Applied mathematics and modeling for chemical engineers / Richard G. Rice, emeritus professor Tazewell, Tennessee, Duong D. Do, University of Queensland, St. Lucia, Queensland, Australia
Contents note continued: 1.10.2.Augmented Matrix -- 1.10.3.Pivoting -- 1.10.4.Scaling -- 1.10.5.Gauss Elimination -- 1.10.6.Gauss-Jordan Elimination: Solving Linear Equations -- 1.10.7.LU Decomposition -- 1.11.Iterative Methods -- 1.11.1.Jacobi Method -- 1.11.2.Gauss-Seidel Iteration Method -- 1.11.3.Successive Overrelaxation Method -- 1.12.Summary of the Model Building Process -- 1.13.Model Hierarchy and its Importance in Analysis -- Problems -- References -- 2.Solution Techniques for Models Yielding Ordinary Differential Equations -- 2.1.Geometric Basis and Functionality -- 2.2.Classification of ODE -- 2.3.First-Order Equations -- 2.3.1.Exact Solutions -- 2.3.2.Equations Composed of Homogeneous Functions -- 2.3.3.Bernoulli's Equation -- 2.3.4.Riccati's Equation -- 2.3.5.Linear Coefficients -- 2.3.6.First-Order Equations of Second Degree -- 2.4.Solution Methods for Second-Order Nonlinear Equations -- 2.4.1.Derivative Substitution Method -- 2.4.2.Homogeneous Function Method --
Contents note continued: 10.1.Introduction -- 10.1.1.Classification and Characteristics of Linear Equations -- 10.2.Particular Solutions for PDEs -- 10.2.1.Boundary and Initial Conditions -- 10.3.Combination of Variables Method -- 10.4.Separation of Variables Method -- 10.4.1.Coated Wall Reactor -- 10.5.Orthogonal Functions and Sturm-Liouville Conditions -- 10.5.1.The Sturm-Liouville Equation -- 10.6.Inhomogeneous Equations -- 10.7.Applications of Laplace Transforms for Solutions of PDEs -- Problems -- References -- 11.Transform Methods for Linear PDEs -- 11.1.Introduction -- 11.2.Transforms in Finite Domain: Sturm-Liouville Transforms -- 11.2.1.Development of Integral Transform Pairs -- 11.2.2.The Eigenvalue Problem and the Orthogonality Condition -- 11.2.3.Inhomogeneous Boundary Conditions -- 11.2.4.Inhomogeneous Equations -- 11.2.5.Time-Dependent Boundary Conditions -- 11.2.6.Elliptic Partial Differential Equations -- 11.3.Generalized Sturm-Liouville Integral Transform --
Contents note continued: 11.3.1.Introduction -- 11.3.2.The Batch Adsorber Problem -- Problems -- References -- 12.Approximate and Numerical Solution Methods for PDEs -- 12.1.Polynomial Approximation -- 12.2.Singular Perturbation -- 12.3.Finite Difference -- 12.3.1.Notations -- 12.3.2.Essence of the Method -- 12.3.3.Tridiagonal Matrix and the Thomas Algorithm -- 12.3.4.Linear Parabolic Partial Differential Equations -- 12.3.5.Nonlinear Parabolic Partial Differential Equations -- 12.3.6.Elliptic Equations -- 12.4.Orthogonal Collocation for Solving PDEs -- 12.4.1.Elliptic PDE -- 12.4.2.Parabolic PDE: Example 1 -- 12.4.3.Coupled Parabolic PDE: Example 2 -- 12.5.Orthogonal Collocation on Finite Elements -- Problems -- References -- Appendix A Review of Methods for Nonlinear Algebraic Equations -- A.1.The Bisection Algorithm -- A.2.The Successive Substitution Method -- A.3.The Newton-Raphson Method -- A.4.Rate of Convergence -- A.5.Multiplicity -- A.6.Accelerating Convergence --
