Mathematics › Numerical Analysis

Numerical methods for differential equations

Works Count: 59531

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Description

This cluster of papers focuses on the development and analysis of numerical integration methods for solving differential equations, with a particular emphasis on exponential integrators, symplectic methods, time-stepping schemes, and variational integrators. The research covers a wide range of applications including the numerical solution of the Schrödinger equation, Hamiltonian systems, and matrix exponential computations.

Keywords

Numerical Integration; Differential Equations; Exponential Integrators; Symplectic Methods; Time-Stepping Schemes; Variational Integrators; Matrix Exponential; Runge-Kutta Methods; Schrödinger Equation; Hamiltonian Systems

Functional Analysis and Semi-groups

1996-02-06

Einar Hille , Robert A. Phillips

Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: … Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: Subadditive functions Semi-modules Addition theorem in a Banach algebra Semi-groups in the strong topology Generator and resolvent Generation of semi-groups Part Three. Advanced Analytical Theory of Semi-Groups: Perturbation theory Adjoint theory Operational calculus Spectral theory Holomorphic semi-groups Applications to ergodic theory Part Four. Special Semi-groups and Applications: Translations and powers Trigonometric semi-groups Semi-groups in $L_p(-\infty,\infty)$ Semi-groups in Hilbert space Miscellaneous applications Part Five. Extensions of the theory: Notes on Banach algebras Lie semi-groups Functions on vectors to vectors Bibliography Index.

Interpolation of Operators

1988-01-01

Curtis D. Bennett , M Sharpley

An Introduction to Infinite-Dimensional Linear Systems Theory

1995-01-01

Ruth F. Curtain , Hans Zwart

Computational Design of the Basic Dynamical Processes of the UCLA General Circulation Model

1977-01-01

Akio Arakawa , V. R. Lamb

Theory and Applications of Partial Functional Differential Equations

1996-01-01

Jian Wu

Partial Differential Equations

2010-03-02

Lawrence Evans

Differential Equations, Dynamical Systems, and Linear Algebra

1974-01-01

Morris W. Hirsch , Stephen T. Smale

Ordinary Differential Equations.

1971-12-01

Ray Redheffer , Jack K. Hale

Applied numerical analysis

2005-06-01

Curtis F. Gerald , Patrick O. Wheatley

The fifth edition of this book continues teaching numerical analysis and techniques. Suitable for students with mathematics and engineering backgrounds, the breadth of topics (partial differential equations, systems of nonlinear … The fifth edition of this book continues teaching numerical analysis and techniques. Suitable for students with mathematics and engineering backgrounds, the breadth of topics (partial differential equations, systems of nonlinear equations, and matrix algebra), provide comprehensive and flexible coverage of numerical analysis.

Solving Ordinary Differential Equations II

1996-01-01

Ernst Hairer , Gerhard Wanner

DIFFERENTIAL-DIFFERENCE EQUATIONS

1987-02-01

Kenneth L. Cooke

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Lectures on Cauchy's Problem in Linear Partial Differential Equations

1953-08-01

Jacques Hadamard , Philip Μ. Morse

On the exponential solution of differential equations for a linear operator

1954-11-01

Wilhelm Magnus

Solving ordinary differential equations I. nonstiff problems

1987-10-01

Ernst Hairer , Syvert P. Nørsett , Gerhard Wanner

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Maximum Principles in Differential Equations

1984-01-01

Murray H. Protter , Hans F. Weinberger

Theory of Ordinary Differential Equations

1956-02-01

Earl A. Coddington , Norman Levinson , T. Teichmann

The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by … The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable. It has been developed from courses given by the authors and probably contains more material than will ordinarily be covered in a one-year course. It is hoped that the book will be a useful text in the application of differential equations as well as for the pure mathematician.

Discrete Variable Methods in Ordinary Differential Equations.

