William E. Schiesser's Books

Numerical Pde Analysis Of The Blood Brain Barrier: Method Of Lines In R

By: William E Schiesser

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Numerical Pde Analysis Of The Blood Brain Barrier: Method Of Lines In R

By: William E Schiesser

The remarkable functionality of the brain is made possible by the metabolism (chemical reaction) of oxygen (O₂) and nutrients in the brain. These metabolism components are supplied to the brain by an intricate blood circulatory system (vasculature). The blood brain barrier (BBB), which is the central topic of this book, determines the rate of transfer from the blood to the brain tissue.In particular, mathematical models are developed for mass transfer across the BBB based on partial differential equations (PDEs) applied to the blood capillaries, the endothelial membrane, and the brain tissue. The PDEs derived from mass balances and computer routines in R are presented for the numerical (computer-based) solution of the PDEs. The computed concentration profiles of the transferred components are functions of time and space within the BBB system, i.e., spatiotemporal solutions.The R routines and the associated numerical algorithms for computing the numerical solutions are discussed in detail. The discussion is introductory, without formal mathematics, e.g., theorems and proofs. The general methodology (algorithm) for numerical PDE solutions is the method of lines (MOL).The models are used to study the transfer of oxygen and nutrients, harmful substances that should not enter the brain such as chemicals and pathogens (viruses, bacteria), and therapeutic drugs. The intent of the book is to provide a quantitative approach to the study of BBB dynamics using a computer-based methodology programmed in R, a quality open-source scientific programming system that is easily downloaded from the Internet for execution on modest computers.

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265

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World Scientific

Published Date

ISBN 10

9813275812

ISBN 13

9789813275812

Mathematical Modeling Approach To Infectious Diseases, A: Cross Diffusion Pde Models For Epidemiology

By: William E Schiesser

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Mathematical Modeling Approach To Infectious Diseases, A: Cross Diffusion Pde Models For Epidemiology

By: William E Schiesser

The intent of this book is to provide a methodology for the analysis of infectious diseases by computer-based mathematical models. The approach is based on ordinary differential equations (ODEs) that provide time variation of the model dependent variables and partial differential equations (PDEs) that provide time and spatial (spatiotemporal) variations of the model dependent variables.The starting point is a basic ODE SIR (Susceptible Infected Recovered) model that defines the S,I,R populations as a function of time. The ODE SIR model is then extended to PDEs that demonstrate the spatiotemporal evolution of the S,I,R populations. A unique feature of the PDE model is the use of cross diffusion between populations, a nonlinear effect that is readily accommodated numerically. A second feature is the use of radial coordinates to represent the geographical distribution of the model populations.The numerical methods for the computer implementation of ODE/PDE models for infectious diseases are illustrated with documented R routines for particular applications, including models for malaria and the Zika virus. The R routines are available from a download so that the reader can reproduce the reported solutions, then extend the applications through computer experimentation, including the addition of postulated effects and associated equations, and the implementation of alternative models of interest.The ODE/PDE methodology is open ended and facilitates the development of computer-based models which hopefully can elucidate the causes/conditions of infectious disease evolution and suggest methods of control.

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460

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World Scientific

Published Date

ISBN 10

9813238801

ISBN 13

9789813238800

A Compendium of Partial Differential Equation Models

Method of Lines Analysis with Matlab

By: William E. Schiesser,Graham W. Griffiths

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A Compendium of Partial Differential Equation Models

Method of Lines Analysis with Matlab

By: William E. Schiesser,Graham W. Griffiths

Presents numerical methods and computer code in Matlab for the solution of ODEs and PDEs with detailed line-by-line discussion.

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491

Publisher

Cambridge University Press

Published Date

ISBN 10

0521519861

ISBN 13

9780521519861

Dynamic Modeling of Transport Process Systems

By: C. A. Silebi,William E. Schiesser

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Dynamic Modeling of Transport Process Systems

By: C. A. Silebi,William E. Schiesser

This book presents a methodology for the development and computer implementation of dynamic models for transport process systems. Rather than developing the general equations of transport phenomena, it develops the equations required specifically for each new example application. These equations are generally of two types: ordinary differential equations (ODEs) and partial differential equations (PDEs) for which time is an independent variable. The computer-based methodology presented is general purpose and can be applied to most applications requiring the numerical integration of initial-value ODEs/PDEs. A set of approximately two hundred applications of ODEs and PDEs developed by the authors are listed in Appendix 8.

