solving differential equations in r use r

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Solving Differential Equations In R

Author : Karline Soetaert
ISBN : 9783642280702
Genre : Computers
File Size : 39. 94 MB
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Mathematics plays an important role in many scientific and engineering disciplines. This book deals with the numerical solution of differential equations, a very important branch of mathematics. Our aim is to give a practical and theoretical account of how to solve a large variety of differential equations, comprising ordinary differential equations, initial value problems and boundary value problems, differential algebraic equations, partial differential equations and delay differential equations. The solution of differential equations using R is the main focus of this book. It is therefore intended for the practitioner, the student and the scientist, who wants to know how to use R for solving differential equations. However, it has been our goal that non-mathematicians should at least understand the basics of the methods, while obtaining entrance into the relevant literature that provides more mathematical background. Therefore, each chapter that deals with R examples is preceded by a chapter where the theory behind the numerical methods being used is introduced. In the sections that deal with the use of R for solving differential equations, we have taken examples from a variety of disciplines, including biology, chemistry, physics, pharmacokinetics. Many examples are well-known test examples, used frequently in the field of numerical analysis.

Computer L Sung Gew Hnlicher Differentialgleichungen

Author : Lawrence F. Shampine
ISBN : 9783322938015
Genre : Mathematics
File Size : 48. 49 MB
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Gew Hnliche Differentialgleichungen

Author : Wolfgang Walter
ISBN : 9783642963179
Genre : Mathematics
File Size : 79. 96 MB
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Using R For Numerical Analysis In Science And Engineering

Author : Victor A. Bloomfield
ISBN : 9781315360492
Genre : Mathematics
File Size : 77. 90 MB
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Instead of presenting the standard theoretical treatments that underlie the various numerical methods used by scientists and engineers, Using R for Numerical Analysis in Science and Engineering shows how to use R and its add-on packages to obtain numerical solutions to the complex mathematical problems commonly faced by scientists and engineers. This practical guide to the capabilities of R demonstrates Monte Carlo, stochastic, deterministic, and other numerical methods through an abundance of worked examples and code, covering the solution of systems of linear algebraic equations and nonlinear equations as well as ordinary differential equations and partial differential equations. It not only shows how to use R’s powerful graphic tools to construct the types of plots most useful in scientific and engineering work, but also: Explains how to statistically analyze and fit data to linear and nonlinear models Explores numerical differentiation, integration, and optimization Describes how to find eigenvalues and eigenfunctions Discusses interpolation and curve fitting Considers the analysis of time series Using R for Numerical Analysis in Science and Engineering provides a solid introduction to the most useful numerical methods for scientific and engineering data analysis using R.

Advanced Spatial Modeling With Stochastic Partial Differential Equations Using R And Inla

Author : Elias T. Krainski
ISBN : 9780429628214
Genre : Mathematics
File Size : 63. 2 MB
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Modeling spatial and spatio-temporal continuous processes is an important and challenging problem in spatial statistics. Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA describes in detail the stochastic partial differential equations (SPDE) approach for modeling continuous spatial processes with a Matérn covariance, which has been implemented using the integrated nested Laplace approximation (INLA) in the R-INLA package. Key concepts about modeling spatial processes and the SPDE approach are explained with examples using simulated data and real applications. This book has been authored by leading experts in spatial statistics, including the main developers of the INLA and SPDE methodologies and the R-INLA package. It also includes a wide range of applications: * Spatial and spatio-temporal models for continuous outcomes * Analysis of spatial and spatio-temporal point patterns * Coregionalization spatial and spatio-temporal models * Measurement error spatial models * Modeling preferential sampling * Spatial and spatio-temporal models with physical barriers * Survival analysis with spatial effects * Dynamic space-time regression * Spatial and spatio-temporal models for extremes * Hurdle models with spatial effects * Penalized Complexity priors for spatial models All the examples in the book are fully reproducible. Further information about this book, as well as the R code and datasets used, is available from the book website at http://www.r-inla.org/spde-book. The tools described in this book will be useful to researchers in many fields such as biostatistics, spatial statistics, environmental sciences, epidemiology, ecology and others. Graduate and Ph.D. students will also find this book and associated files a valuable resource to learn INLA and the SPDE approach for spatial modeling.

