ref_ODE.bib

@book{agarwalUniquenessNonuniquenessCriteria1993,
  title = {Uniqueness and {{Nonuniqueness Criteria}} for {{Ordinary Differential Equations}}},
  author = {Agarwal, R. P. and Lakshmikantham, V.},
  year = {1993},
  month = mar,
  publisher = {WSPC},
  address = {Singapore ; River Edge, NJ},
  url = {https://www.worldscientific.com/worldscibooks/10.1142/1988},
  isbn = {978-981-02-1357-2},
  langid = {english}
}

@book{ascherComputerMethodsOrdinary1998,
  title = {Computer {{Methods}} for {{Ordinary Differential Equations}} and {{Differential-Algebraic Equations}}},
  author = {Ascher, Uri M. and Petzold, Linda R.},
  year = {1998},
  month = jul,
  publisher = {{SIAM: Society for Industrial and Applied Mathematics}},
  address = {Philadelphia},
  abstract = {Designed for those people who want to gain a practical knowledge of modern techniques, this book contains all the material necessary for a course on the numerical solution of differential equations. Written by two of the field's leading authorities, it provides a unified presentation of initial value and boundary value problems in ODEs as well as differential-algebraic equations. The approach is aimed at a thorough understanding of the issues and methods for practical computation while avoiding an extensive theorem-proof type of exposition. It also addresses reasons why existing software succeeds or fails. This is a practical and mathematically well informed introduction that emphasizes basic methods and theory, issues in the use and development of mathematical software, and examples from scientific engineering applications. Topics requiring an extensive amount of mathematical development are introduced, motivated, and included in the exercises, but a complete and rigorous mathematical presentation is referenced rather than included.},
  isbn = {978-0-89871-412-8},
  langid = {english}
}

@book{ascherNumericalMethodsEvolutionary2008,
  title = {Numerical {{Methods}} for {{Evolutionary Differential Equations}}},
  author = {Ascher, Uri M.},
  year = {2008},
  month = jan,
  series = {Computational {{Science}} \& {{Engineering}}},
  publisher = {SIAM},
  abstract = {Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applications such as fluid flow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical finance. This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. Audience: suitable for researchers and graduate students from a variety of fields including computer science, applied mathematics, physics, earth and ocean sciences, and various engineering disciplines. Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method.},
  googlebooks = {\_bodx5uHkuYC},
  isbn = {978-0-89871-891-1},
  langid = {english}
}

@article{bilesCaratheodoryConditionsInitial1997,
  title = {On {{Carath{\'e}odory}}'s Conditions for the Initial Value Problem},
  author = {Biles, D. and Binding, P.},
  year = {1997},
  journal = {Proceedings of the American Mathematical Society},
  volume = {125},
  number = {5},
  pages = {1371--1376},
  issn = {0002-9939, 1088-6826},
  doi = {10.1090/S0002-9939-97-03942-7},
  url = {https://www.ams.org/proc/1997-125-05/S0002-9939-97-03942-7/},
  urldate = {2023-11-02},
  abstract = {Advancing research. Creating connections.},
  langid = {english}
}

@book{blanesConciseIntroductionGeometric2016,
  title = {A {{Concise Introduction}} to {{Geometric Numerical Integration}}},
  author = {Blanes, Sergio and Casas, Fernando},
  year = {23 kv{\v e}tna 2016},
  series = {Monographs and Research Notes in Mathematics},
  publisher = {Chapman \& Hall / CRC},
  address = {Boca Raton},
  url = {https://www.routledge.com/A-Concise-Introduction-to-Geometric-Numerical-Integration/Blanes-Casas/p/book/9781482263428},
  abstract = {A Concise Introduction to Geometric Numerical Integration presents the main themes, techniques, and applications of geometric integrators for researchers in mathematics, physics, astronomy, and chemistry who are already familiar with numerical tools for solving differential equations. It also offers a bridge from traditional training in the numerical analysis of differential equations to understanding recent, advanced research literature on numerical geometric integration. The book first examines high-order classical integration methods from the structure preservation point of view. It then illustrates how to construct high-order integrators via the composition of basic low-order methods and analyzes the idea of splitting. It next reviews symplectic integrators constructed directly from the theory of generating functions as well as the important category of variational integrators. The authors also explain the relationship between the preservation of the geometric properties of a numerical method and the observed favorable error propagation in long-time integration. The book concludes with an analysis of the applicability of splitting and composition methods to certain classes of partial differential equations, such as the Schr{\"o}dinger equation and other evolution equations. The motivation of geometric numerical integration is not only to develop numerical methods with improved qualitative behavior but also to provide more accurate long-time integration results than those obtained by general-purpose algorithms. Accessible to researchers and post-graduate students from diverse backgrounds, this introductory book gets readers up to speed on the ideas, methods, and applications of this field. Readers can reproduce the figures and results given in the text using the MATLAB{\textregistered} programs and model files available online.},
  isbn = {978-1-4822-6342-8}
}