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533

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Elsevier

Published Date

ISBN 10

0080925820

ISBN 13

9780080925820

PDE Models for Atherosclerosis Computer Implementation in R

By: William E. Schiesser

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PDE Models for Atherosclerosis Computer Implementation in R

By: William E. Schiesser

Atherosclerosis is a pathological condition of the arteries in which plaque buildup and stiffening (hardening) can lead to stroke, myocardial infarction (heart attacks), and even death. Cholesterol in the blood is a key marker for atherosclerosis, with two forms: (1) LDL - low density lipoproteins and (2) HDL - high density lipoproteins. Low LDL and high HDL concentrations are generally considered essential for limited atherosclerosis and good health. This book pertains to a mathematical model for the spatiotemporal distribution of LDL and HDL in the arterial endothelial inner layer (EIL, intima). The model consists of a system of six partial differential equations (PDEs) with the dependent variables 1. (,): concentration of modified LDL 2. h(,): concentration of HDL 3. (,): concentration of chemoattractants 4. (,): concentration of ES cytokines 5. (,): density of monocytes/macrophages 6. (,): density of foam cells and independent variables 1. : distance from the inner arterial wall 2. : time The focus of this book is a discussion of the methodology for placing the model on modest computers for study of the numerical solutions. The foam cell density (,) as a function of the bloodstream LDL and HDL concentrations is of particular interest as a precursor for arterial plaque formation and stiffening. The numerical algorithm for the solution of the model PDEs is the method of lines (MOL), a general procedure for the computer-based numerical solution of PDEs. The MOL coding (programming) is in R, a quality, open-source scientific computing system that is readily available from the Internet. The R routines for the PDE model are discussed in detail, and are available from a download link so that the reader/analyst/researcher can execute the model to duplicate the solutions reported in the book, then experiment with the model, for example, by changing the parameters (constants) and extending the model with additional equations. /div

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136

Publisher

Springer Nature

Published Date

ISBN 10

3031024141

ISBN 13

9783031024146

Numerical PDE Analysis of Retinal Neovascularization

Mathematical Model Computer Implementation in R

By: William E. Schiesser

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Numerical PDE Analysis of Retinal Neovascularization

Mathematical Model Computer Implementation in R

By: William E. Schiesser

Numerical PDE Analysis of Retinal Neovascularization Mathematical Model Computer Implementation in R provides a methodology for the analysis of neovascularization (formation of blood capillaries) in the retina. It describes the starting point—a system of three partial differential equations (PDEs)—that define the evolution of (1) capillary tip density, (2) blood capillary density and (3) concentration of vascular endothelial growth factor (VEGF) in the retina as a function of space (distance along the retina), x, and time, t, the three PDE dependent variables for (1), (2) and (3), and designated as u1(x, t), u2(x, t), u3(x, t), amongst other topics. - Includes PDE routines based on the method of lines (MOL) for computer-based implementation of PDE models - Offers transportable computer source codes for readers in R, with line-by-line code descriptions as it relates to the mathematical model and algorithms - Authored by a leading researcher and educator in PDE models

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146

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Academic Press

Published Date

ISBN 10

0128184531

ISBN 13

9780128184530

Method of Lines PDE Analysis in Biomedical Science and Engineering

By: William E. Schiesser

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Method of Lines PDE Analysis in Biomedical Science and Engineering

By: William E. Schiesser

Presents the methodology and applications of ODE and PDE models within biomedical science and engineering With an emphasis on the method of lines (MOL) for partial differential equation (PDE) numerical integration, Method of Lines PDE Analysis in Biomedical Science and Engineering demonstrates the use of numerical methods for the computer solution of PDEs as applied to biomedical science and engineering (BMSE). Written by a well-known researcher in the field, the book provides an introduction to basic numerical methods for initial/boundary value PDEs before moving on to specific BMSE applications of PDEs. Featuring a straightforward approach, the book’s chapters follow a consistent and comprehensive format. First, each chapter begins by presenting the model as an ordinary differential equation (ODE)/PDE system, including the initial and boundary conditions. Next, the programming of the model equations is introduced through a series of R routines that primarily implement MOL for PDEs. Subsequently, the resulting numerical and graphical solution is discussed and interpreted with respect to the model equations. Finally, each chapter concludes with a review of the numerical algorithm performance, general observations and results, and possible extensions of the model. Method of Lines PDE Analysis in Biomedical Science and Engineering also includes: Examples of MOL analysis of PDEs, including BMSE applications in wave front resolution in chromatography, VEGF angiogenesis, thermographic tumor location, blood-tissue transport, two fluid and membrane mass transfer, artificial liver support system, cross diffusion epidemiology, oncolytic virotherapy, tumor cell density in glioblastomas, and variable grids Discussions on the use of R software, which facilitates immediate solutions to differential equation problems without having to first learn the basic concepts of numerical analysis for PDEs and the programming of PDE algorithms A companion website that provides source code for the R routines Method of Lines PDE Analysis in Biomedical Science and Engineering is an introductory reference for researchers, scientists, clinicians, medical researchers, mathematicians, statisticians, chemical engineers, epidemiologists, and pharmacokineticists as well as anyone interested in clinical applications and the interpretation of experimental data with differential equation models. The book is also an ideal textbook for graduate-level courses in applied mathematics, BMSE, biology, biophysics, biochemistry, medicine, and engineering.