1962-11-01

A. B. Farnell , Peter Henrici

A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method

1993-06-01

Jaeung Chung , Gregory M. Hulbert

A new family of time integration algorithms is presented for solving structural dynamics problems. The new method, denoted as the generalized-α method, possesses numerical dissipation that can be controlled by … A new family of time integration algorithms is presented for solving structural dynamics problems. The new method, denoted as the generalized-α method, possesses numerical dissipation that can be controlled by the user. In particular, it is shown that the generalized-α method achieves high-frequency dissipation while minimizing unwanted low-frequency dissipation. Comparisons are given of the generalized-α method with other numerically dissipative time integration methods; these results highlight the improved performance of the new algorithm. The new algorithm can be easily implemented into programs that already include the Newmark and Hilber-Hughes-Taylor-α time integration methods.

Strong Stability-Preserving High-Order Time Discretization Methods

2001-01-01

Sigal Gottlieb , Chi‐Wang Shu , Eitan Tadmor

In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD … In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

The MATLAB ODE Suite

1997-01-01

L. F. Shampine , Mark W. Reichelt

This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.MSC codes65L0665L0565Y9934A65Keywordsordinary differential equationsstiff systemsBDFGear methodRosenbrock methodnonstiff systemsRunge--Kutta methodAdams methodsoftware This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.MSC codes65L0665L0565Y9934A65Keywordsordinary differential equationsstiff systemsBDFGear methodRosenbrock methodnonstiff systemsRunge--Kutta methodAdams methodsoftware

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Geometrical Methods in the Theory of Ordinary Differential Equations

1988-01-01

V. I. Arnold

A family of embedded Runge-Kutta formulae

1980-03-01

John R. Dormand , P.J. Prince

Group Analysis of Differential Equations.

1985-02-01

Willard Miller , Л.В. Овсянников

Numerical Initial Value Problems in Ordinary Differential Equations

1973-07-01

E. I. , C. W. Gear

Construction of higher order symplectic integrators

1990-11-01

Haruo Yoshida

Improved numerical dissipation for time integration algorithms in structural dynamics

1977-07-01

Hans M. Hilber , Thomas J.R. Hughes , Robert L. Taylor

Abstract A new family of unconditionally stable one‐step methods for the direct integration of the equations of structural dynamics is introduced and is shown to possess improved algorithmic damping properties … Abstract A new family of unconditionally stable one‐step methods for the direct integration of the equations of structural dynamics is introduced and is shown to possess improved algorithmic damping properties which can be continuously controlled. The new methods are compared with members of the Newmark family, and the Houbolt and Wilson methods.

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SUNDIALS

2005-09-01

A.C. Hindmarsh , Peter N. Brown , Keith Eric Grant +4 more

SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential … SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.

Numerical Initial Value Problems in Ordinary Differential Equations

1972-05-01

James Watt

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Foundations of Optimal Control Theory.

1969-01-01

D. M. G. Wishart , E. B. Lee , Lawrence Markus

Abstract : This complete and authoritative presentation of the current status of control theory offers a useful foundation for both study and research. With emphasis on general nonlinear differential systems, … Abstract : This complete and authoritative presentation of the current status of control theory offers a useful foundation for both study and research. With emphasis on general nonlinear differential systems, the book is carefully and systematically developed from elementary motivating examples, through the most comprehensive theory, to the final numerical solution of serious scientific and engineering control problems. The book features reviews of the most recent researches on processes described by partial differential equations, functional- differential, and delay-differential equations; the most recent treatment of impulse controllers, bounded rate controllers, feedback controllers, and bounded phase problems; and many unpublished new research results of the authors. In addition to an exhaustive treatment of the quantitative problems of optimal control, the qualitative concepts of stability, controllability, observability, and plant recognition receive a complete exposition. (Author)

Symmetries and Differential Equations

1989-01-01

George W. Bluman , Sukeyuki Kumei

Stochastic Differential Equations: Theory and Applications.