Contents note continued: 2.5.Linear Equations of Higher Order -- 2.5.1.Second-Order Unforced Equations: Complementary Solutions -- 2.5.2.Particular Solution Methods for Forced Equations -- 2.5.3.Summary of Particular Solution Methods -- 2.6.Coupled Simultaneous ODE -- 2.7.Eigenproblems -- 2.8.Coupled Linear Differential Equations -- 2.9.Summary of Solution Methods for ODE -- Problems -- References -- 3.Series Solution Methods and Special Functions -- 3.1.Introduction to Series Methods -- 3.2.Properties of Infinite Series -- 3.3.Method of Frobenius -- 3.3.1.Indicial Equation and Recurrence Relation -- 3.4.Summary of the Frobenius Method -- 3.5.Special Functions -- 3.5.1.Bessel's Equation -- 3.5.2.Modified Bessel's Equation -- 3.5.3.Generalized Bessel's Equation -- 3.5.4.Properties of Bessel Functions -- 3.5.5.Differential, Integral, and Recurrence Relations -- Problems -- References -- 4.Integral Functions -- 4.1.Introduction -- 4.2.The Error Function --
Contents note continued: 4.2.1.Properties of Error Function -- 4.3.The Gamma and Beta Functions -- 4.3.1.The Gamma Function -- 4.3.2.The Beta Function -- 4.4.The Elliptic Integrals -- 4.5.The Exponential and Trigonometric Integrals -- Problems -- References -- 5.Staged-Process Models: The Calculus of Finite Differences -- 5.1.Introduction -- 5.1.1.Modeling Multiple Stages -- 5.2.Solution Methods for Linear Finite Difference Equations -- 5.2.1.Complementary Solutions -- 5.3.Particular Solution Methods -- 5.3.1.Method of Undetermined Coefficients -- 5.3.2.Inverse Operator Method -- 5.4.Nonlinear Equations (Riccati Equation) -- Problems -- References -- 6.Approximate Solution Methods for ODE: Perturbation Methods -- 6.1.Perturbation Methods -- 6.1.1.Introduction -- 6.2.The Basic Concepts -- 6.2.1.Gauge Functions -- 6.2.2.Order Symbols -- 6.2.3.Asymptotic Expansions and Sequences -- 6.2.4.Sources of Nonuniformity -- 6.3.The Method of Matched Asymptotic Expansion --
Contents note continued: 6.3.1.Outer Solutions -- 6.3.2.Inner Solutions -- 6.3.3.Matching -- 6.3.4.Composite Solutions -- 6.3.5.General Matching Principle -- 6.3.6.Composite Solution of Higher Order -- 6.4.Matched Asymptotic Expansions for Coupled Equations -- 6.4.1.Outer Expansion -- 6.4.2.Inner Expansion -- 6.4.3.Matching -- Problems -- References -- Part II -- 7.Numerical-Solution Methods (Initial Value Problems) -- 7.1.Introduction -- 7.2.Type of Method -- 7.3.Stability -- 7.4.Stiffness -- 7.5.Interpolation and Quadrature -- 7.6.Explicit Integration Methods -- 7.7.Implicit Integration Methods -- 7.8.Predictor-Corrector Methods and Runge-Kutta Methods -- 7.8.1.Predictor-Corrector Methods -- 7.9.Runge-Kutta Methods -- 7.10.Extrapolation -- 7.11.Step Size Control -- 7.12.Higher Order Integration Methods -- Problems -- References -- 8.Approximate Methods for Boundary Value Problems: Weighted Residuals -- 8.1.The Method of Weighted Residuals --
Contents note continued: 8.1.1.Variations on a Theme of Weighted Residuals -- 8.2.Jacobi Polynomials -- 8.2.1.Rodrigues Formula -- 8.2.2.Orthogonality Conditions -- 8.3.Lagrange Interpolation Polynomials -- 8.4.Orthogonal Collocation Method -- 8.4.1.Differentiation of a Lagrange Interpolation Polynomial -- 8.4.2.Gauss-Jacobi Quadrature -- 8.4.3.Radau and Lobatto Quadrature -- 8.5.Linear Boundary Value Problem: Dirichlet Boundary Condition -- 8.6.Linear Boundary Value Problem: Robin Boundary Condition -- 8.7.Nonlinear Boundary Value Problem: Dirichlet Boundary Condition -- 8.8.One-Point Collocation -- 8.9.Summary of Collocation Methods -- 8.10.Concluding Remarks -- Problems -- References -- 9.Introduction to Complex Variables and Laplace Transforms -- 9.1.Introduction -- 9.2.Elements of Complex Variables -- 9.3.Elementary Functions of Complex Variables -- 9.4.Multivalued Functions -- 9.5.Continuity Properties for Complex Variables: Analyticity -- 9.5.1.Exploiting Singularities --