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413

Publisher

John Wiley & Sons

Published Date

ISBN 10

1119130514

ISBN 13

9781119130512

PDE Modeling of Tissue Engineering and Regenerative Medicine

Computer Analysis in R

By: William E. Schiesser

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PDE Modeling of Tissue Engineering and Regenerative Medicine

Computer Analysis in R

By: William E. Schiesser

PDE Modeling of Tissue Engineering and Regenerative Medicine: Computer Analysis in R presents the formulation and computer implementation of mathematical models for the forefront research areas of tissue engineering and regenerative medicine. The mathematical model discussed in this book consists of a system of eight partial differential equations (PDEs) with dependent variables. The computer-based example models are presented through routines coded in R—a quality, open-source scientific computing system that is readily available from the Internet. Formal mathematics is minimized, e.g., no theorems and proofs. Includes detailed examples that the reader can execute on modest computers. - Includes PDE routines based on the method of lines (MOL) for computer-based implementation of PDE models - Offers transportable computer source codes for readers in R, with line-by-line code descriptions as it relates to the mathematical model and algorithms - Authored by a leading researcher and educator in PDE models

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190

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Elsevier

Published Date

ISBN 10

044318741X

ISBN 13

9780443187414

Computational Modeling Of The Covid-19 Disease: Numerical Ode Analysis With R Programming

By: William E Schiesser

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Computational Modeling Of The Covid-19 Disease: Numerical Ode Analysis With R Programming

By: William E Schiesser

The book is intended for readers who are interested in learning about the use of computer-based modelling of the COVID-19 disease. It provides a basic introduction to a five-ordinary differential equation (ODE) model by providing a complete statement of the model, including a detailed discussion of the ODEs, initial conditions and parameters, followed by a line-by-line explanation of a set of R routines (R is a quality, scientific programming system readily available from the Internet). The reader can access and execute these routines without having to first study numerical algorithms and computer coding (programming) and can perform numerical experimentation with the model on modest computers.

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109

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World Scientific

Published Date

ISBN 10

9811222894

ISBN 13

9789811222894

Differential Equation Analysis in Biomedical Science and Engineering

Partial Differential Equation Applications with R

By: William E. Schiesser

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Differential Equation Analysis in Biomedical Science and Engineering

Partial Differential Equation Applications with R

By: William E. Schiesser

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.

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280

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John Wiley & Sons

Published Date

ISBN 10

1118705165

ISBN 13

9781118705162

Numerical Integration of Space Fractional Partial Differential Equations

Vol 2 - Applications from Classical Integer PDEs

By: Younes Salehi,William E. Schiesser

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Numerical Integration of Space Fractional Partial Differential Equations

Vol 2 - Applications from Classical Integer PDEs

By: Younes Salehi,William E. Schiesser

Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: Vol 1: Introduction to Algorithms and Computer Coding in R Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the Riemann-Liouville derivative. In the second volume, the emphasis is on applications of SFPDEs developed mainly through the extension of classical integer PDEs to SFPDEs. The example applications are: Fractional diffusion equation with Dirichlet, Neumann and Robin boundary conditions Fisher-Kolmogorov SFPDE Burgers SFPDE Fokker-Planck SFPDE Burgers-Huxley SFPDE Fitzhugh-Nagumo SFPDE /div These SFPDEs were selected because they are integer first order in time and integer second order in space. The variation in the spatial derivative from order two (parabolic) to order one (first order hyperbolic) demonstrates the effect of the spatial fractional order with 1 ≤ ≤ 2. All of the example SFPDEs are one dimensional in Cartesian coordinates. Extensions to higher dimensions and other coordinate systems, in principle, follow from the examples in this second volume. The examples start with a statement of the integer PDEs that are then extended to SFPDEs. The format of each chapter is the same as in the first volume. The R routines can be downloaded and executed on a modest computer (R is readily available from the Internet).