1976-01-01

P. Whittle , Ludwig Arnold

Ordinary Differential Equations

1945-10-01

A. T. Lonseth , E. L. Ince

Mathematical methods of classical mechanics

1978-01-01

Владимир Игоревич Арнольд

Dummy Endogenous Variables in a Simultaneous Equation System

1977-05-01

James J. Heckman

This paper considers the formulation and estimation of simultaneous equation models with both discrete and continuous endogenous variables.The statistical model proposed here is sufficiently rich to encompass the álassjcai simultaneous … This paper considers the formulation and estimation of simultaneous equation models with both discrete and continuous endogenous variables.The statistical model proposed here is sufficiently rich to encompass the álassjcai simultaneous equation model for continuous endogenous variables and more recent models for purely discrete endogenous variables as special cases of a more general model.Interest in discrete data has been ftsledby a rapid growth in the availability of microeconomic data sets coupled with a growing awareness of the importance of discrete choice models for the analysis of uiicroeconomic problems (see McFadden, 1976).To date, the only available statistical models for the analysis of discrete endogenous variables have been developed for the purely discrete case.The log-linear or logistic model of Goodman (1970) as expanded by Raberman (1974) and Nerlove and Press (1976) is one Vol.II, 1967; Lord and Novick, cbs.16-20, 1967.)It is argued in this paper that this class of statistical models provides a natural framework for generating simultaneous equation models with both discrete and continuous random variables.In contrast, the framework of Goodman, while convenient for formulating descriptive models for discrete data, offers a much less natural apparatus for analyzing econometric structural equation models.This is so primarily because the simultaneous equation model is inherently an unconditional representation of behavioral equations while the model of Goodman is designed to facilitate the analysis of conditional representations, and does not lend itself to the unconditional formulations required in simultaneous equation theory.The structure of this paper is in four parts.in part one general models are discussed.Dummy endogenous variables are introduced in two distinct roles: (1) as proxies for unobserved latent variables and (2) as direct shifters of behavioral equations.Five models incorporating such dummy variables are discussed.Part two, also the longest section, presents a complete analysis of the most novel and most general of the five models presented in part one.This is a model with both continuous and discrete endogenous variables.The issues of identification and estimation are discussed together by proving the existence of consistent estimators.Maximum likelihood estimators and alternative estimators are discussed.In part three, a brief discussion of a multivariate probit model with structural shift is presented.Part four presents a comparison between the models developed in this paper and the models of Goodman and Nerlove and Press.

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Numerical Solution of Partial Differential Equations, Finite Difference Methods.

1987-04-01

V. T. , G. D. Smith

Ordinary Differential Equations

2017-10-12

John M. Stewart

This handbook is the fourth volume in a series of volumes devoted to self contained and up-to-date surveys in the theory of ordinary differential equations, with an additional effort to … This handbook is the fourth volume in a series of volumes devoted to self contained and up-to-date surveys in the theory of ordinary differential equations, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience. It covers a variety of problems in ordinary differential equations. It provides pure mathematical and real world applications. It is written for mathematicians and scientists of many related fields.

Introduction to Functional Differential Equations

1993-01-01

Jack K. Hale , Sjoerd M. Verduyn Lunel

On the Partial Difference Equations of Mathematical Physics

1967-03-01

Richard Courant , Kurt Friedrichs , Hans Lewy

Problems involving the classical linear partial differential equations of mathematical physics can be reduced to algebraic ones of a very much simpler structure by replacing the differentials by difference quotients … Problems involving the classical linear partial differential equations of mathematical physics can be reduced to algebraic ones of a very much simpler structure by replacing the differentials by difference quotients on some (say rectilinear) mesh. This paper will undertake an elementary discussion of these algebraic problems, in particular of the behavior of the solution as the mesh width tends to zero. For present purposes we limit ourselves mainly to simple but typical cases, and treat them in such a way that the applicability of the method to more general difference equations and to those with arbitrarily many independent variables is made clear.

