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Numerical Methods for Solving Partial Differential Equations: A Comprehensive Introduction for Scientists and Engineers [Kõva köide]

(Princeton University)
  • Formaat: Hardback, 304 pages, kõrgus x laius x paksus: 259x180x20 mm, kaal: 771 g
  • Ilmumisaeg: 12-Feb-2019
  • Kirjastus: John Wiley & Sons Inc
  • ISBN-10: 1119316111
  • ISBN-13: 9781119316114
Teised raamatud teemal:
Numerical Methods for Solving Partial Differential Equations: A Comprehensive Introduction for Scientists and EngineersSuurem pilt
  • Formaat: Hardback, 304 pages, kõrgus x laius x paksus: 259x180x20 mm, kaal: 771 g
  • Ilmumisaeg: 12-Feb-2019
  • Kirjastus: John Wiley & Sons Inc
  • ISBN-10: 1119316111
  • ISBN-13: 9781119316114
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781119316114.html
Märksõnad:

A comprehensive guide to numerical methods for simulating physical-chemical systems 

This book offers a systematic, highly accessible presentation of numerical methods used to simulate the behavior of physical-chemical systems. Unlike most books on the subject, it focuses on methodology rather than specific applications. Written for students and professionals across an array of scientific and engineering disciplines and with varying levels of experience with applied mathematics, it provides comprehensive descriptions of numerical methods without requiring an advanced mathematical background.

Based on its author’s more than forty years of experience teaching numerical methods to engineering students, Numerical Methods for Solving Partial Differential Equations presents the fundamentals of all of the commonly used numerical methods for solving differential equations at a level appropriate for advanced undergraduates and first-year graduate students in science and engineering. Throughout, elementary examples show how numerical methods are used to solve generic versions of equations that arise in many scientific and engineering disciplines. In writing it, the author took pains to ensure that no assumptions were made about the background discipline of the reader.

  • Covers the spectrum of numerical methods that are used to simulate the behavior of physical-chemical systems that occur in science and engineering
  • Written by a professor of engineering with more than forty years of experience teaching numerical methods to engineers
  • Requires only elementary knowledge of differential equations and matrix algebra to master the material
  • Designed to teach students to understand, appreciate and apply the basic mathematics and equations on which Mathcad and similar commercial software packages are based

Comprehensive yet accessible to readers with limited mathematical knowledge, Numerical Methods for Solving Partial Differential Equations is an excellent text for advanced undergraduates and first-year graduate students in the sciences and engineering. It is also a valuable working reference for professionals in engineering, physics, chemistry, computer science, and applied mathematics.