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192

Publisher

Springer Nature

Published Date

ISBN 10

3031024125

ISBN 13

9783031024122

Computational Chemotaxis Models For Neurodegenerative Disease

By: William E Schiesser

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Computational Chemotaxis Models For Neurodegenerative Disease

By: William E Schiesser

The mathematical model presented in this book, based on partial differential equations (PDEs) describing attractant-repellent chemotaxis, is offered for a quantitative analysis of neurodegenerative disease (ND), e.g., Alzheimer's disease (AD). The model is a representation of basic phenomena (mechanisms) for diffusive transport and biochemical kinetics that provides the spatiotemporal distribution of components which could explain the evolution of ND, and is offered with the intended purpose of providing a small step toward the understanding, and possible treatment of ND.The format and emphasis of the presentation is based on the following elements:In other words, a methodology for numerical PDE modeling is presented that is flexible, open ended and readily implemented on modest computers. If the reader is interested in an alternate model, it might possibly be implemented by: (1) modifying and/or extending the current model (for example, by adding terms to the PDEs or adding additional PDEs), or (2) using the reported routines as a prototype for the model of interest.These suggestions illustrate an important feature of computer-based modeling, that is, the readily available procedure of numerically experimenting with a model. The current model is offered as only a first step toward the resolution of this urgent medical problem.

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172

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World Scientific Publishing Company

Published Date

ISBN 10

9813207477

ISBN 13

9789813207479

The Numerical Method of Lines

Integration of Partial Differential Equations

By: William E. Schiesser

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The Numerical Method of Lines

Integration of Partial Differential Equations

By: William E. Schiesser

This is the first book on the numerical method of lines, a relatively new method for solving partial differential equations. The Numerical Method of Lines is also the first book to accommodate all major classes of partial differential equations. This is essentially an applications book for computer scientists. The author will separately offer a disk of FORTRAN 77 programs with 250 specific applications, ranging from "Shuttle Launch Simulation" to "Temperature Control of a Nuclear Fuel Rod.

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341

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Elsevier

Published Date

ISBN 10

0128015519

ISBN 13

9780128015513

ODE/PDE α-synuclein Models for Parkinson's Disease

By: William E. Schiesser

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ODE/PDE α-synuclein Models for Parkinson's Disease

By: William E. Schiesser

ODE/PDE Alpha-Synuclein Models for Parkinson's Disease discusses a mechanism for the evolution of Parkinson's Disease (PD) based on the dynamics of the protein a-synuclein, a monomer that has been implicated in this disease. Specifically, a-synuclein morphs and aggregates into a polymer that can interfere with functioning neurons and lead to neurodegenerative pathology. This book first demonstrates computer-based implementation of a prototype ODE/PDE model for the dynamics of the a-synuclein monomer and polymer, and then details the methodology for the numerical integration of ODE/PDE systems which can be applied to computer-based analyses of alternative models based on the reader's interest. This book facilitates immediate computer use for research without the necessity to first learn the basic concepts of numerical analysis for ODE/PDEs and programming algorithms - Includes PDE routines based on the method of lines (MOL) for computer-based implementation of ODE/PDE models - Offers transportable computer source codes for readers, with line-by-line code descriptions relating to the mathematical model and algorithms - Authored by a leading researcher and educator in ODE/PDE models

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230

Publisher

Academic Press

Published Date

ISBN 10

0128148020

ISBN 13

9780128148020

Mathematical Modeling Of Virus Infection: Ode/pde Analysis In R

By: William E Schiesser

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Mathematical Modeling Of Virus Infection: Ode/pde Analysis In R

By: William E Schiesser

Two models for the spread and control of a virus are detailed in this book: The Lung/Respiratory System Model (LSM) and the SVIR (Susceptible-Vaccinated-Infected-Recovered) Model.The LSM gives the spatiotemporal distribution of four viral-related proteins: virus population density along the lung air passage, host cell primary infection protein (viral genetic material (VGM)) concentration, host cell secondary infection protein (VGM) concentration, and air stream virion population density.The model is executed for a single inhalation, and a series of inhalation/exhalation cycles. For the latter, the progression of the viral infection into the lung is a principal result.The SVIR is first formulated as a system of ordinary differential equations (ODEs) in time, then extended to a system of PDEs to account for spatial effects (spatiotemporal modeling).Principal outputs from the ODE/PDE models are the levels of vaccinations and infections. For the latter, the efficacy of the vaccine is a parameter that can be varied in a computer-based analysis of a vaccine therapy.The coding of the models is in R, a quality, open-source scientific computing system, and can be executed on modest computers. The R routines are available from a download link so that the example models can be executed without having to first study numerical methods and computer coding. The routines can then be applied to variations and extensions of the ODE/PDE models, such as changes in the parameters and the form of the model equations.

