Applied Mathematics and Modeling for Chemical Engineers: Richard G. Rice (Louisiana State Univ.); Duong D. Do (Univ. of Queensland, St. Lucia, Australia): 9780471303770: Book: Technology: eCampus.com

Bridges the gap between classical analysis and modern applications. Following the chapter on the model building stage, it introduces traditional techniques for solving ordinary differential equations, adding new material on approximate solution methods such as perturbation techniques and elementary numerical solutions. Also includes analytical methods to deal with important classes of finite-difference equations. The last half discusses numerical solution techniques and partial differential equations.

Formulation of Physicochemical Problems

(34)

Introduction

(1)

Illustration of the Formulation Process (Cooling of Fluids)

(6)

Combining Rate and Equilibrium Concepts (Packed Bed Adsorber)

(3)

Boundary Conditions and Sign Conventions

(3)

Summary of the Model Building Process

(1)

Model Hierarchy and Its Importance in Analysis

(11)

References

(1)

Problems

(9)

Solution Techniques for Models Yielding Ordinary Differential Equations (ODE)

(67)

Geometric Basis and Functionality

(2)

Classification of ODE

(1)

First Order Equations

(12)

Exact Solutions

(2)

Equations Composed of Homogeneous Functions

(2)

Bernoulli's Equation

(1)

Riccati's Equation

(4)

Linear Coefficients

(1)

First Order Equations of Second Degree

(1)

Solution Methods for Second Order Nonlinear Equations

(10)

Derivative Substitution Method

(6)

Homogeneous Function Method

(3)

Linear Equations of Higher Order

(28)

Second Order Unforced Equations: Complementary Solutions

(9)

Particular Solution Methods for Forced Equations

(16)

Summary of Particular Solution Methods

(1)

Coupled Simultaneous ODE

(7)

Summary of Solution Methods for ODE

(1)

References

(1)

Problems

(7)

Series Solution Methods and Special Functions

104

(44)

Introduction to Series Methods

104

(2)

Properties of Infinite Series

106

(2)

Method of Frobenius

108

(18)

Indicial Equation and Recurrence Relation

109

(17)

Summary of the Frobenius Method

126

(1)

Special Functions

127

(14)

Bessel's Equation

128

(2)

Modified Bessel's Equation

130

(1)

Generalized Bessel Equation

131

(4)

Properties of Bessel Functions

135

(2)

Differential, Integral and Recurrence Relations

137

(4)

References

141

(1)

Problems

142

(6)

Integral Functions

148

(16)

Introduction

148

(1)

The Error Function

148

(2)

Properties of Error Function

149

(1)

The Gamma and Beta Functions

150

(2)

The Gamma Function

150

(2)

The Beta Function

152

(1)

The Elliptic Integrals

152

(4)

The Exponential and Trigonometric Integrals

156

(2)

References

158

(1)

Problems

158

(6)

Staged-Process Models: The Calculus of Finite Differences

164

(20)

Introduction

164

(2)

Modeling Multiple Stages

165

(1)

Solution Methods for Linear Finite Difference Equations

166

(6)

Complementary Solutions

167

(5)

Particular Solution Methods

172

(4)

Method of Undetermined Coefficients

172

(2)

Inverse Operator Method

174

(2)

Nonlinear Equations (Riccati Equation)

176

(3)

References

179

(1)

Problems

179

(5)

Approximate Solution Methods for ODE: Perturbation Methods

184

(41)

Perturbation Methods

184

(5)

Introduction

184

(5)

The Basic Concepts

189

(6)

Gauge Functions

189

(1)

Order Symbols

190

(1)

Asymptotic Expansions and Sequences

191

(2)

Sources of Nonuniformity

193

(2)

The Method of Matched Asymptotic Expansion

195

(12)

Matched Asymptotic Expansions for Coupled Equations

202

(5)

References

207

(1)

Problems

208

(17)

Numerical Solution Methods (Initial Value Problems)

225

(43)

Introduction

225

(5)

Type of Method

230

(2)

Stability

232

(11)

Stiffness

243

(3)

Interpolation and Quadrature

246

(3)

Explicit Integration Methods

249

(3)

Implicit Integration Methods

252

(1)

Predictor-Corrector Methods and Runge--Kutta Methods

253

(5)

Predictor-Corrector Methods

253

(1)

Runge--Kutta Methods

254

(4)

Extrapolation

258

(1)

Step Size Control

258

(2)

Higher Order Integration Methods

260

(1)

References

260

(1)

Problems

261

(7)

Approximate Methods for Boundary Value Problems: Weighted Residuals

268

(63)

The Method of Weighted Residuals

268

(17)

Variations on a Theme of Weighted Residuals

271

(14)

Jacobi Polynomials

285

(4)

Rodrigues Formula

285

(1)

Orthogonality Conditions

286

(3)

Lagrange Interpolation Polynomials

289

(1)

Orthogonal Collocation Method

290

(6)

Differentiation of a Lagrange Interpolation Polynomial

291

(2)

Gauss--Jacobi Quadrature

293

(2)

Radau and Lobatto Quadrature

295

(1)

Linear Boundary Value Problem---Dirichlet Boundary Condition

296

(5)

Linear Boundary Value Problem---Robin Boundary Condition

301

(3)

Nonlinear Boundary Value Problem---Dirichlet Boundary Condition

304

(5)

One-Point Collocation

309

(2)

Summary of Collocation Methods

311

(2)

Concluding Remarks

313

(1)

References

313

(1)

Problems

314

(17)

Introduction to Complex Variables and Laplace Transforms

331

(66)

Introduction

331

(1)

Elements of Complex Variables

332

(2)

