Nonstandard Finite Difference Models Of Differential Equations
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Nonstandard Finite Difference Models Of Differential Equations
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Author : Ronald E. Mickens
language : en
Publisher: World Scientific
Release Date : 1994
Nonstandard Finite Difference Models Of Differential Equations written by Ronald E. Mickens and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Mathematics categories.
This book provides a clear summary of the work of the author on the construction of nonstandard finite difference schemes for the numerical integration of differential equations. The major thrust of the book is to show that discrete models of differential equations exist such that the elementary types of numerical instabilities do not occur. A consequence of this result is that in general bigger step-sizes can often be used in actual calculations and/or finite difference schemes can be constructed that are conditionally stable in many instances whereas in using standard techniques no such schemes exist. The theoretical basis of this work is centered on the concepts of ?exact? and ?best? finite difference schemes. In addition, a set of rules is given for the discrete modeling of derivatives and nonlinear expressions that occur in differential equations. These rules often lead to a unique nonstandard finite difference model for a given differential equation.
Nonstandard Finite Difference Schemes Methodology And Applications
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Author : Ronald E Mickens
language : en
Publisher: World Scientific
Release Date : 2020-11-11
Nonstandard Finite Difference Schemes Methodology And Applications written by Ronald E Mickens and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-11-11 with Mathematics categories.
This second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the 'principle of dynamic consistency' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.
Applications Of Nonstandard Finite Difference Schemes
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Author : Ronald E. Mickens
language : en
Publisher: World Scientific
Release Date : 2000
Applications Of Nonstandard Finite Difference Schemes written by Ronald E. Mickens and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
The main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter I gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations, that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain mathematical issues related to the nonstandard discretization of competitive and cooperative models for ecology. The application chapters illustrate well the power of nonstandard methods. In particular, for the same accuracy as obtained by standard techniques, larger step sizes can be used. This volume will satisfy the needs of scientists, engineers, and mathematicians who wish to know how to construct nonstandard schemes and see how these are applied to obtain numerical solutions of the differential equations which arise in the study of nonlinear dynamical systems modeling important physical phenomena.
Advances In The Applications Of Nonstandard Finite Diffference Schemes
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Author : Ronald E. Mickens
language : en
Publisher: World Scientific
Release Date : 2005
Advances In The Applications Of Nonstandard Finite Diffference Schemes written by Ronald E. Mickens and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990's and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic.
Advances In The Applications Of Nonstandard Finite Difference Schemes
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Author : Ronald E Mickens
language : en
Publisher: World Scientific
Release Date : 2005-10-25
Advances In The Applications Of Nonstandard Finite Difference Schemes written by Ronald E Mickens and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-10-25 with Mathematics categories.
This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990's and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic. Contents:Nonstandard Finite Difference Methods (R E Mickens)Application of Nonstandard Finite Difference Schemes to the Simulation Studies of Robotic Systems (R F Abo-Shanab et al.)Applications of Mickens Finite Differences to Several Related Boundary Value Problems (R Buckmire)High Accuracy Nonstandard Finite-Difference Time-Domain Algorithms for Computational Electromagnetics: Applications to Optics and Photonics (J B Cole)Nonstandard Finite Difference Schemes for Solving Nonlinear Micro Heat Transport Equations in Double-Layered Metal Thin Films Exposed to Ultrashort Pulsed Lasers (W Dai)Reliable Finite Difference Schemes with Applications in Mathematical Ecology (D T Dimitrov et al.)Applications of the Nonstandard Finite Difference Method in Non-Smooth Mechanics (Y Dumont)Finite Difference Schemes on Unbounded Domains (M Ehrhardt)Asymptotically Consistent Nonstandard Finite-Difference Methods for Solving Mathematical Models Arising in Population Biology (A B Gumel et al.)Nonstandard Finite Difference Methods and Biological Models (S R-J Jang)Robust Discretizations versus Increase of the Time Step for Chaotic Systems (C Letellier & E M A M Mendes)Contributions to the Theory of Nonstandard Finite-Difference Methods and Applications to Singular Perturbation Problems (J M-S Lubuma & K C Patidar)Frequency Accurate Finite Difference Methods (A L Perkins et al.)Nonstandard Discretization Methods on Lotka-Volterra Differential Equations (L-I W Roeger) Readership: Applied mathematicians, and researchers in numerical & computational mathematics and analysis & differential equations. Usable as a secondary text to a standard undergraduate or graduate course on numerical methods for differential equations. Keywords:Numerical Integration Methods;Finite Differences;Nonstandard Finite Difference Schemes;Differential Equations;Discrete Models;Numerical and Computational MathematicsKey Features:A collection of papers from renowned experts in their respective fieldsProvides the most recent work on the application of NSFD schemes and some of the mathematical analysis related to these schemes
Mathematical Modelling And Nonstandard Schemes For The Corona Virus Pandemic
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Author : Sarah Marie Treibert
language : en
Publisher: Springer Nature
Release Date : 2021-12-11
Mathematical Modelling And Nonstandard Schemes For The Corona Virus Pandemic written by Sarah Marie Treibert and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-12-11 with Mathematics categories.