Applications of Lie Groups to Differential Equations

1993-01-01

Peter J. Olver

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Differential Equations with Discontinuous Righthand Sides

1988-01-01

Alexander Filippov

Nonlinear semigroups and differential equations in Banach spaces

1976-01-01

Viorel Barbu

A generalization of Browder’s Theorem to parametric DND-condensing mappings

2025-06-26

G. García | Rendiconti del Circolo Matematico di Palermo Series 2

High-Order Adams–Bashforth Methods for First-Order IVPs with Oscillating Solutions

2025-06-25

Mei Hong , Chia‐Liang Lin , Theodore E. Simos | Mediterranean Journal of Mathematics

Hybrid continuous multi-step method for second order problems in ordinary differential equations

2025-06-24

Oluwasayo Esther Taiwo , N. S. Yakusak , Muideen O. Ogunniran | Istanbul Journal of Mathematics

Deep neural network approach to solve coupled differential equations with multiple delays

2025-06-24

Mohd Noor , Muslim Malik | Physica Scripta

Abstract This paper proposes a scientific machine learning approach based on Physics Informed Neural Networks (PINNs) to solve the delay differential equations (DDEs) of higher order with multiple daely, neutral … Abstract This paper proposes a scientific machine learning approach based on Physics Informed Neural Networks (PINNs) to solve the delay differential equations (DDEs) of higher order with multiple daely, neutral DDEs (NDDEs) and system of DDEs. Using optimization algorithms and automatic differentiation, we modify the network parameters in PINN to minimize the loss function. As a result, the delay differential equations (DDEs) can be numerically solved without the grid dependency and polynomial interpolation that come with conventional numerical techniques. Numerous numerical examples have demonstrated our method's high precision in a variety of challenges, demonstrating its efficacy and great potential in addressing various DDEs.

Stochastic parareal algorithm for stochastic differential equations

2025-06-23

H. P. Wang , Jing Lyu , Zhigang Peng +1 more | Numerical Algorithms

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Explicit radial basis function Runge–Kutta methods

2025-06-23

Jiaxi Gu , Xinjuan Chen , Jae-Hun Jung | Numerical Algorithms

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High order boundary value linear multistep method for the numerical solution of IVPs in ODEs

2025-06-21

S. E. Ogunfeyitimi , Monday Ikhile , P O Olatunji | Journal of the Nigerian Society of Physical Sciences

In this paper, we introduce High order boundary value linear multistep method (HOBVLMM) for the numerical solution of stiff systems of initial value problems (IVPs). The order, error constant, zero … In this paper, we introduce High order boundary value linear multistep method (HOBVLMM) for the numerical solution of stiff systems of initial value problems (IVPs). The order, error constant, zero stability and the region of absolute stability for the HOBVLMM are discussed. The proposed scheme posses 0k,k-1 -stability and (Ak,k-1 )-stability, achieving a high order of p = 2k - 1, where k represents the step number of the LMM. The methods prove to be effective for stiff systems of IVPs in ordinary differential equations (ODEs), as evidenced by our numerical experiments, which shows superior performance compared to some existing methods.

Perimetric contraction principle on quadrilaterals in semimetric spaces with triangle functions

2025-06-20

Ravindra K. Bisht | Acta Mathematica Academiae Scientiarum Hungaricae

Combining the Least-Squares Method with Touchard Polynomials for Solving Mixed Integro-Differential Equation

2025-06-20

Hameeda Oda Al-Humedi , Zahraa Adnan Jameel | Baghdad Science Journal

Dynamic Modeling and Analysis of Geometrically Exact Beams Based on Geometric Algebra

2025-06-20

Qingli Li , Ye Ding | Journal of Computational and Nonlinear Dynamics

Abstract In this paper, we explore a geometric algebra-based approach to model the dynamics of a geometrically exact beam with large deflections. The configuration space of the beam is parametrized … Abstract In this paper, we explore a geometric algebra-based approach to model the dynamics of a geometrically exact beam with large deflections. The configuration space of the beam is parametrized with geometric algebra Cl3,0,1, which indicates the rotation and translation fields are interpolated in a coupled manner. We derive the dynamic equations by applying the Lagrangian method and employing a full discretization approach. This methodology is wellsuited for modeling the large deflection deformations of the beam while preserving the symplectic structure in the long-term simulations. Finally, we validate the proposed approach through some numerical examples, demonstrating its computational effciency compared to matrix-based methods.

DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH BOUNDARY TERMS

2025-06-19

Zhanar Artykbayeva , Baltabek Kanguzhin | International Journal of Mathematics and Physics

In the referenced study, a differential equation (DE) exhibiting a hybrid structure is examined. The principal objective of this manuscript is to determine the feasibility of substituting the given supplementary … In the referenced study, a differential equation (DE) exhibiting a hybrid structure is examined. The principal objective of this manuscript is to determine the feasibility of substituting the given supplementary boundary conditions with alternative equivalent conditions. This is achieved through the establishment and proof of four theorems, providing a rigorous foundation for the proposed substitutions. Incipiently, the existing (3) conditions are considered in a nonhomogeneous context. Subsequently, new conditions, denoted as (7), are introduced. These newly formulated conditions are demonstrated to be equivalent to the original ones, ensuring the unique solvability of the hybrid-structured system labeled as (1). The system under consideration is characterized as hybrid due to the presence of both unknown and algebraic components. This dual nature necessitates a nuanced approach to boundary condition formulation and analysis. The methodology employed in this study underscores the importance of flexibility in boundary condition specification, particularly in complex or hybrid configurations. By establishing the equivalence of different boundary conditions, article provides valuable insights into the solvability and analysis of such frameworks. Furthermore, the study meticulously details prior research in this domain, delineating the specific conditions and configurations previously explored. This comprehensive review situates the current manuscript within the broader context of hybrid DE analysis. Key words: boundary, hybrid, dissipative, DAE, BVP.

SOLVING THE NEGATIVE ORDER KORTEWEG-DE VRIES EQUATION WITH A SELF-CONSISTENT SOURCE CORRESPONDING TO MOVING EIGENVALUES

2025-06-19

G. U. Urazboev , I. I. Baltaeva | International Journal of Apllied Mathematics

A kind of adaptive variable stepsize embedded Runge–Kutta pairs coupled with the Sinc collocation method for solving the KdV equation

2025-06-16

Cheng Chen , Wenting Shao | Results in Applied Mathematics

Insight to Multi-derivative Hybrid Linear Multistep Formula for directly Solving Third-order Initial Value Problems of Ordinary Differential Equations

2025-06-16

E. A. Areo , Kolade M. Owolabi , A. L. Momoh +1 more | Journal of Mathematical Analysis and Modeling

This article focuses on the multi-derivative hybrid linear multistep formula (MHLMF) for the numericalsolution of third-order ordinary differential equations. Power series was used as the basis function in thederivation of … This article focuses on the multi-derivative hybrid linear multistep formula (MHLMF) for the numericalsolution of third-order ordinary differential equations. Power series was used as the basis function in thederivation of the formula. An approximate solution from the basis function was interpolated at some selected off-grid points. In contrast, the third derivative of the approximate solution was located at all points in the grid and outside the grid to generate a system of linear equations to determine the unknown parameters. The derived method was examined to be consistent, convergent, and zero stable. The method was implemented to solve third order ordinary differential equations, including the Genesio equation, demonstrating that the derived methods efficiently handle nonlinear problems. Absolute errors obtained in the numerical experiments established the good performance of the proposed method when compared with other cited methods in the literature.

