Finite difference method

Abstract: * Applications of Mickens Discretizations to Boundary Value Problems of Bratu, Gelfand and Others (R Buckmire) * Nonstandard Finite Difference Time Domain Algorithms for Computational Electromagnetics: Applications to Current Topics in Optics and Photonics (J B Cole) * Reliable Finite Difference Schemes with Applications in Mathematical Ecology (D T Dimitrov et al.) * Application of the Nonstandard Finite Difference Method in Non-Smooth Mechanics (Y Dumont) * Finite Difference Schemes on Unbounded Domains (M Ehrhardt) * Dynamically-Consistent Nonstandard Finite Difference Methods for Epidemic Models (A Gumel & K C Patidar) * Nonstandard Finite Difference Methods and Biological Models (S R-J Jang) * Contribution to the Theory of Nonstandard Finite Difference Methods and Applications to Singular Perturbation Problems (J M-S Lubuma & K C Patidar) * Nonstandard Discretization Methods on Lotka-Volterra Differential Equations (L-I W Roeger)

Abstract: Heat Conduction Fundamentals.- Fourier Law.- Mass and Energy Balance Equations.- The Reduction of Transient Heat Conduction Equations and Boundary Conditions.- Substituting Heat Conduction Equation by Two-Equations System.- Variable Change.- Exercises. Solving Heat Conduction Problems.- Heat Transfer Fundamentals.- Two-Dimensional Steady-State Heat Conduction. Analytical Solutions.- Analytical Approximation Methods. Integral Heat Balance Method.- Two-Dimensional Steady-State Heat Conduction. Graphical Method.- Two-Dimensional Steady-State Problems. The Shape Coefficient.- Solving Steady-State Heat Conduction Problems by Means of Numerical Methods.- Finite Element Balance Method and Boundary Element Method.- Transient Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings.- Transient Heat Conduction in Half-Space.- Transient Heat Conduction in Simple-Shape Elements.- Superposition Method in One-Dimensional Transient Heat Conduction Problems.- Transient Heat Conduction in a Semi-Infinite Body. The Inverse Problem.- Inverse Transient Heat Conduction Problems.- Multidimensional Problems. The Superposition Method.- Approximate Analytical Methods for Solving Transient Heat Conduction Problems.- Finite Difference Method.- Solving Transient Heat Conduction Problems by Means of Finite Element Method (FEM).- Numerical-Analytical Methods.- Solving Inverse Heat Conduction Problems by Means of Numerical Methods.- Heat Sources.- Melting and Solidification (Freezing).

Abstract: The newly developed finite-difference vector beam propagation method (FD-VBPM) is analyzed and assessed for application to two-dimensional waveguide structures. The general formulations for the FD-VBPM are derived from the vector wave equations for the electric fields. The stability criteria, the numerical dissipation, and the dispersion of the finite-difference schemes are analyzed by applying the von Neumann method. Important issues regarding the implementation, such as the choice of reference refractive index, the application of numerical boundary conditions, and the use of numerical solution schemes, are discussed. The FD-VBPM is assessed by calculating the attenuation coefficients and the percentage errors of the propagation constants of the TE and TM modes of a step-index slab waveguide. Several salient features of the FD-VBPM are illustrated. >