@book{butcherNumericalMethodsOrdinary2016,
  title = {Numerical {{Methods}} for {{Ordinary Differential Equations}}},
  author = {Butcher, J. C.},
  year = {2016},
  month = aug,
  edition = {3},
  publisher = {Wiley},
  address = {Chichester, West Sussex, United Kingdom},
  url = {https://onlinelibrary.wiley.com/doi/book/10.1002/9781119121534},
  isbn = {978-1-119-12150-3},
  langid = {english}
}

@book{coddingtonTheoryOrdinaryDifferential1984,
  title = {Theory of {{Ordinary Differential Equations}}},
  author = {Coddington, Earl A. and Levinson, Norman},
  year = {1984},
  month = mar,
  edition = {1st edition},
  publisher = {McGraw-Hill},
  address = {New Delhi},
  isbn = {978-0-07-099256-6},
  langid = {english}
}

@book{griffithsNumericalMethodsOrdinary2010,
  title = {Numerical {{Methods}} for {{Ordinary Differential Equations}}: {{Initial Value Problems}}},
  shorttitle = {Numerical {{Methods}} for {{Ordinary Differential Equations}}},
  author = {Griffiths, David F. and Higham, Desmond J.},
  year = {2010},
  month = nov,
  series = {Springer {{Undergraduate Mathematics Series}}},
  publisher = {Springer},
  abstract = {Numerical Methods for Ordinary Differential Equations is a self-contained introduction to a fundamental field of numerical analysis and scientific computation. Written for undergraduate students with a mathematical background, this book focuses on the analysis of numerical methods without losing sight of the practical nature of the subject. It covers the topics traditionally treated in a first course, but also highlights new and emerging themes. Chapters are broken down into `lecture' sized pieces, motivated and illustrated by numerous theoretical and computational examples. Over 200 exercises are provided and these are starred according to their degree of difficulty. Solutions to all exercises are available to authorized instructors. The book covers key foundation topics: o Taylor series methods o Runge--Kutta methods o Linear multistep methods o Convergence o Stability and a range of modern themes: o Adaptive stepsize selection o Long term dynamics o Modified equations o Geometric integration o Stochastic differential equations The prerequisite of a basic university-level calculus class is assumed, although appropriate background results are also summarized in appendices. A dedicated website for the book containing extra information can be found via www.springer.com},
  isbn = {978-0-85729-147-9},
  langid = {english}
}

@book{hairerGeometricNumericalIntegration2006,
  title = {Geometric {{Numerical Integration}}: {{Structure-Preserving Algorithms}} for {{Ordinary Differential Equations}}},
  shorttitle = {Geometric {{Numerical Integration}}},
  author = {Hairer, Ernst and Lubich, Christian and Wanner, Gerhard},
  year = {2006},
  series = {Springer {{Series}} in {{Computational Mathematics}}},
  edition = {2},
  publisher = {Springer-Verlag},
  address = {Berlin Heidelberg},
  doi = {10.1007/3-540-30666-8},
  url = {https://www.springer.com/gp/book/9783540306634},
  urldate = {2021-04-15},
  abstract = {Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.},
  isbn = {978-3-540-30663-4},
  langid = {english}
}