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178

Publisher

World Scientific

Published Date

ISBN 10

9811236658

ISBN 13

9789811236655

Spline Collocation Methods for Partial Differential Equations

With Applications in R

By: William E. Schiesser

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Spline Collocation Methods for Partial Differential Equations

With Applications in R

By: William E. Schiesser

A comprehensive approach to numerical partial differential equations Spline Collocation Methods for Partial Differential Equations combines the collocation analysis of partial differential equations (PDEs) with the method of lines (MOL) in order to simplify the solution process. Using a series of example applications, the author delineates the main features of the approach in detail, including an established mathematical framework. The book also clearly demonstrates that spline collocation can offer a comprehensive method for numerical integration of PDEs when it is used with the MOL in which spatial (boundary value) derivatives are approximated with splines, including the boundary conditions. R, an open-source scientific programming system, is used throughout for programming the PDEs and numerical algorithms, and each section of code is clearly explained. As a result, readers gain a complete picture of the model and its computer implementation without having to fill in the details of the numerical analysis, algorithms, or programming. The presentation is not heavily mathematical, and in place of theorems and proofs, detailed example applications are provided. Appropriate for scientists, engineers, and applied mathematicians, Spline Collocation Methods for Partial Differential Equations: Introduces numerical methods by first presenting basic examples followed by more complicated applications Employs R to illustrate accurate and efficient solutions of the PDE models Presents spline collocation as a comprehensive approach to the numerical integration of PDEs and an effective alternative to other, well established methods Discusses how to reproduce and extend the presented numerical solutions Identifies the use of selected algorithms, such as the solution of nonlinear equations and banded or sparse matrix processing Features a companion website that provides the related R routines Spline Collocation Methods for Partial Differential Equations is a valuable reference and/or self-study guide for academics, researchers, and practitioners in applied mathematics and engineering, as well as for advanced undergraduates and graduate-level students.

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570

Publisher

John Wiley & Sons

Published Date

ISBN 10

111930105X

ISBN 13

9781119301059

Spatiotemporal Modeling of Influenza

Partial Differential Equation Analysis in R

By: William E. Schiesser

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Spatiotemporal Modeling of Influenza

Partial Differential Equation Analysis in R

By: William E. Schiesser

This book has a two-fold purpose: An introduction to the computer-based modeling of influenza, a continuing major worldwide communicable disease. The use of (1) as an illustration of a methodology for the computer-based modeling of communicable diseases. For the purposes of (1) and (2), a basic influenza model is formulated as a system of partial differential equations (PDEs) that define the spatiotemporal evolution of four populations: susceptibles, untreated and treated infecteds, and recovereds. The requirements of a well-posed PDE model are considered, including the initial and boundary conditions. The terms of the PDEs are explained. The computer implementation of the model is illustrated with a detailed line-by-line explanation of a system of routines in R (a quality, open-source scientific computing system that is readily available from the Internet). The R routines demonstrate the straightforward numerical solution of a system of nonlinear PDEs by the method of lines (MOL), an established general algorithm for PDEs. The presentation of the PDE modeling methodology is introductory with a minumum of formal mathematics (no theorems and proofs), and with emphasis on example applications. The intent of the book is to assist in the initial understanding and use of PDE mathematical modeling of communicable diseases, and the explanation and interpretation of the computed model solutions, as illustrated with the influenza model.

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113

Publisher

Morgan & Claypool Publishers

Published Date

ISBN 10

1681735709

ISBN 13

9781681735702

Dynamic Modeling of Transport Process Systems

By: C. A. Silebi,William E. Schiesser

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Dynamic Modeling of Transport Process Systems

By: C. A. Silebi,William E. Schiesser

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533

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Elsevier

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ISBN 10

0080925820

ISBN 13

9780080925820

Differential Equation Analysis in Biomedical Science and Engineering

Partial Differential Equation Applications with R

By: William E. Schiesser

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Differential Equation Analysis in Biomedical Science and Engineering

Partial Differential Equation Applications with R

By: William E. Schiesser

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280

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John Wiley & Sons

Published Date

ISBN 10

1118705165

ISBN 13

9781118705162

Moving Boundary PDE Analysis

Biomedical Applications in R

By: William Schiesser

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Moving Boundary PDE Analysis

Biomedical Applications in R

By: William Schiesser

Mathematical models stated as systems of partial differential equations (PDEs) are broadly used in biology, chemistry, physics and medicine (physiology). These models describe the spatial and temporial variations of the problem system dependent variables, such as temperature, chemical and biochemical concentrations and cell densities, as a function of space and time (spatiotemporal distributions). For a complete PDE model, initial conditions (ICs) specifying how the problem system starts and boundary conditions (BCs) specifying how the system is defined at its spatial boundaries, must also be included for a well-posed PDE model. In this book, PDE models are considered for which the physical boundaries move with time. For example, as a tumor grows, its boundary moves outward. In atherosclerosis, the plaque formation on the arterial wall moves inward, thereby restricting blood flow with serious consequences such as stroke and myocardial infarction (heart attack). These two examples are considered as applications of the reported moving boundary PDE (MBPDE) numerical method (algorithm). The method is programmed in a set of documented routines coded in R, a quality, open-source scientific programming system. The routines are provided as a download so that the reader/analyst/researcher can use MFPDE models without having to first study numerical methods and computer programming.

