing its performance and efficiency in real-world applications.
JOURNAL OF MATHEMATICAL CHEMISTRY
(2023)
Article
Mathematics, Applied
Mufutau Ajani Rufai, Francesca Mazzia, Higinio Ramos
Summary: This research presents an adaptive optimized one-step Nystrom method for solving second-order initial value problems of ODEs and time-dependent partial differential equations. The method employs a new approach for selecting collocation points and utilizes an embedding-like procedure to estimate the error. Numerical experiments demonstrate that the introduced error estimation and step-size control strategy outperform other existing numerical methods.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Computer Science, Interdisciplinary Applications
Saurabh Tomar, Mehakpreet Singh, Kuppalapalle Vajravelu, Higinio Ramos
Summary: This paper presents a novel method for calculating the Lagrange multiplier, which improves the efficiency of the variational iteration method in solving nonlinear problems.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Morteza Bisheh-Niasar, Higinio Ramos
Summary: This paper presents an efficient numerical approach for solving first-order delay differential equations with a piece-wise constant delay. The approach is based on a five-point hybrid block method designed for ordinary differential equations. Interpolation technique is utilized to evaluate delay terms at grid points. The paper investigates the method's characteristics such as zero stability, local truncation errors, convergence, and stability region. Numerical experiments demonstrate the efficiency and accuracy of the proposed method compared to other existing methods in the literature.
DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Ram Shiromani, Vembu Shanthi, Higinio Ramos
Summary: This article investigates a class of singularly perturbed two-dimensional steady-state convection-diffusion problems with Robin boundary conditions. The solutions to these problems exhibit regular boundary layers and corner layers. A numerical approach is used to provide a good approximation of the exact solutions using a finite-difference technique with a layer-adapted piecewise-uniform Shishkin mesh. Numerical examples demonstrate the accuracy of the approximations and their agreement with theoretical results.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics
Manikandan Mariappan, Chandru Muthusamy, Higinio Ramos
Summary: This article develops and analyzes a numerical scheme to solve a singularly perturbed parabolic system of n reaction-diffusion equations. The scheme considers m equations with a perturbation parameter and the rest without it. It uses finite difference approximations on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable. Convergence properties and error analyses are derived, and numerical experiments are presented to support the theoretical results.
Article
Mathematics
Mufutau Ajani Rufai, Higinio Ramos
Summary: This research article introduces an efficient method for integrating Lane-Emden-Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nystrom technique with a set of formulas applied at the initial step to circumvent the singularity at the beginning of the interval. The variable step-size formulation is implemented using an embedded-type approach, resulting in an efficient technique that outperforms its counterpart methods that used fixed step-size implementation. The numerical simulations confirm the better performance of the variable step-size implementation.
Article
Mathematics, Applied
R. I. Abdulganiy, H. Ramos, J. A. Osilagun, S. A. Okunuga, Sania Qureshi
Summary: This research presents a fourth-order convergent functionally-fitted block hybrid Falkner method based on interpolation and collocation for solving the approximate solution of the Kepler equations and related problems. The proposed method achieves high stability and accuracy.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Multidisciplinary Sciences
Higinio Ramos, Mufutau Ajani Rufai, Bruno Carpentieri
Summary: This paper introduces an efficient approach for solving Lane-Emden-Fowler problems, which utilizes two Nystrom schemes for integration and generates simultaneous approximations by solving an algebraic system of equations, outperforming existing numerical methods.
Article
Mathematics, Interdisciplinary Applications
Mufutau Ajani Rufai, Bruno Carpentieri, Higinio Ramos
Summary: This paper presents a new hybrid block method for solving initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method uses an adaptive stepsize strategy to control the estimated error, and numerical simulations show that it is more efficient than other existing numerical methods.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Interdisciplinary Applications
Ahmed Mohamed Hassan, Higinio Ramos, Osama Moaaz
Summary: The present article aims to study the oscillatory properties of a class of second-order dynamic equations on time scales. The noncanonical case, which received less attention compared to the canonical dynamic equations, is considered in this study. The approach adopted involves converting the noncanonical equation to a corresponding canonical equation. By using this transformation and employing various techniques, new, more effective, and sharp oscillation criteria are created. The effectiveness and importance of the results are explained by applying them to specific cases of the studied equation.
FRACTAL AND FRACTIONAL
(2023)
Article
Computer Science, Interdisciplinary Applications
Nikhil Sriwastav, Amit K. Barnwal, Higinio Ramos, Ravi P. Agarwal, Mehakpreet Singh
Summary: In this article, a novel approach based on the shooting projection method and the Legendre wavelet operational matrix formulation is proposed for approximating a class of two-point singular boundary value problems (SBVPs) with Dirichlet and Neumann-Robin boundary conditions. The approach involves making an initial guess in contrast to the boundary conditions, solving the initial value problem using the Legendre wavelet operational matrix method, and iteratively improving the initial condition using a shooting projection method until the desired accuracy is achieved.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2024)