Contents note continued: 9.6.Integration: Cauchy's Theorem -- 9.7.Cauchy's Theory of Residues -- 9.7.1.Practical Evaluation of Residues -- 9.7.2.Residues at Multiple Poles -- 9.8.Inversion of Laplace Transforms by Contour Integration -- 9.8.1.Summary of Inversion Theorem for Pole Singularities -- 9.9.Laplace Transformations: Building Blocks -- 9.9.1.Taking the Transform -- 9.9.2.Transforms of Derivatives and Integrals -- 9.9.3.The Shifting Theorem -- 9.9.4.Transform of Distribution Functions -- 9.10.Practical Inversion Methods -- 9.10.1.Partial Fractions -- 9.10.2.Convolution Theorem -- 9.11.Applications of Laplace Transforms for Solutions of ODE -- 9.12.Inversion Theory for Multivalued Functions: the Second Bromwich Path -- 9.12.1.Inversion When Poles and Branch Points Exist -- 9.13.Numerical Inversion Techniques -- 9.13.1.The Zakian Method -- 9.13.2.The Fourier Series Approximation -- Problems -- References -- 10.Solution Techniques for Models Producing PDEs --
Contents note continued: References -- Appendix B Derivation of the Fourier-Mellin Inversion Theorem -- Appendix C Table of Laplace Transforms -- Appendix D Numerical Integration -- D.1.Basic Idea of Numerical Integration -- D.2.Newton Forward Difference Polynomial -- D.3.Basic Integration Procedure -- D.3.1.Trapezoid Rule -- D.3.2.Simpson's Rule -- D.4.Error Control and Extrapolation -- D.5.Gaussian Quadrature -- D.6.Radau Quadrature -- D.7.Lobatto Quadrature -- D.8.Concluding Remarks -- References -- Appendix E Nomenclature
Machine generated contents note: Part I -- 1.Formulation of Physicochemical Problems -- 1.1.Introduction -- 1.2.Illustration of the Formulation Process (Cooling of Fluids) -- 1.2.1.Model I: Plug Flow -- 1.2.2.Model II: Parabolic Velocity -- 1.3.Combining Rate and Equilibrium Concepts (Packed Bed Adsorber) -- 1.4.Boundary Conditions and Sign Conventions -- 1.5.Models with Many Variables: Vectors and Matrices -- 1.6.Matrix Definition -- 1.6.1.The Matrix -- 1.6.2.The Vector -- 1.7.Types of Matrices -- 1.7.1.Square Matrix -- 1.7.2.Diagonal Matrix -- 1.7.3.Triangular Matrix -- 1.7.4.Tridiagonal Matrix -- 1.7.5.Symmetric Matrix -- 1.7.6.Sparse Matrix -- 1.7.7.Diagonally Dominant Matrix -- 1.8.Matrix Algebra -- 1.8.1.Addition and Subtraction -- 1.8.2.Multiplication -- 1.8.3.Inverse -- 1.8.4.Matrix Decomposition or Factorization -- 1.9.Useful Row Operations -- 1.9.1.Scaling -- 1.9.2.Pivoting -- 1.9.3.Elimination -- 1.10.Direct Elimination Methods -- 1.10.1.Basic Procedure --
Summary
This book combines the classical analysis and modern applications of applied mathematics for chemical engineers. The book introduces traditional techniques for solving ordinary differential equations (ODEs), adding new material on approximate solution methods such as perturbation techniques and elementary numerical solutions. It also includes analytical methods to deal with important classes of finite-difference equations. The last half discusses numerical solution techniques and partial differential equations (PDEs). The reader will then be equipped to apply mathematics in the formulation of problems in chemical engineering. Like the first edition, there are many examples provided as homework and worked examples