Elementary Functions of Complex Variables

334

(1)

Multivalued Functions

335

(2)

Continuity Properties for Complex Variables: Analyticity

337

(4)

Exploiting Singularities

341

(1)

Integration: Cauchy's Theorem

341

(4)

Cauchy's Theory of Residues

345

(5)

Practical Evaluation of Residues

347

(2)

Residues at Multiple Poles

349

(1)

Inversion of Laplace Transforms by Contour Integration

350

(4)

Summary of Inversion Theorem for Pole Singularities

353

(1)

Laplace Transformations: Building Blocks

354

(9)

Taking the Transform

354

(3)

Transforms of Derivatives and Integrals

357

(3)

The Shifting Theorem

360

(1)

Transform of Distribution Functions

361

(2)

Practical Inversion Methods

363

(5)

Partial Fractions

363

(3)

Convolution Theorem

366

(2)

Applications of Laplace Transforms for Solutions of ODE

368

(10)

Inversion Theory for Multivalued Functions: The Second Bromwich Path

378

(5)

Inversion when Poles and Branch Points Exist

382

(1)

Numerical Inversion Techniques

383

(7)

The Zakian Method

383

(5)

The Fourier Series Approximation

388

(2)

References

390

(1)

Problems

390

(7)

Solution Techniques for Models Producing PDEs

397

(89)

Introduction

397

(8)

Classification and Characteristics of Linear Equations

402

(3)

Particular Solutions for PDEs

405

(4)

Boundary and Initial Conditions

406

(3)

Combination of Variables Method

409

(11)

Separation of Variables Method

420

(6)

Coated Wall Reactor

421

(5)

Orthogonal Functions and Sturm--Liouville Conditions

426

(8)

The Sturm--Liouville Equation

426

(8)

Inhomogeneous Equations

434

(9)

Applications of Laplace Transforms for Solutions of PDEs

443

(11)

References

454

(1)

Problems

455

(31)

Transform Methods for Linear PDEs

486

(60)

Introduction

486

(1)

Transforms in Finite Domain: Sturm--Liouville Transforms

487

(34)

Development of Integral Transform Pairs

487

(7)

The Eigenvalue Problem and the Orthogonality Condition

494

(10)

Inhomogeneous Boundary Conditions

504

(7)

Inhomogeneous Equations

511

(2)

Time-Dependent Boundary Conditions

513

(3)

Elliptic Partial Differential Equations

516

(5)

Generalized Sturm--Liouville Integral Transform

521

(16)

Introduction

521

(1)

The Batch Adsorber Problem

521

(16)

References

537

(1)

Problems

538

(8)

Approximate and Numerical Solution Methods for PDEs

546

(84)

Polynomial Approximation

546

(16)

Singular Perturbation

562

(10)

Finite Difference

572

(20)

Notations

573

(1)

Essence of the Method

574

(2)

Tridiagonal Matrix and the Thomas Algorithm

576

(2)

Linear Parabolic Partial Differential Equations

578

(8)

Nonlinear Parabolic Partial Differential Equations

586

(2)

Elliptic Equations

588

(4)

Orthogonal Collocation for Solving PDEs

592

(11)

Elliptic PDE

592

(6)

Parabolic PDE: Example 1

598

(2)

Coupled Parabolic PDE: Example 2

600

(3)

Orthogonal Collocation on Finite Elements

603

(12)

References

615

(1)

Problems

616

(14)

Appendix A: Review of Methods for Nonlinear Algebraic Equations

630

(14)

A.1 The Bisection Algorithm

630

(2)

A.2 The Successive Substitution Method

632

(3)

A.3 The Newton--Raphson Method

635

(4)

A.4 Rate of Convergence

639

(2)

A.5 Multiplicity

641

(1)

A.6 Accelerating Convergence

642

(1)

A.7 References

643

(1)

Appendix B: Vectors and Matrices

644

(19)

B.1 Matrix Definition

644

(2)

B.2 Types of Matrices

646

(1)

B.3 Matrix Algebra

647

(2)

B.4 Useful Row Operations

649

(2)

B.5 Direct Elimination Methods

651

(8)

B.5.1 Basic Procedure

651

(1)

B.5.2 Augmented Matrix

652

(2)

B.5.3 Pivoting

654

(1)

B.5.4 Scaling

655

(1)

B.5.5 Gauss Elimination

656

(1)

B.5.6 Gauss--Jordan Elimination

656

(2)

B.5.7 LU Decomposition

658

(1)

B.6 Iterative Methods

659

(1)

B.6.1 Jacobi Method

659

(1)

B.6.2 Gauss--Seidel Iteration Method

660

(1)

B.6.3 Successive Overrelaxation Method

660

(1)

B.7 Eigenproblems

660

(1)

B.8 Coupled Linear Differential Equations

661

(1)

B.9 References

662

(1)

Appendix C: Derivation of the Fourier--Mellin Inversion Theorem

663

(8)

Appendix D: Table of Laplace Transforms

671

(5)

Appendix E: Numerical Integration

676

(18)

E.1 Basic Idea of Numerical Integration

676

(1)

E.2 Newton Forward Difference Polynomial

677

(1)

E.3 Basic Integration Procedure

678

(4)

E.3.1 Trapezoid Rule

678

(2)

E.3.2 Simpson's Rule

680

(2)

E.4 Error Control and Extrapolation

682

(1)

E.5 Gaussian Quadrature

683

(4)

E.6 Radau Quadrature

687

(3)

E.7 Lobatto Quadrature

690

(3)

E.8 Concluding Remarks

693

(1)

E.9 References

693

(1)

Nomenclature

694

(4)

Postface

698

(3)

Index

701


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