This book deals with the prediction of possible future scenarios concerning the COVID-19 pandemic. Based on the well-known SIR model by Kermack and McKendrick a compartment model is established. This model comprises its own assumptions, transition rates and transmission dynamics, as well as a corresponding system of ordinary differential equations. Making use of numerical methods and a nonstandard-finite-difference scheme, two submodels are implemented in Matlab in order to make parameter estimations and compare different scenarios with each other.
Exact Finite Difference Schemes
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Author : Sergey Lemeshevsky
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2016-09-26
Exact Finite Difference Schemes written by Sergey Lemeshevsky and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-09-26 with Mathematics categories.
Exact Finite-Difference Schemes is a first overview of the topic also describing the state-of-the-art in this field of numerical analysis. Construction of exact difference schemes for various parabolic and elliptic partial differential equations are discussed, including vibrations and transport problems. After this, applications are discussed, such as the discretisation of ODEs and PDEs and numerical methods for stochastic differential equations. Contents: Basic notation Preliminary results Hyperbolic equations Parabolic equations Use of exact difference schemes to construct NSFD discretizations of differential equations Exact and truncated difference schemes for boundary-value problem Exact difference schemes for stochastic differential equations Numerical blow-up time Bibliography
Finite Difference Methods For Ordinary And Partial Differential Equations
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Author : Randall J. LeVeque
language : en
Publisher: SIAM
Release Date : 2007-01-01
Finite Difference Methods For Ordinary And Partial Differential Equations written by Randall J. LeVeque and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-01-01 with Mathematics categories.
This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.
Fractional Partial Differential Equations And Their Numerical Solutions
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Author : Boling Guo
language : en
Publisher: World Scientific
Release Date : 2015-03-09
Fractional Partial Differential Equations And Their Numerical Solutions written by Boling Guo and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-03-09 with Mathematics categories.
This book aims to introduce some new trends and results on the study of the fractional differential equations, and to provide a good understanding of this field to beginners who are interested in this field, which is the authors' beautiful hope. This book describes theoretical and numerical aspects of the fractional partial differential equations, including the authors' researches in this field, such as the fractional Nonlinear Schrödinger equations, fractional Landau–Lifshitz equations and fractional Ginzburg–Landau equations. It also covers enough fundamental knowledge on the fractional derivatives and fractional integrals, and enough background of the fractional PDEs. Contents:Physics BackgroundFractional Calculus and Fractional Differential EquationsFractional Partial Differential EquationsNumerical Approximations in Fractional CalculusNumerical Methods for the Fractional Ordinary Differential EquationsNumerical Methods for Fractional Partial Differential Equations Readership: Graduate students and researchers in mathematical physics, numerical analysis and computational mathematics. Key Features:This book covers the fundamentals of this field, especially for the beginnersThe book covers new trends and results in this fieldThe book covers numerical results, which will be of broad interests to researchersKeywords:Fractional Partial Differential Equations;Numerical Solutions
Advanced Numerical Methods For Differential Equations
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Author : Harendra Singh
language : en
Publisher: CRC Press
Release Date : 2021-07-29
Advanced Numerical Methods For Differential Equations written by Harendra Singh and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-07-29 with Mathematics categories.
Mathematical models are used to convert real-life problems using mathematical concepts and language. These models are governed by differential equations whose solutions make it easy to understand real-life problems and can be applied to engineering and science disciplines. This book presents numerical methods for solving various mathematical models. This book offers real-life applications, includes research problems on numerical treatment, and shows how to develop the numerical methods for solving problems. The book also covers theory and applications in engineering and science. Engineers, mathematicians, scientists, and researchers working on real-life mathematical problems will find this book useful.