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Page Count

186

Publisher

CRC Press

Published Date

ISBN 10

100000788X

ISBN 13

9781000007886

The Numerical Method of Lines

Integration of Partial Differential Equations

By: William E. Schiesser

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The Numerical Method of Lines

Integration of Partial Differential Equations

By: William E. Schiesser

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341

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Elsevier

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ISBN 10

0128015519

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9780128015513

Traveling Wave Analysis of Partial Differential Equations

Numerical and Analytical Methods with Matlab and Maple

By: Graham W. Griffiths,W. E. Schiesser

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Traveling Wave Analysis of Partial Differential Equations

Numerical and Analytical Methods with Matlab and Maple

By: Graham W. Griffiths,W. E. Schiesser

Partial Differential Equations have been developed and used in science and engineering for more than 200 years, yet they remain a very active area of research, both because of their role in mathematics and their application to virtually all areas of science and engineering. This research is due relatively recently to the development of computer solution methods for PDEs that have extended PDE applications in quantifying board areas of physical, chemical, and biological phenomena. This book surveys some of these new development in analytical and numerical method, and relates the two through a series of PDF examples. The PDFs that have been selected are largely, "named" in thee sense that they have the names of their original contributors. These names usually reflect that the PDFs are widely recognized and used in many application areas. The development of analytical solutions directly supports the development of numerical methods by providing a spectrum of test problem that can be used to evaluate numerical methods.

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447

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ISBN 10

0123846528

ISBN 13

9780123846525

Spatiotemporal Modeling of Influenza

Partial Differential Equation Analysis in R

By: William E. Schiesser

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Spatiotemporal Modeling of Influenza

Partial Differential Equation Analysis in R

By: William E. Schiesser

This book has a two-fold purpose: (1) An introduction to the computer-based modeling of influenza, a continuing major worldwide communicable disease. (2) The use of (1) as an illustration of a methodology for the computer-based modeling of communicable diseases. For the purposes of (1) and (2), a basic influenza model is formulated as a system of partial differential equations (PDEs) that define the spatiotemporal evolution of four populations: susceptibles, untreated and treated infecteds, and recovereds. The requirements of a well-posed PDE model are considered, including the initial and boundary conditions. The terms of the PDEs are explained. The computer implementation of the model is illustrated with a detailed line-by-line explanation of a system of routines in R (a quality, open-source scientific computing system that is readily available from the Internet). The R routines demonstrate the straightforward numerical solution of a system of nonlinear PDEs by the method of lines (MOL), an established general algorithm for PDEs. The presentation of the PDE modeling methodology is introductory with a minumum of formal mathematics (no theorems and proofs), and with emphasis on example applications. The intent of the book is to assist in the initial understanding and use of PDE mathematical modeling of communicable diseases, and the explanation and interpretation of the computed model solutions, as illustrated with the influenza model.

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Springer Nature

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ISBN 10

3031016653

ISBN 13

9783031016653

A Compendium of Partial Differential Equation Models

Method of Lines Analysis with Matlab

By: William E. Schiesser,Graham W. Griffiths

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A Compendium of Partial Differential Equation Models

Method of Lines Analysis with Matlab

By: William E. Schiesser,Graham W. Griffiths

Presents numerical methods and computer code in Matlab for the solution of ODEs and PDEs with detailed line-by-line discussion.

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491

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Cambridge University Press

Published Date

ISBN 10

0521519861

ISBN 13

9780521519861

Method of Lines PDE Analysis in Biomedical Science and Engineering

By: William E. Schiesser

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Method of Lines PDE Analysis in Biomedical Science and Engineering

By: William E. Schiesser

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372

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John Wiley & Sons

Published Date

ISBN 10

1119130484

ISBN 13

9781119130482

Differential Equation Analysis in Biomedical Science and Engineering

Ordinary Differential Equation Applications with R

By: William E. Schiesser

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Differential Equation Analysis in Biomedical Science and Engineering

Ordinary Differential Equation Applications with R

By: William E. Schiesser

Features a solid foundation of mathematical and computational tools to formulate and solve real-world ODE problems across various fields With a step-by-step approach to solving ordinary differential equations (ODEs), Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R successfully applies computational techniques for solving real-world ODE 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 ordinary differential equations. The author’s primary focus is on models expressed as systems of ODEs, which generally result by neglecting spatial effects so that the ODE dependent variables are uniform in space. Therefore, time is the independent variable in most applications of ODE systems. 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 ODEs Models as systems of ODEs with explanations of the associated chemistry, physics, biology, and physiology as well as the algebraic equations used to calculate intermediate variables Numerical solutions of the presented model equations with a discussion of the important features of the solutions Aspects of general ODE computation through various biomolecular science and engineering applications Differential Equation Analysis in Biomedical Science and Engineering: Ordinary 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.

