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Random Ordinary Differential Equations and Their Numerical Solution

  • Xiaoying Han
  • Peter E. Kloeden

Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 85)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Random and Stochastic Ordinary Differential Equations

    1. Front Matter
      Pages 1-1
    2. Xiaoying Han, Peter E. Kloeden
      Pages 3-13
    3. Xiaoying Han, Peter E. Kloeden
      Pages 15-27
    4. Xiaoying Han, Peter E. Kloeden
      Pages 29-36
    5. Xiaoying Han, Peter E. Kloeden
      Pages 37-47
    6. Xiaoying Han, Peter E. Kloeden
      Pages 49-58
  3. Taylor Expansions

    1. Front Matter
      Pages 59-59
    2. Xiaoying Han, Peter E. Kloeden
      Pages 73-79
  4. Numerical Schemes for Random Ordinary Differential Equations

    1. Front Matter
      Pages 99-99
    2. Xiaoying Han, Peter E. Kloeden
      Pages 101-108
    3. Xiaoying Han, Peter E. Kloeden
      Pages 109-128
    4. Xiaoying Han, Peter E. Kloeden
      Pages 129-142
    5. Xiaoying Han, Peter E. Kloeden
      Pages 143-153
    6. Xiaoying Han, Peter E. Kloeden
      Pages 155-161
    7. Xiaoying Han, Peter E. Kloeden
      Pages 163-179
  5. Random Ordinary Differential Equations in the Life Sciences

    1. Front Matter
      Pages 181-181
    2. Xiaoying Han, Peter E. Kloeden
      Pages 183-192
    3. Xiaoying Han, Peter E. Kloeden
      Pages 193-203
    4. Xiaoying Han, Peter E. Kloeden
      Pages 205-214
    5. Xiaoying Han, Peter E. Kloeden
      Pages 215-222
  6. Back Matter
    Pages 223-250

About this book

Introduction

This book is intended to make recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a broader readership, and to familiarize readers with RODEs themselves as well as the closely associated theory of random dynamical systems. In addition, it demonstrates how RODEs are being used in the biological sciences, where non-Gaussian and bounded noise are often more realistic than the Gaussian white noise in stochastic differential equations (SODEs).

 

RODEs are used in many important applications and play a fundamental role in the theory of random dynamical systems.  They can be analyzed pathwise with deterministic calculus, but require further treatment beyond that of classical ODE theory due to the lack of smoothness in their time variable. Although classical numerical schemes for ODEs can be used pathwise for RODEs, they rarely attain their traditional order since the solutions of RODEs do not have sufficient smoothness to have Taylor expansions in the usual sense.  However, Taylor-like expansions can be derived for RODEs using an iterated application of the appropriate chain rule in integral form, and represent the starting point for the systematic derivation of consistent higher order numerical schemes for RODEs.

 

The book is directed at a wide range of readers in applied and computational mathematics and related areas as well as readers who are interested in the applications of mathematical models involving random effects, in particular in the biological sciences.The level of this book is suitable for graduate students in applied mathematics and related areas, computational sciences and systems biology.  A basic knowledge of ordinary differential equations and numerical analysis is required. 

Keywords

37H10, 60H10, 60H35 34F05, 37H70, 60H30, 65LC30, 65L05, 65L06, 65L20, 92-08 random dynamical systems biological sciences Numerical Schemes for RODEs Taylor expansions for ODEs and SODEs Stochastic Ordinary Differential Equations Life Sciences

Authors and affiliations

  • Xiaoying Han
    • 1
  • Peter E. Kloeden
    • 2
  1. 1.Auburn USA
  2. 2.Huazhong University of Science and TechnologySchool of Mathematics and StatisticsWuhanChina

Bibliographic information

  • DOI https://doi.org/10.1007/978-981-10-6265-0
  • Copyright Information Springer Nature Singapore Pte Ltd. 2017
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-981-10-6264-3
  • Online ISBN 978-981-10-6265-0
  • Series Print ISSN 2199-3130
  • Series Online ISSN 2199-3149
  • Buy this book on publisher's site
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