@book{hairerSolvingOrdinaryDifferential1993,
  title = {Solving {{Ordinary Differential Equations I}}: {{Nonstiff Problems}}},
  shorttitle = {Solving {{Ordinary Differential Equations I}}},
  author = {Hairer, Ernst and N{\o}rsett, Syvert P. and Wanner, Gerhard},
  year = {1993},
  series = {Springer {{Series}} in {{Computational Mathematics}}},
  edition = {2},
  publisher = {Springer-Verlag},
  address = {Berlin Heidelberg},
  doi = {10.1007/978-3-540-78862-1},
  url = {https://www.springer.com/gp/book/9783540566700},
  urldate = {2021-04-15},
  abstract = {From the reviews "This is the revised version of the first edition of Vol. I published in 1987. {\dots}.Vols. I and II (SSCM 14) of Solving Ordinary Differential Equations together are the standard text on numerical methods for ODEs. ...This book is well written and is together with Vol. II, the most comprehensive modern text on numerical integration methods for ODEs. It may serve a a text book for graduate courses, ...and also as a reference book for all those who have to solve ODE problems numerically." Zeitschrift f{\"u}r Angewandte Mathematik und Physik "{\dots} This book is a valuable tool for students of mathematics and specialists concerned with numerical analysis, mathematical physics, mechanics, system engineering, and the application of computers for design and planning{\dots}" Optimization "{\dots} This book is highly recommended as a text for courses in numerical methods for ordinary differential equations and as a reference for the worker. It should be in every library, both academic and industrial." Mathematics and Computers},
  isbn = {978-3-540-56670-0},
  langid = {english}
}

@book{hairerSolvingOrdinaryDifferential1996,
  title = {Solving {{Ordinary Differential Equations II}}: {{Stiff}} and {{Differential-Algebraic Problems}}},
  shorttitle = {Solving {{Ordinary Differential Equations II}}},
  author = {Hairer, Ernst and Wanner, Gerhard},
  year = {1996},
  series = {Springer {{Series}} in {{Computational Mathematics}}, {{Springer Ser}}.{{Comp}}.{{Mathem}}. {{Hairer}},{{E}}.:{{Solving Ordinary Diff}}.},
  edition = {2},
  publisher = {Springer-Verlag},
  address = {Berlin Heidelberg},
  doi = {10.1007/978-3-642-05221-7},
  url = {https://www.springer.com/gp/book/9783540604525},
  urldate = {2021-04-15},
  abstract = {The subject of this book is the solution of stiff differential equations and of differential-algebraic systems (differential equations with constraints). There is a chapter on one-step and extrapolation methods for stiff problems, another on multistep methods and general linear methods for stiff problems, a third on the treatment of singular perturbation problems, and a last one on differential-algebraic problems with applications to constrained mechanical systems. The beginning of each chapter is of introductory nature, followed by practical applications, the discussion of numerical results, theoretical investigations on the order and accuracy, linear and nonlinear stability, convergence and asymptotic expansions. Stiff and differential-algebraic problems arise everywhere in scientific computations (e.g. in physics, chemistry, biology, control engineering, electrical network analysis, mechanical systems). Many applications as well as computer programs are presented. Ernst Hairer and Gerhard Wanner were jointly awarded the 2003 Peter Henrici Prize at ICIAM 2003 in Sydney, Australia.},
  isbn = {978-3-540-60452-5},
  langid = {english}
}

@book{haleOrdinaryDifferentialEquations2009,
  title = {Ordinary {{Differential Equations}}},
  author = {Hale, Prof Jack K.},
  year = {2009},
  month = may,
  edition = {2},
  publisher = {Dover Publications},
  address = {Mineola, N.Y},
  url = {https://store.doverpublications.com/0486472116.html},
  isbn = {978-0-486-47211-9},
  langid = {english}
}