Differential Equations

Author : Lester R. Ford
ISBN : UOM:39015000500960
Genre : Mathematics
File Size : 55. 19 MB
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Differential Equations Problem Solver

Author : David R. Arterburn
ISBN : 9780738668307
Genre : Mathematics
File Size : 56. 51 MB
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Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. Nothing remotely as comprehensive or as helpful exists in their subject anywhere. Perfect for undergraduate and graduate studies. Here in this highly useful reference is the finest overview of differential equations currently available, with hundreds of differential equations problems that cover everything from integrating factors and Bernoulli's equation to variation of parameters and undetermined coefficients. Each problem is clearly solved with step-by-step detailed solutions. DETAILS - The PROBLEM SOLVERS are unique - the ultimate in study guides. - They are ideal for helping students cope with the toughest subjects. - They greatly simplify study and learning tasks. - They enable students to come to grips with difficult problems by showing them the way, step-by-step, toward solving problems. As a result, they save hours of frustration and time spent on groping for answers and understanding. - They cover material ranging from the elementary to the advanced in each subject. - They work exceptionally well with any text in its field. - PROBLEM SOLVERS are available in 41 subjects. - Each PROBLEM SOLVER is prepared by supremely knowledgeable experts. - Most are over 1000 pages. - PROBLEM SOLVERS are not meant to be read cover to cover. They offer whatever may be needed at a given time. An excellent index helps to locate specific problems rapidly. TABLE OF CONTENTS Introduction Units Conversion Factors Chapter 1: Classification of Differential Equations Chapter 2: Separable Differential Equations Variable Transformation u = ax + by Variable Transformation y = vx Chapter 3: Exact Differential Equations Definitions and Examples Solving Exact Differential Equations Making a Non-exact Differential Equation Exact Chapter 4: Homogenous Differential Equations Identifying Homogenous Differential Equations Solving Homogenous Differential Equations by Substitution and Separation Chapter 5: Integrating Factors General Theory of Integrating Factors Equations of Form dy/dx + p(x)y = q(x) Grouping to Simplify Solutions Solution Directly From M(x, y)dx + N(x, y)dy = 0 Chapter 6: Method of Grouping Chapter 7: Linear Differential Equations Integrating Factors Bernoulli's Equation Chapter 8: Riccati's Equation Chapter 9: Clairaut's Equation Geometrical Construction Problems Chapter 10: Orthogonal Trajectories Elimination of Constants Orthogonal Trajectories Differential Equations Derived from Considerations of Analytical Geometry Chapter 11: First Order Differential Equations: Applications I Gravity and Projectile Hooke's Law, Springs Angular Motion Over-hanging Chain Chapter 12: First Order Differential Equations: Applications II Absorption of Radiation Population Dynamics Radioactive Decay Temperature Flow from an Orifice Mixing Solutions Chemical Reactions Economics One-Dimensional Neutron Transport Suspended Cable Chapter 13: The Wronskian and Linear Independence Determining Linear Independence of a Set of Functions Using the Wronskian in Solving Differential Equations Chapter 14: Second Order Homogenous Differential Equations with Constant Coefficients Roots of Auxiliary Equations: Real Roots of Auxiliary: Complex Initial Value Higher Order Differential Equations Chapter 15: Method of Undetermined Coefficients First Order Differential Equations Second Order Differential Equations Higher Order Differential Equations Chapter 16: Variation of Parameters Solution of Second Order Constant Coefficient Differential Equations Solution of Higher Order Constant