Preface xi
1 Interpolation
1(32)
1.1 Purpose
1(1)
1.2 Definitions
1(1)
1.3 Example
2(1)
1.4 Weirstraus Approximation Theorem
3(1)
1.5 Lagrange Interpolation
4(4)
1.5.1 Example
6(2)
1.6 Compare P2(θ) and f(θ)
8(1)
1.7 Error of Approximation
9(5)
1.8 Multiple Elements
14(6)
1.8.1 Example
17(3)
1.9 Hermite Polynomials
20(3)
1.10 Error in Approximation by Hermites
23(1)
1.11
Chapter Summary
24(1)
1.12 Problems
24(9)
2 Numerical Differentiation
33(22)
2.1 General Theory
33(1)
2.2 Two-Point Difference Formulae
34(3)
2.2.1 Forward Difference Formula
35(1)
2.2.2 Backward Difference Formula
36(1)
2.2.3 Example
36(1)
2.2.4 Error of the Approximation
36(1)
2.3 Two-Point Formulae from Taylor Series
37(3)
2.4 Three-point Difference Formulae
40(6)
2.4.1 First-Order Derivative Difference Formulae
41(2)
2.4.2 Second-Order Derivatives
43(3)
2.5
Chapter Summary
46(1)
2.6 Problems
46(9)
3 Numerical Integration
55(12)
3.1 Newton-Cotes Quadrature Formula
55(7)
3.1.1 Lagrange Interpolation
55(1)
3.1.2 Trapezoidal Rule
56(1)
3.1.3 Simpson's Rule
57(1)
3.1.4 General Form
58(1)
3.1.5 Example using Simpson's Rule
59(1)
3.1.6 Gauss Legendre Quadrature
59(3)
3.2
Chapter Summary
62(1)
3.3 Problems
63(4)
4 Initial Value Problems
67(16)
4.1 Euler Forward Integration Method Example
68(1)
4.2 Convergence
69(3)
4.3 Consistency
72(1)
4.4 Stability
73(1)
4.4.1 Example of Stability
74(1)
4.5 Lax Equivalence Theorem
74(1)
4.6 Runge--Kutta Type Formulae
75(3)
4.6.1 General Form
75(1)
4.6.2 Runge--Kutta First-Order Form (Euler's Method)
75(1)
4.6.3 Runge--Kutta Second-Order Form
75(3)
4.7
Chapter Summary
78(1)
4.8 Problems
78(5)
5 Weighted Residuals Methods
83(56)
5.1 Finite Volume or Subdomain Method
84(10)
5.1.1 Example
86(7)
5.1.2 Finite Difference Interpretation of the Finite Volume Method
93(1)
5.2 Galerkin Method for First Order Equations
94(8)
5.2.1 Finite-Difference Interpretation of the Galerkin Approximation
102(1)
5.3 Galerkin Method for Second-Order Equations
102(10)
5.3.1 Finite Difference Interpretation of Second-Order Galerkin Method
111(1)
5.4 Finite Volume Method for Second-Order Equations
112(11)
5.4.1 Example of Finite Volume Solution of a Second-Order Equation
116(6)
5.4.2 Finite Difference Representation of the Finite-Volume Method for Second-Order Equations
122(1)
5.5 Collocation Method
123(10)
5.5.1 Collocation Method for First-Order Equations
123(3)
5.5.2 Collocation Method for Second-Order Equations
126(7)
5.6
Chapter Summary
133(1)
5.7 Problems
133(6)
6 Initial Boundary-Value Problems
139(30)
6.1 Introduction
139(1)
6.2 Two Dimensional Polynomial Approximations
139(2)
6.2.1 Example of a Two Dimensional Polynomial Approximation
140(1)
6.3 Finite Difference Approximation
141(8)
6.3.1 Example of Implicit First-Order Accurate Finite Difference Calculation
144(2)
6.3.2 Example of Second Order Accurate Finite Difference Approximation in Space
146(3)
6.4 Stability of Finite Difference Approximations
149(9)
6.4.1 Example of Stability
153(3)
6.4.2 Example Simulation
156(2)
6.5 Galerkin Finite Element Approximations in Time
158(4)
6.5.1 Strategy One: Forward Difference Approximation
160(1)
6.5.2 Strategy Two: Backward Difference Approximation
161(1)
6.6
Chapter Summary
162(1)
6.7 Problems
162(7)
7 Finite Difference Methods in Two Space
169(12)
7.1 Example Problem
174(1)
7.2
Chapter Summary
175(1)
7.3 Problems
176(5)
8 Finite Element Methods in Two Space
181(58)
8.1 Finite Element Approximations over Rectangles
181(14)
8.2 Finite Element Approximations over Triangles
195(16)
8.2.1 Formulation of Triangular Basis Functions
196(4)
8.2.2 Example Problem of Finite Element Approximation over Triangles
200(6)
8.2.3 Second Type or Neumann Boundary-Value Problem
206(5)
8.3 Isoparametric Finite Element Approximation
211(19)
8.3.1 Natural Coordinate Systems
211(6)
8.3.2 Basis Functions
217(2)
8.3.3 Calculation of the Jacobian
219(4)
8.3.4 Example of Isoparametric Formulation
223(7)
8.4
Chapter Summary
230(1)
8.5 Problems
230(9)
9 Finite Volume Approximation in Two Space
239(34)
9.1 Finite Volume Formulation
239(7)
9.2 Finite Volume Example Problem 1
246(16)
9.2.1 Problem Definition
246(1)
9.2.2 Weighted Residual Formulation
246(2)
9.2.3 Element Coefficient Matrices
248(1)
9.2.4 Evaluation of the Line Integral
249(7)
9.2.5 Evaluation of the Area Integral
256(4)
9.2.6 Global Matrix Assembly
260(2)
9.3 Finite Volume Example Problem Two
262(4)
9.3.1 Problem Definition
262(1)
9.3.2 Weighted Residual Formulation
262(1)
9.3.3 Element Coefficient Matrices
263(2)
9.3.4 Evaluation of the Source Term
265(1)
9.4
Chapter Summary
266(1)
9.5 Problems
266(7)
10 Initial Boundary-Value Problems
273(6)
10.1 Mass Lumping
276(1)
10.2
Chapter Summary
276(1)
10.3 Problems
276(3)
11 Boundary-Value Problems in Three Space
279(10)
11.1 Finite Difference Approximations
279(1)
11.2 Finite Element Approximations
280(5)
11.3
Chapter Summary
285(4)
12 Nomenclature
289(4)
Index 293
George F. Pinder, PhD, is a Distinguished Professor of Civil and Environmental Engineering with a secondary appointments in Mathematics and Statistics and Computer Science at the University of Vermont, Burlington, Vermont. He is the author or co-author of ten books in numerical mathematics and engineering. Dr. Pinder is the recipient of numerous national and international honors and is a member of the National Academy of Engineering.