Highly accurate compact difference schemes for multidimensional delay Schrödinger equations

2025-06-16

Allaberen Ashyralyev , Deniz Ağırseven , Baris Erköse | Georgian Mathematical Journal

Numerical Stability Analysis of Implicit Solution Methods for Harmonic Balance Equations

2025-06-16

Yuxuan Zhang , Dingxi Wang , Sen Zhang +1 more | Journal of Turbomachinery

Abstract This paper investigates the numerical stability of various implicit solution methods for efficiently solving harmonic balance equations for turbomachinery unsteady flows. Those solution methods implicitly integrate the time spectral … Abstract This paper investigates the numerical stability of various implicit solution methods for efficiently solving harmonic balance equations for turbomachinery unsteady flows. Those solution methods implicitly integrate the time spectral source term to enhance stability and accelerate convergence. They include the block Jacobi method(BJ), the Jacobi iteration method(JI), and their variants, i.e. the modified block Jacobi method(MBJ) and the modified Jacobi iteration method(MJI). These implicit treatments are typically combined with the lower upper symmetric Gauss-Seidel method(LU-SGS) as a preconditioner of a Runge-Kutta scheme. The von Neumann analysis is applied to evaluate the stability and damping properties of all these methods. The findings reveal that the LU-SGS/BJ and LU-SGS/MJI schemes allow larger Courant numbers, in the order of hundreds, significantly improving convergence speed, while the LU-SGS/MBJ and LU-SGS/JI schemes fail to stabilize the solution, resulting in a Courant number below 10 as the grid-reduced frequency increases. The influence of Jacobi iterations on stabilization is also investigated. It is found that the minimum allowable relaxation factor does not change monotonically with the number of Jacobi iterations. Typically 2-4 Jacobi iterations are suggested for stability and efficiency, while more than 4 offers no benefit. The stability analysis results are verified by solving the harmonic balance equation system for two cases: inviscid flow over a 2-D bump with a pressure disturbance at the outlet and turbulent flow in a 3-D transonic compressor stage.

Numerical third-order Hopf form for ordinary differential equations

2025-06-13

Dávid András Horváth , Tamás Kalmár‐Nagy | Nonlinear Dynamics

Generalized enright-type symmetric methods for the solution of boundary value problems

2025-06-12

T. Okor , GC Nwachukwu | Journal of Computational and Applied Mathematics

Construction and Application of Some Optimal Algorithms of Multistep Type

2025-06-12

Vagif Ibrahimov , Mehri̇ban İmanova | WSEAS TRANSACTIONS ON SYSTEMS

Many problems in natural science are reduced to solving integral equations, among which the Volterra integral equation occupies a significant place. As is known, the Volterra integral equations are represented … Many problems in natural science are reduced to solving integral equations, among which the Volterra integral equation occupies a significant place. As is known, the Volterra integral equations are represented in two forms, which are called the Volterra integral equations of the first and second kind. Given that there is a transition from one kind of integral equation to another, here consider to investigation of the Volterra integral equation of the second kind. There are many approximate methods for solving Volterra integral equations of the second kind. Here, have studied the numerical solution of the integral equation of the Volterra type. For this aim, here is a recommendation using the multistep methods with constant coefficients, which are very popular in solving initial-value problems for ODEs of the first order. Some authors for the construction of multistep methods have used the quadrature methods which are used to calculate definite integrals. Here by showing the disadvantages of the quadrature methods, suggested constructing the multistep methods for solving the Volterra integral equations of the second kind. For the construction of multistep methods with the best properties, suggest using advanced multistep and multistep second derivative methods. By defining the maximum order of accuracy for the stable methods of the above-mentioned types, here have recommended some optimal methods and by using them have constructed an optimal algorithm for solving the Volterra integral equations.