Coefficient Differential Equations Solution of Variable Coefficient Differential Equations Chapter 17: Reduction of Order Chapter 18: Differential Operators Algebra of Differential Operators Properties of Differential Operators Simple Solutions Solutions Using Exponential Shift Solutions by Inverse Method Solution of a System of Differential Equations Chapter 19: Change of Variables Equation of Type (ax + by + c)dx + (dx + ey + f)dy = 0 Substitutions for Euler Type Differential Equations Trigonometric Substitutions Other Useful Substitutions Chapter 20: Adjoint of a Differential Equation Chapter 21: Applications of Second Order Differential Equations Harmonic Oscillator Simple Pendulum Coupled Oscillator and Pendulum Motion Beam and Cantilever Hanging Cable Rotational Motion Chemistry Population Dynamics Curve of Pursuit Chapter 22: Electrical Circuits Simple Circuits RL Circuits RC Circuits LC Circuits Complex Networks Chapter 23: Power Series Some Simple Power Series Solutions May Be Expanded Finding Power Series Solutions Power Series Solutions for Initial Value Problems Chapter 24: Power Series about an Ordinary Point Initial Value Problems Special Equations Taylor Series Solution to Initial Value Problem Chapter 25: Power Series about a Singular Point Singular Points and Indicial Equations Frobenius Method Modified Frobenius Method Indicial Roots: Equal Special Equations Chapter 26: Laplace Transforms Exponential Order Simple Functions Combination of Simple Functions Definite Integral Step Functions Periodic Functions Chapter 27: Inverse Laplace Transforms Partial Fractions Completing the Square Infinite Series Convolution Chapter 28: Solving Initial Value Problems by Laplace Transforms Solutions of First Order Initial Value Problems Solutions of Second Order Initial Value Problems Solutions of Initial Value Problems Involving Step Functions Solutions of Third Order Initial Value Problems Solutions of Systems of Simultaneous Equations Chapter 29: Second Order Boundary Value Problems Eigenfunctions and Eigenvalues of Boundary Value Problem Chapter 30: Sturm-Liouville Problems Definitions Some Simple Solutions Properties of Sturm-Liouville Equations Orthonormal Sets of Functions Properties of the Eigenvalues Properties of the Eigenfunctions Eigenfunction Expansion of Functions Chapter 31: Fourier Series Properties of the Fourier Series Fourier Series Exppansions Sine and Cosine Expansions Chapter 32: Bessel and Gamma Functions Properties of the Gamma Function Solutions to Bessel's Equation Chapter 33: Systems of Ordinary Differential Equations Converting Systems of Ordinary Differential Equations Solutions of Ordinary Differential Equation Systems Matrix Mathematics Finding Eigenvalues of a Matrix Converting Systems of Ordinary Differential Equations into Matrix Form Calculating the Exponential of a Matrix Solving Systems by Matrix Methods Chapter 34: Simultaneous Linear Differential Equations Definitions Solutions of 2 x 2 Systems Checking Solution and Linear Independence in Matrix Form Solution of 3 x 3 Homogenous System Solution of Non-homogenous System Chapter 35: Method of Perturbation Chapter 36: Non-Linear Differential Equations Reduction of Order Dependent Variable Missing Independent Variable Missing Dependent and Independent Variable Missing Factorization Critical Points Linear Systems Non-Linear Systems Liapunov Function Analysis Second Order Equation Perturbation Series Chapter 37: Approximation Techniques Graphical Methods Successive Approximation Euler's Method Modified Euler's Method Chapter 38: Partial Differential Equations Solutions of General Partial Differential Equations Heat Equation Laplace's Equation One-Dimensional Wave Equation Chapter 39: Calculus of Variations Index WHAT THIS BOOK IS FOR Students have generally found differential equations a difficult subject to understand and learn. Despite the pub.