@article{hoseaAnalysisImplementationTRBDF21996,
  title = {Analysis and Implementation of {{TR-BDF2}}},
  author = {Hosea, M. E. and Shampine, L. F.},
  year = {1996},
  month = feb,
  journal = {Applied Numerical Mathematics},
  series = {Method of {{Lines}} for {{Time-Dependent Problems}}},
  volume = {20},
  number = {1},
  pages = {21--37},
  issn = {0168-9274},
  doi = {10.1016/0168-9274(95)00115-8},
  url = {https://www.sciencedirect.com/science/article/pii/0168927495001158},
  urldate = {2023-08-26},
  abstract = {Bank et al. (1985) developed a one-step method, TR-BDF2, for the simulation of circuits and semiconductor devices based on the trapezoidal rule and the backward differentiation formula of order 2 that provides some of the important advantages of BDF2 without the disadvantages of a memory. Its success and popularity in the context justify its study and further development for general-purpose codes. Here the method is shown to be strongly S-stable. It is shown to be optimal in a class of practical one-step methods. An efficient, globally C1 interpolation scheme is developed. The truncation error estimate of Bank et al. (1985) is not effective when the problem is very stiff. Coming to an understanding of this leads to a way of correcting the estimate and to a more effective implementation. These developments improve greatly the effectiveness of the method for very stiff problems.}
}

@book{inceOrdinaryDifferentialEquations1956,
  title = {Ordinary {{Differential Equations}}},
  author = {Ince, Edward L.},
  year = {1956},
  month = jun,
  edition = {Reprint edition},
  publisher = {Dover Publications},
  address = {New York, NY},
  isbn = {978-0-486-60349-0},
  langid = {english}
}

@book{iserlesFirstCourseNumerical2008a,
  title = {A {{First Course}} in the {{Numerical Analysis}} of {{Differential Equations}}},
  author = {Iserles, Arieh},
  year = {2008},
  month = dec,
  series = {Cambridge {{Texts}} in {{Applied Mathematics}}},
  edition = {2},
  number = {44},
  publisher = {Cambridge University Press},
  address = {Cambridge ; New York},
  url = {https://doi.org/10.1017/CBO9780511995569},
  abstract = {Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This second edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.},
  isbn = {978-0-521-73490-5},
  langid = {english}
}

@misc{jayLobattoMethods2015,
  title = {Lobatto {{Methods}}},
  author = {Jay, Laurent O.},
  editor = {Engquist, B.},
  year = {2015},
  journal = {Encyclopedia of Applied and Computational Mathematics},
  pages = {817--826},
  publisher = {Springer},
  address = {Berling, Heidelberg},
  url = {https://doi.org/10.1007/978-3-540-70529-1_123},
  urldate = {2023-04-11},
  isbn = {978-3-540-70528-4}
}

@phdthesis{joldesRigorousPolynomialApproximations2011,
  title = {Rigorous {{Polynomial Approximations}} and {{Applications}}},
  author = {Joldes, Mioara Maria},
  year = {2011},
  month = sep,
  url = {https://theses.hal.science/tel-00657843},
  urldate = {2023-11-23},
  abstract = {For purposes of evaluation and manipulation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floating-point implementations of elementary functions, integration, ordinary differential equations (ODE) solving. For that, a wide range of numerical methods exists. We consider the application of such methods in the context of rigorous computing, where we need guarantees on the accuracy of the result, with respect to both the truncation and rounding errors.A rigorous polynomial approximation (RPA) for a function f defined over an interval [a,b] is a couple (P, Delta) where P is a polynomial and Delta is an interval such that f(x)-P(x) belongs to Delta, for all x in [a,b]. In this work we analyse and bring forth several ways of obtaining RPAs for univariate functions. Firstly, we analyse and refine an existing approach based on Taylor expansions. Secondly, we replace them with better approximations such as minimax approximations, Chebyshev truncated series or interpolation polynomials.Several applications are presented: one from standard functions implementation in mathematical libraries (libm), another regarding the computation of Chebyshev series expansions solutions of linear ODEs with polynomial coefficients, and finally an automatic process for function evaluation with guaranteed accuracy in reconfigurable hardware.},
  langid = {english},
  school = {Ecole normale sup{\'e}rieure de lyon - ENS LYON}
}

@article{molerAreWeThere1997,
  title = {Are We There yet? {{Zero}} Crossing and Event Handling for Differential Equations},
  author = {Moler, Cleve},
  year = {1997},
  journal = {Matlab News \& Notes},
  pages = {16--17},
  url = {https://www.mathworks.com/content/dam/mathworks/tag-team/Objects/a/77503_97slCleve.pdf},
  langid = {english}
}