Ordinary and calibrated differential operators, connections and application to curvilinear webs

2025-06-02

Daniel Lehmann | Proceedings of the International Geometry Center

Abelian relations of a curvilinear web are the solutions of the partial differential equation defined by a certain differential operator that we prove to be always "ordinary" and "calibrated". Thus … Abelian relations of a curvilinear web are the solutions of the partial differential equation defined by a certain differential operator that we prove to be always "ordinary" and "calibrated". Thus we begin by explaning why the space of solutions of the partial differential equation defined by such an ordinary and calibrated homogeneous linear differential operator (of arbitrary order) is isomorphic to the space of the sections of a certain vector bundle with vanishing covariant derivative for a certain tautological connection. We then apply this result to the case of webs by curves, recovering the upper-bound for the rank of such a web given in [D. Damiano, PhD Thesis, Brown University, 1986] та [D. Damiano, Amer. J. Math. (1983) 105:6, 1325-1345], and defining finally the "curvature" of the web which vanishes iff the web has maximal rank.

An efficient procedure to compute the continuous Baker–Campbell–Hausdorff formula

2025-06-02

Ana Arnal , Fernando Casas , Cristina Chiralt | Applied Mathematics and Computation

Simulating images of radio galaxies with diffusion models (Corrigendum)

2025-06-01

T. Vičánek Martínez , Nicolás Barón Pérez , M. Brüggen | Astronomy and Astrophysics

A Computational Approach to Developing Two-Derivative ODE-Solving Formulations: γβI-(2+3)P Method

2025-06-01

Mehdi Babaei | Journal of Computational Science

On discontinuous differential equations and the method of solution regions

2025-06-01

Jorge Rodríguez López | Journal of Mathematical Analysis and Applications

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Acceleration of implicit schemes for large systems of delay differential equations

2025-06-01

Mouhamad Al Sayed Ali , Miloud Sadkane | Journal of Computational and Applied Mathematics

PositiveIntegrators.jl: A Julia library of positivity-preserving time integration methods

2025-05-31

Stefan Kopecz , Joshua Lampert , Hendrik Ranocha | The Journal of Open Source Software

Efficient 5-Point Block Method for Oscillatory ODEs with Phase Lag and Amplification Error Control

2025-05-30

Rubayyi T. Alqahtani , Anurag Kaur , Theodore E. Simos

This research presents innovative modified explicit block methods with fifth-order algebraic accuracy to address initial value problems (IVPs). The derivation of the methods employs fitting coefficients that eliminate phase lag … This research presents innovative modified explicit block methods with fifth-order algebraic accuracy to address initial value problems (IVPs). The derivation of the methods employs fitting coefficients that eliminate phase lag and amplification error, as well as their derivatives. A thorough stability analysis of the new approach is conducted. Comparative assessments with existing methods highlight the superior effectiveness of the proposed algorithms. Numerical tests verify that this technique significantly surpasses conventional methods for solving IVPs, particularly those exhibiting oscillatory solutions.

An Ohta–Kawasaki model set on the space

2025-05-29

Lorena Aguirre Salazar , Xin Lü , Jian Wei

Abstract We examine a nonlocal diffuse interface energy with Coulomb repulsion in three dimensions inspired by the Thomas–Fermi–Dirac–von Weizsäcker, and the Ohta–Kawasaki models. We consider the corresponding mass‐constrained variational problem … Abstract We examine a nonlocal diffuse interface energy with Coulomb repulsion in three dimensions inspired by the Thomas–Fermi–Dirac–von Weizsäcker, and the Ohta–Kawasaki models. We consider the corresponding mass‐constrained variational problem and show the existence of minimizers for small masses, and the absence of minimizers for large masses.

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New stability conditions for the discrete-time and continuous-time linear Roesser models

2025-05-29

Tadeusz Kaczorek

New simple sufficient conditions of the asymptotic stability of the discrete-time and continuous-time linear 2D Roesser models are proposed. The effectiveness of the new conditions are demonstrated on simple numerical … New simple sufficient conditions of the asymptotic stability of the discrete-time and continuous-time linear 2D Roesser models are proposed. The effectiveness of the new conditions are demonstrated on simple numerical examples.