Spline Collocation Methods For Partial Differential Equations

Author : William E. Schiesser
ISBN : 9781119301035
Genre : MATHEMATICS
File Size : 85. 40 MB
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One-dimensional PDEs -- Multidimensional PDEs -- Navier-Stokes, Burgers equations -- Korteweg-deVries equation -- Maxwell equations -- Poisson-Nernst-Planck equations -- Fokker-Planck equation -- Fisher-Kolmogorov equation -- Klein-Gordon equation -- Boussinesq equation -- Cahn-Hilliard equation -- Camassa-Holm equation -- Burgers-Huxley equation -- Gierer-Meinhardt equations -- Keller-Segel equations -- Fitzhugh-Nagumo equations -- Euler-Poisson-Darboux equation -- Kuramoto-Sivashinsky equation -- Einstein-Maxwell equations

Differential Equation Analysis In Biomedical Science And Engineering

Author : William E. Schiesser
ISBN : 9781118705162
Genre : Mathematics
File Size : 40. 20 MB
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Features a solid foundation of mathematical and computationaltools to formulate and solve real-world PDE problems across variousfields With a step-by-step approach to solving partial differentialequations (PDEs), Differential Equation Analysis in BiomedicalScience and Engineering: Partial Differential Equation Applicationswith R successfully applies computational techniques forsolving real-world PDE problems that are found in a variety offields, including chemistry, physics, biology, and physiology. Thebook provides readers with the necessary knowledge to reproduce andextend the computed numerical solutions and is a valuable resourcefor dealing with a broad class of linear and nonlinear partialdifferential equations. The author’s primary focus is on models expressed assystems of PDEs, which generally result from including spatialeffects so that the PDE dependent variables are functions of bothspace and time, unlike ordinary differential equation (ODE) systemsthat pertain to time only. As such, the book emphasizes details ofthe numerical algorithms and how the solutions were computed.Featuring computer-based mathematical models for solving real-worldproblems in the biological and biomedical sciences and engineering,the book also includes: R routines to facilitate the immediate use of computation forsolving differential equation problems without having to firstlearn the basic concepts of numerical analysis and programming forPDEs Models as systems of PDEs and associated initial and boundaryconditions with explanations of the associated chemistry, physics,biology, and physiology Numerical solutions of the presented model equations with adiscussion of the important features of the solutions Aspects of general PDE computation through various biomedicalscience and engineering applications Differential Equation Analysis in Biomedical Science andEngineering: Partial Differential Equation Applications with Ris an excellent reference for researchers, scientists, clinicians,medical researchers, engineers, statisticians, epidemiologists, andpharmacokineticists who are interested in both clinicalapplications and interpretation of experimental data withmathematical models in order to efficiently solve the associateddifferential equations. The book is also useful as a textbook forgraduate-level courses in mathematics, biomedical science andengineering, biology, biophysics, biochemistry, medicine, andengineering.

Numerical Analysis Of Partial Differential Equations

Author : S. H, Lui
ISBN : 9781118111116
Genre : Mathematics
File Size : 40. 51 MB
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A balanced guide to the essential techniques for solving elliptic partial differential equations Numerical Analysis of Partial Differential Equations provides a comprehensive, self-contained treatment of the quantitative methods used to solve elliptic partial differential equations (PDEs), with a focus on the efficiency as well as the error of the presented methods. The author utilizes coverage of theoretical PDEs, along with the nu merical solution of linear systems and various examples and exercises, to supply readers with an introduction to the essential concepts in the numerical analysis of PDEs. The book presents the three main discretization methods of elliptic PDEs: finite difference, finite elements, and spectral methods. Each topic has its own devoted chapters and is discussed alongside additional key topics, including: The mathematical theory of elliptic PDEs Numerical linear algebra Time-dependent PDEs Multigrid and domain decomposition PDEs posed on infinite domains The book concludes with a discussion of the methods for nonlinear problems, such as Newton's method, and addresses the importance of hands-on work to facilitate learning. Each chapter concludes with a set of exercises, including theoretical and programming problems, that allows readers to test their understanding of the presented theories and techniques. In addition, the book discusses important nonlinear problems in many fields of science and engineering, providing information as to how they can serve as computing projects across various disciplines. Requiring only a preliminary understanding of analysis, Numerical Analysis of Partial Differential Equations is suitable for courses on numerical PDEs at the upper-undergraduate and graduate levels. The book is also appropriate for students majoring in the mathematical sciences and engineering.

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