@misc{rackauckasComparisonDifferentialEquation2017,
  title = {A {{Comparison Between Differential Equation Solver Suites In MATLAB}}, {{R}}, {{Julia}}, {{Python}}, {{C}}, {{Mathematica}}, {{Maple}}, and {{Fortran}}},
  author = {Rackauckas, Christopher},
  year = {2017},
  month = sep,
  journal = {Stochastic Lifestyle},
  url = {https://www.stochasticlifestyle.com/comparison-differential-equation-solver-suites-matlab-r-julia-python-c-fortran/},
  urldate = {2022-10-04},
  abstract = {Many times a scientist is choosing a programming language or a software for a specific purpose. For the field of scientific computing, the methods for solving differential equations are one of the important areas. What I would like to do is take the time to compare and contrast between the most popular offerings. This is a good way to reflect upon what's available and find out where there is room for improvement. I hope that by giving you the details for how each suite was put together (and the "why", as gathered from software publications) you can come to your own conclusion as to which suites are right for you. (Full disclosure, I am the lead developer of DifferentialEquations.jl. You will see at the end that DifferentialEquations.jl does offer pretty much everything from the other suite combined, but that's no accident: ... READ MORE},
  langid = {american}
}

@book{shampineSolvingODEsMATLAB2003,
  title = {Solving {{ODEs}} with {{MATLAB}}},
  author = {Shampine, L. F. and Gladwell, I. and Thompson, S.},
  year = {2003},
  publisher = {Cambridge University Press},
  address = {Cambridge},
  doi = {10.1017/CBO9780511615542},
  url = {https://www.cambridge.org/core/books/solving-odes-with-matlab/6F10D15E84B24899D331EEB49EE79FE4},
  urldate = {2023-04-11},
  abstract = {This concise text, first published in 2003, is for a one-semester course for upper-level undergraduates and beginning graduate students in engineering, science, and mathematics, and can also serve as a quick reference for professionals. The major topics in ordinary differential equations, initial value problems, boundary value problems, and delay differential equations, are usually taught in three separate semester-long courses. This single book provides a sound treatment of all three in fewer than 300 pages. Each chapter begins with a discussion of the 'facts of life' for the problem, mainly by means of examples. Numerical methods for the problem are then developed, but only those methods most widely used. The treatment of each method is brief and technical issues are minimized, but all the issues important in practice and for understanding the codes are discussed.  The last part of each chapter is a tutorial that shows how to solve problems by means of small, but realistic, examples.},
  isbn = {978-0-521-82404-0}
}

@book{trefethenExploringODEs2017,
  title = {Exploring {{ODEs}}},
  author = {Trefethen, Lloyd N. and Birkisson, {\'A}sgeir and Driscoll, Tobin A.},
  year = {2017},
  month = dec,
  publisher = {{SIAM-Society for Industrial and Applied Mathematics}},
  address = {Philadelphia},
  url = {http://people.maths.ox.ac.uk/trefethen/ExplODE/},
  isbn = {978-1-61197-515-4},
  langid = {english}
}

@article{tsitourasRungeKuttaPairs2011,
  title = {Runge--{{Kutta}} Pairs of Order 5(4) Satisfying Only the First Column Simplifying Assumption},
  author = {Tsitouras, {\relax Ch}.},
  year = {2011},
  month = jul,
  journal = {Computers \& Mathematics with Applications},
  volume = {62},
  number = {2},
  pages = {770--775},
  issn = {0898-1221},
  doi = {10.1016/j.camwa.2011.06.002},
  url = {https://www.sciencedirect.com/science/article/pii/S0898122111004706},
  urldate = {2022-10-04},
  abstract = {Among the most popular methods for the solution of the Initial Value Problem are the Runge--Kutta pairs of orders 5 and 4. These methods can be derived solving a system of nonlinear equations for its coefficients. To achieve this, we usually admit various simplifying assumptions. The most common of them are the so-called row simplifying assumptions. Here we neglect them and present an algorithm for the construction of Runge--Kutta pairs of orders 5 and 4 based only in the first column simplifying assumption. The result is a pair that outperforms other known pairs in the bibliography when tested to the standard set of problems of DETEST. A cost free fourth order formula is also derived for handling dense output.},
  langid = {english}
}