SOLUCIÓN A LA ECUACIÓN DE ONDA PROPAGADA RADIALMENTE PARA EL CAMPO ELÉCTRICO EN COORDENADAS CILÍNDRICAS

2025-05-29

E. A. ZARATE , Mateo Márquez Arias , I. Jiménez +1 more

An Efficient A-Stable Linear Multistep Hybrid Block Method for Solving Fifth-Order Initial Value Problems in Ordinary Differential Equations

2025-05-29

Duromola Kolawole

Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and … Furthering research on linear multistep hybrid block methods is essential to enhance accuracy, stability, and efficiency in solving ordinary differential equations, enabling advanced modelling of complex systems across science and engineering for better predictive analysis and real-world applications. In this study, an efficient linear multistep hybrid block method with a single-step and seven off-step points for the direct numerical integration of fifth-order initial value problems (IVPs) in ordinary differential equations (ODEs), eliminating the need for reduction to a system of first-order ODEs is proposed. The method is constructed using a collocation approach at both grid and off-grid points, alongside interpolation at five off-grid points, to approximate the solution via a power series polynomial. The resulting system of equations is solved to obtain the necessary discrete and additional formulae that constitute the block approach. A comprehensive theoretical analysis confirms that the method possesses desirable numerical properties, including a well-defined order, zero stability, consistency, convergence, and absolute stability. Comparative numerical experiments against existing methods demonstrate that the proposed approach achieves superior accuracy and efficiency, making it a promising tool for solving both linear and nonlinear fifth-order ODEs.

Development of an Efficient Optimized Hybrid One Step Methods for Solving Fourth-Order Initial Value Problems (IVPs)

2025-05-29

D. Raymond , Pantuvo T.P. , Masanari Kida +1 more

This paper examines the development of a one-step optimized block schemes for solving-fourth order initial value problems (IVP) of ordinary differential equations. Interpolation and collocation techniques are considered using approximated … This paper examines the development of a one-step optimized block schemes for solving-fourth order initial value problems (IVP) of ordinary differential equations. Interpolation and collocation techniques are considered using approximated power series as the basis function and its fourth derivative as the collocating equation. Two hybrid points and are chosen to achieve optimized errors. The method's properties are analyzed and it is zero stable, consistent and convergent. The application of the method to some fourth-order initial value problems shows the derived method performed better in terms of accuracy and effectiveness; outperforming some existing methods when compared.

Partial differential equations

2025-05-28

Chul Ki Song , Jong-Ryeol Kim

DEVELOPMENT OF A CONTINUOUS BACKWARD DIFFERENTIATION FORMULAE FOR SOLVING FIRST-ORDER AND SECOND-ORDER INITIAL VALUE PROBLEM

2025-05-28

Mahmoud Muhammad Yahaya , A Abubakar , Yusuf Dauda Jikantoro

Numerical methods for solving ordinary differential equations (ODEs) are essential in modeling dynamical systems across science and engineering. While specialized methods exist for first-order and second-order ODEs, developing a unified … Numerical methods for solving ordinary differential equations (ODEs) are essential in modeling dynamical systems across science and engineering. While specialized methods exist for first-order and second-order ODEs, developing a unified approach that efficiently handles both classes remain an active area of research. In this paper, we present a novel two-step hybrid block method based on the backward differentiation formula (BDF), capable of approximating solutions for both first- and second-order ODEs without requiring separate derivations. The method is constructed using interpolation and collocation techniques, and its numerical analysis confirms consistency, zero-stability, and convergence. Furthermore, stability analysis via the general linear method demonstrates that the scheme is A-stable, making it suitable for stiff systems. Numerical experiments including applications to the SIR epidemic model, Riccati differential equations, nonlinear stiff chemical systems, and second-order nonlinear ODEs—validate the method’s accuracy and computational efficiency. Comparative results with existing methods in the literature highlight its superior performance in terms of error reduction and stability. This work contributes a versatile, high-precision tool for ODE solutions, bridging gaps in the adaptability of traditional BDF-based approaches.