Table of Contents: Part I: Numerical methods
Chapter 1: Iteration, pp. 1-30
1.1 Introduction
1.2 Iteration
1.2.1 Algebraic equations
1.2.2 Improved Iteration
1.2.3 Newton-Raphson method
1.2.3.1 Newton-Raphson method graphical interpretation
1.2.4 The Secant method and the method of false position
1.2.5 Improved the Newton-Raphlson method
1.2.6 Example & Computer program
1.3 Linear system
1.3.1 Methods of solution
1.3.2 The row-sum criteria for convergence by total steps
1.4 Algebraic equations
1.4.1 Iteration for algebraic system
1.4.2 Newton-Raphson method for a system of algebraic equations
1.4.3 Numerical example: system of two nonlinear equations
Chapter 2: Interpolation, Differentiation & Integration, pp. 31-46
2.1 Finite differences
2.1.1 Newton’s formula for forward interpolation
2.1.2 Newton’s formula for backward Interpolation
2.1.3 Inverse interpolation
2.1.4 Numerical Differentiation & Integration
2.2 Integration
2.2.1 Simpson’s rule
2.2.2 Quadrature: error analysis - Trapezoidal rule
2.2.3 Product integration
2.3 Finite difference approximations by Taylor expansions
Chapter 3: Ordinary Differential Equations, pp. 47-115
3.1 Initial value problems pp.519-632
3.1.1 Existence and uniqueness: Contraction mapping
3.1.2 Euler’s method (FORTRAN program)
3.1.3 Unstable central difference solution (FORTRAN program)
3.1.4 Error analysis for finite difference (Solution to differential equations)
3.1.5 Truncation error
3.1.6 Stability analysis for a non-linear ordinary differential equation
3.1.7 Finite difference formula with higher order truncation errors
3.1.8 Stability
3.1.9 Consistency, convergence and stability theory
3.1.10 Equivalence of convergence and stability for a consistent scheme
3.1.11 Stability of second difference equation
3.1.12 Nonlinear spring problem
3.1.13 Stability of the nonlinear spring problem
3.1.14 Instability of Patankar’s scheme
3.1.15 Accurate, stable finite difference schemes
3.1.16 Runge-Kutta method
3.1.17 Runge-Kutta for simultaneous equations
3.1.18 Stiff differential equations
3.2 Boundary value problems
3.2.1 Introduction
3.2.2 Illustration of a solution by finite differences
3.2.3 A non-linear boundary value problem
3.2.4 Solution of boundary value problems by Runge-Kutta and Newton-Raphson Methods
3.2.5 Quasilinearization
3.2.5.1 Solution of the Blasius problem-laminar flow over a flat plate
3.2.5.2 Solution by superposition and initial value techniques
3.2.5.3 The Thomas method: tridiagonal matrix inversion
Appendix A: Adiabatic Humidification in Boundary Layer Flow for Nonlinear Boundary Value Problems by Runge-Kutta and Newton-Raphson methods.
Appendix B: Difference Equations
Chapter 4: Numerical Solution of Partial Differential Equations (Solution by finite differences), pp. 117-143
4.1 Solution of the diffusion equation by finite differences
4.1.1 An explicit computational method
4.1.2 Schmidt method
4.1.3 Restriction on the ratio “R” example
4.1.4 Consistency & Errors
4.1.5 Explicit molecule – Stability - Von Neumann Method
4.1.6 Stability of explicit scheme (Alternative method)
4.1.7 An implicit method
4.1.8 Explicit methods using more than two grids
4.1.9 The Crank – Nicolson method – implicit
4.1.10 ADI-method for heat equation
4.1.11 Conditional consistency for Du Fort-Frankel scheme
4.2 Stability theory: System of diffusion equations
4.3 Stability of Navier-Stokes equations
4.3.1 Free convective in a 2-D cavity by finite differences
4.3.2 Finite difference form of the parabolic equations – explicit form
4.3.3 Von Neumann stability for simultaneous PDE
Chapter 5. - Numerical Solution of Partial Differential Equations: Hyperbolic Partial Differential Equations: The Theory of Characteristics, pp. 145-223
5.1 First order partial differential equation
5.1.1 Integration theory
5.1.2 Unsteady plug flow reactor
5.1.3 Separation in a chromatographic column
5.1.4 Linear isotherm: Henry’s law
5.1.5 Dispersion
5.1.6 Shock formation
5.1.7 General first order partial differential equation by characteristics
5.2 Numerical solution of a hyperbolic system of first order partial differential equations
5.2.1 Decoupling of quasi-linear first order PDE
5.2.2 Transient pipe flow
5.2.3 Dimensionless representation of pipe flow
5.2.4 Frictionless flow example
5.2.5 Numerical solution by characteristics
5.2.6 Uniqueness of solution and boundary conditions
5.3 Relation of characteristics to stability
5.4 Partial differential equations theory
5.4.1 Well-posedness of the system
5.4.2 Lax’s equivalence theorem
5.4.3 Relationship between characteristics and stability
5.4.4 Two-step Lax-Wendroff method
5.5 Hyperbolic systems: classical approach
5.5.1 Method of characteristic –finite differences
5.5.2 Classification of a second order partial differential equation-quasi linear
5.5.3 An introduction to the theory of characteristics & classification of PDE
5.6 System of hyperbolic partial differential equations in more than two independent variables
Appendix A: First order partial differential equation
Appendix B: Wave equation
Appendix C: Decoupling method example
Part II Computational Fluidization and Reactor Design
Chapter 1: Experimental Foundation, pp. 227-277
Chapter 2: Elementary Multiphase Kinetic Theory, pp. 279-306
Chapter 3: Multiphase Kinetic Theory of Mixtures, pp. 307-322
Chapter 4: Computation of Flow Regimes for Fluidization, pp. 323-349
Chapter 5: Finite Volume Method for Navier-Stokes Equations, pp. 351-380
Chapter 6: Manual for Computer Programs and Tutorial, pp. 381-469
Chapter 7: Efficient Coal Gasifier-Fuel Cell with CO2 Sequestration, pp. 471-494
Chapter 8: Computation of Mass Transfer Coefficients with Reactions, pp. 495-516
Part III Green’s Functions and Functional Analysis
Chapter 1 : Green’s Functions for Ordinary Differential Equations, pp. 519-546
Chapter 2 : Nonlinear Integral Equations, pp. 547-583
Chapter 3 : Green’s functions for the Laplace’s equation, pp. 585-608
Chapter 4 : Green’s Functions for the diffusion equation, pp. 609-632
Index |