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346

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John Wiley & Sons

Published Date

ISBN 10

1118705238

ISBN 13

9781118705230

Traveling Wave Analysis of Partial Differential Equations

Numerical and Analytical Methods with Matlab and Maple

By: Graham Griffiths,William E. Schiesser

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Traveling Wave Analysis of Partial Differential Equations

Numerical and Analytical Methods with Matlab and Maple

By: Graham Griffiths,William E. Schiesser

Although the Partial Differential Equations (PDE) models that are now studied are usually beyond traditional mathematical analysis, the numerical methods that are being developed and used require testing and validation. This is often done with PDEs that have known, exact, analytical solutions. The development of analytical solutions is also an active area of research, with many advances being reported recently, particularly traveling wave solutions for nonlinear evolutionary PDEs. Thus, the current development of analytical solutions directly supports the development of numerical methods by providing a spectrum of test problems that can be used to evaluate numerical methods. This book surveys some of these new developments in analytical and numerical methods, and relates the two through a series of PDE examples. The PDEs that have been selected are largely "named'' since they carry the names of their original contributors. These names usually signify that the PDEs are widely recognized and used in many application areas. The authors' intention is to provide a set of numerical and analytical methods based on the concept of a traveling wave, with a central feature of conversion of the PDEs to ODEs. The Matlab and Maple software will be available for download from this website shortly. www.pdecomp.net - Includes a spectrum of applications in science, engineering, applied mathematics - Presents a combination of numerical and analytical methods - Provides transportable computer codes in Matlab and Maple

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463

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Academic Press

Published Date

ISBN 10

0123846536

ISBN 13

9780123846532

PDE Modeling of Tissue Engineering and Regenerative Medicine

Computer Analysis in R

By: William E. Schiesser

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PDE Modeling of Tissue Engineering and Regenerative Medicine

Computer Analysis in R

By: William E. Schiesser

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190

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Elsevier

Published Date

ISBN 10

044318741X

ISBN 13

9780443187414

Numerical PDE Analysis of Retinal Neovascularization

Mathematical Model Computer Implementation in R

By: William E. Schiesser

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Numerical PDE Analysis of Retinal Neovascularization

Mathematical Model Computer Implementation in R

By: William E. Schiesser

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146

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Academic Press

Published Date

ISBN 10

0128184531

ISBN 13

9780128184530

Numerical Modeling of COVID-19 Neurological Effects

ODE/PDE Analysis in R

By: William Schiesser

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Numerical Modeling of COVID-19 Neurological Effects

ODE/PDE Analysis in R

By: William Schiesser

Covid-19 is primarily a respiratory disease which results in impaired oxygenation of blood. The O2-deficient blood then moves through the body, and for the study in this book, the focus is on the blood flowing to the brain. The dynamics of blood flow along the brain capillaries and tissue is modeled as systems of ordinary and partial differential equations (ODE/PDEs). The ODE/PDE methodology is presented through a series of examples, 1. A basic one PDE model for O2 concentration in the brain capillary blood. 2. A two PDE model for O2 concentration in the brain capillary blood and in the brain tissue, with O2 transport across the blood brain barrier (BBB). 3. The two model extended to three PDEs to include the brain functional neuron cell density. Cognitive impairment could result from reduced neuron cell density in time and space (in the brain) that follows from lowered O2 concentration (hypoxia). The computer-based implementation of the example models is presented through routines coded (programmed) in R, a quality, open-source scientific computing system that is readily available from the Internet. Formal mathematics is minimized, e.g., no theorems and proofs. Rather, the presentation is through detailed examples that the reader/researcher/analyst can execute on modest computers. The PDE analysis is based on the method of lines (MOL), an established general algorithm for PDEs, implemented with finite differences. The routines are available from a download link so that the example models can be executed without having to first study numerical methods and computer coding. The routines can then be applied to variations and extensions of the blood/brain hypoxia models, such as changes in the ODE/PDE parameters (constants) and form of the model equations.

