Finite difference method

Abstract: In this paper the MHD of a Non-Newtonian unsteady flow of an incompressible fluid under the effect of couple stresses and a uniform external magnetic field is analysed by using the Eyring Powell model. In the first approximation the solution is obtained by using the Mathematica computational program with assuming a pulsatile pressure gradient in the direction of the motion. In the second order approximation a numerical solution of the non-linear partial differential equation is obtained by using a finite difference method. The effects of different parameters are discussed with the help of graphs in the two cases.

Abstract: The nonlinear partial differential equation governing spacing of horizontal, parallel, subsurface, agricultural drains is solved numerically by finite difference methods. The differential equation is based on the Dupuit-Forchheimer assumption. Since use of Hooghoudt’s equivalent depth is suggested as a means of accounting for the effect of convergence, a simplified formula is derived from computing that quantity. Solutions are presented in the form of dimensionless curve families whose abscissas are a time parameter and whose ordinates represent: (1) Maximum water table height between drains, (2) rate of discharge, and (3) volume of water removed. Individual curve parameters establish the depth from drains to an impermeable barrier which can vary from zero to infinity. An example of application is given.

Abstract: Fundamental Concepts Introduction Review of Electromagnetic Theory Classification of EM Problems Some Important Theorems Analytical Methods Introduction Separation of Variables Separation of Variables in Rectangular Coordinates Separation of Variables in Cylindrical Coordinates Separation of Variables in Spherical Coordinates Some Useful Orthogonal Functions Series Expansion Practical Applications Attenuation Due to Raindrops Concluding Remarks Finite Difference Methods Introduction Finite Difference Schemes Finite Differencing of Parabolic PDEs Finite Differencing of Hyperbolic PDEs Finite Differencing of Elliptic PDEs Accuracy and Stability of FD Solutions Practical Applications I - Guided Structures Practical Applications II - Wave Scattering (FDTD) Absorbing Boundary Conditions for FDTD Finite Differencing for Nonrectangular Systems Numerical Integration Concluding Remarks Variational Methods Introduction Operators in Linear Spaces Calculus of Variations Construction of Functionals from PDEs Rayleigh-Ritz Method Weighted Residual Method Eigenvalue Problems Practical Applications Concluding Remarks Moment Methods Introduction Integral Equations Green's Functions Applications I - Quasi-Static Problems Applications II - Scattering Problems Applications III- Radiation Problems Applications IV - EM Absorption in the Human Body Concluding Remarks Finite Element Method Introduction Solution of Laplace's Equation Solution of Poisson's Equation Solution of the Wave Equation Automatic Mesh Generation I - Rectangular Domains Automatic Mesh Generation II - Arbitrary Domains Bandwidth Reduction Higher Order Elements Three-Dimensional Elements Finite Element Methods for Exterior Problems Finite-Element Time-Domain Method Concluding Remarks Transmission-line-matrix Method Introduction Transmission-line Equations Solution of Diffusion Equation Solution of Wave Equations Inhomogeneous and Lossy Media in TLM Three-Dimensional TLM Mesh Error Sources and Correction Absorbing Boundary Conditions Concluding Remarks Monte Carlo Methods Introduction Generation of Random Numbers and Variables Evaluation of Error Numerical Integration Solution of Potential Problems Regional Monte Carlo Methods Time-Dependent Problems Concluding Remarks Method of Lines Introduction Solution of Laplace's Equation Solution of Wave Equation Time-Domain Solution Concluding Remarks References Problems APPENDICES Vector Relations Vector Identities Vector Theorems Orthogonal Coordinates Programming in MATLAB MATLAB Fundamentals Using MATLAB to Plot Programming with MATLAB Functions Solving Equations Programming Hints Other Useful MATLAB Commands Solution of Simultaneous Equations Elimination Methods Iterative Methods Matrix Inversion Eigenvalue Problems Answers to Odd-Numbered Problems