Differential Equation Analysis in Biomedical Science and Engineering ：Partial Differential Equation Applications with R

Publication subTitle ：Partial Differential Equation Applications with R

Author： William E. Schiesser

Publisher： John Wiley & Sons Inc‎

Publication year： 2014

E-ISBN: 9781118705322

P-ISBN(Hardback): 9781118705186

Subject： Q811 bionics

Language： ENG

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Description

Features a solid foundation of mathematical and computational tools to formulate and solve real-world PDE problems across various fields

With a step-by-step approach to solving partial differential equations (PDEs), Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R successfully applies computational techniques for solving real-world PDE problems that are found in a variety of fields, including chemistry, physics, biology, and physiology. The book provides readers with the necessary knowledge to reproduce and extend the computed numerical solutions and is a valuable resource for dealing with a broad class of linear and nonlinear partial differential equations.

The author’s primary focus is on models expressed as systems of PDEs, which generally result from including spatial effects so that the PDE dependent variables are functions of both space and time, unlike ordinary differential equation (ODE) systems that pertain to time only. As such, the book emphasizes details of the numerical algorithms and how the solutions were computed. Featuring computer-based mathematical models for solving real-world problems in the biological and biomedical sciences and engineering, the book also includes:

R routines to facilitate the immediate use of computation for solving differential equation problems without having to first learn the basic concepts of numerical analysis and programming for PDEs
Models as systems of PDEs and associated initial and boundary conditions with explanations of the associated chemistry, physics, biology, and physiology
Numerical solutions of the presented model equations with a discussion of the important features of the solutions
Aspects of general PDE computation through various biomedical science and engineering applications

Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R is an excellent reference for researchers, scientists, clinicians, medical researchers, engineers, statisticians, epidemiologists, and pharmacokineticists who are interested in both clinical applications and interpretation of experimental data with mathematical models in order to efficiently solve the associated differential equations. The book is also useful as a textbook for graduate-level courses in mathematics, biomedical science and engineering, biology, biophysics, biochemistry, medicine, and engineering.

Chapter

Chapter 1 Introduction to Partial Differentiation Equation Analysis: Chemotaxis

1.1 Introduction

1.2 Linear Diffusion Model

1.3 Nonlinear Chemotaxis Model

1.4 Method of Lines Solution of 2-PDE Chemotaxis Model

1.4.1 Main Program

1.4.2 ODE Routine

1.5 Model Output

1.6 Computation of PDE Terms

1.7 Conclusions

References

Chapter 2 Pattern Formation

2.1 Introduction

2.2 Two PDE Model

2.2.1 ODE Routine

2.2.2 Main Program

2.2.3 Numerical Solution

2.3 Three PDE Model

2.3.1 ODE Routine

2.3.2 Main Program

2.3.3 Supplemental Routine

2.3.4 Numerical Solution

2.4 Conclusions

References

Chapter 3 Belousov-Zhabotinskii Reaction System

3.1 Introduction

3.2 The Belousov-Zhabotinskii Reaction System

3.2.1 ODE Routine

3.2.2 Main Program

3.2.3 Model Output

3.3 Conclusions

Reference

Chapter 4 Hodgkin-Huxley and Fitzhugh-Nagumo Models

4.1 Introduction

4.2 PDE Model

4.3 MOL Routines

4.3.1 ODE Routine

4.3.2 Main Program

4.4 Model Output

4.5 Discontinuous Initial Condition

4.6 Conclusions

References

Chapter 5 Anesthesia Spatiotemporal Distribution

5.1 Introduction

5.2 Two PDE Model

5.2.1 Main Program

5.2.2 ODE Routine

5.2.3 Other PDE Routines

5.2.4 Model Output

5.3 Conclusions

5.4 Model Extension

5.4.1 ODE Routine for the Extended Model

5.4.2 Main Program for the Extended Model

5.4.3 Output for the Extended Model

Acknowledgment

References

Chapter 6 Influenza with Vaccination and Diffusion

6.1 Introduction

6.2 Five PDE Model

6.2.1 ODE Routine

6.2.2 Main Program

6.2.3 Model Output

6.3 Summary

References

Chapter 7 Drug Release Tracking

7.1 Introduction

7.2 Three PDE Model

7.2.1 ODE Routine

7.2.2 Rate Functions

7.2.3 Main Program

7.3 Model Output

7.3.1 Base Case

7.3.2 Variation of Spatial Differentiation

7.3.3 Variation of Polymer Stress

7.4 Alternate Coordinate Systems

7.5 Summary

References

Chapter 8 Temperature Distributions in Cryosurgery

8.1 Introduction

8.2 PDE Model

8.3 Method of Lines Analysis

8.3.1 ODE Routine

8.3.2 Physical Property Routines

8.3.3 Main Program

8.4 Numerical Output

8.5 Variations in the Model

8.6 Summary

References

Index

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