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196

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CRC Press

Published Date

ISBN 10

1000509958

ISBN 13

9781000509953

Numerical Integration of Space Fractional Partial Differential Equations

Vol 1 - Introduction to Algorithms and Computer Coding in R

By: Younes Salehi,William E. Schiesser

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Numerical Integration of Space Fractional Partial Differential Equations

Vol 1 - Introduction to Algorithms and Computer Coding in R

By: Younes Salehi,William E. Schiesser

Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: Vol 1: Introduction to Algorithms and Computer Coding in R Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the Riemann-Liouville derivative. Partial differential equations (PDEs) are one of the most used widely forms of mathematics in science and engineering. PDEs can have partial derivatives with respect to (1) an initial value variable, typically time, and (2) boundary value variables, typically spatial variables. Therefore, two fractional PDEs can be considered, (1) fractional in time (TFPDEs), and (2) fractional in space (SFPDEs). The two volumes are directed to the development and use of SFPDEs, with the discussion divided as: Vol 1: Introduction to Algorithms and Computer Coding in R Vol 2: Applications from Classical Integer PDEs. Various definitions of space fractional derivatives have been proposed. We focus on the Caputo derivative, with occasional reference to the Riemann-Liouville derivative. The Caputo derivative is defined as a convolution integral. Thus, rather than being local (with a value at a particular point in space), the Caputo derivative is non-local (it is based on an integration in space), which is one of the reasons that it has properties not shared by integer derivatives. A principal objective of the two volumes is to provide the reader with a set of documented R routines that are discussed in detail, and can be downloaded and executed without having to first study the details of the relevant numerical analysis and then code a set of routines. In the first volume, the emphasis is on basic concepts of SFPDEs and the associated numerical algorithms. The presentation is not as formal mathematics, e.g., theorems and proofs. Rather, the presentation is by examples of SFPDEs, including a detailed discussion of the algorithms for computing numerical solutions to SFPDEs and a detailed explanation of the associated source code.

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202

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Morgan & Claypool Publishers

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ISBN 10

1681732084

ISBN 13

9781681732084

Numerical Integration of Space Fractional Partial Differential Equations

Vol 2 - Applications from Classical Integer PDEs

By: Younes Salehi,William E. Schiesser

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Numerical Integration of Space Fractional Partial Differential Equations

Vol 2 - Applications from Classical Integer PDEs

By: Younes Salehi,William E. Schiesser

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192

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Springer Nature

Published Date

ISBN 10

3031024125

ISBN 13

9783031024122

ODE/PDE Analysis of Antibiotic/Antimicrobial Resistance

Programming in R

By: William E. Schiesser

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ODE/PDE Analysis of Antibiotic/Antimicrobial Resistance

Programming in R

By: William E. Schiesser

ODE/PDE Analysis of Antibiotic/Antimicrobial Resistance: Programming in R presents mathematical models for antibiotic/antimicrobial resistance based on ordinary and partial differential equations (ODE/PDEs). Sections cover the basic ODE model, the detailed PDE model that gives the spatiotemporal distribution of four dependent variable components, including susceptible bacteria population density, resistant bacteria population density, plasmid number, and antibiotic concentration. The computer-based implementation of the example models is presented through routines coded (programmed) in R, a quality, open-source scientific computing system that is readily available from the Internet. As such, formal mathematics is minimized and no theorems and proofs are required. The PDE analysis is based on the method of lines (MOL), an established general algorithm for PDEs that is implemented with finite differences. Routines are available from a download link so that the example models can be executed without having to first study numerical methods and computer coding. Routines can then be applied to variations and extensions of the antibiotic/antimicrobial models, such as changes in the ODE/PDE parameters (constants) and the form of the model equations. - Includes PDE routines based on the method of lines (MOL) for computer-based implementation of PDE models - Offers transportable computer source codes for readers in R, with line-by-line code descriptions as it relates to mathematical model and algorithms - Authored by a leading researcher and educator in PDE models

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124

Publisher

Academic Press

Published Date

ISBN 10

0128242507

ISBN 13

9780128242506

Spatiotemporal Modeling of Stem Cell Differentiation

Partial Differentiation Equation Analysis in R

By: William E. Schiesser

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Spatiotemporal Modeling of Stem Cell Differentiation

Partial Differentiation Equation Analysis in R

By: William E. Schiesser

Spatiotemporal Modeling of Stem Cell Differentiation: Partial Differentiation Equation Analysis in R covers topics surrounding how stem cells evolve into specialized cells during tissue formation and in diseased tissue regeneration. As the process of stem cell differentiation occurs in space and time, the mathematical modeling of spatiotemporal development is expressed in this book as systems of partial differential equations (PDEs). In addition, the book explores important feature of six PDE model which can represent, for example, the development of tissue in organs. In addition, the book covers the computer-based implementation of example models through routines coded (programmed) in R. The routines described in the book are available from a download link so that example models can be executed without having to first study numerical methods and computer coding. The routines can then be applied to variations and extensions of the stem differentiation models, such as changes in the PDE parameters (constants) and the form of the model equations. - Includes PDE routines based on the method of lines (MOL) for computer-based implementation of PDE models - Offers transportable computer source codes for readers in R, with line-by-line code descriptions for mathematical models and algorithms - Authored by a leading researcher and educator in PDE models

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