This book provides a good introduction to modern computational methods for Partial Differential Equations in Mechanics. Finite-difference methods for parabolic, hyperbolic as well as elliptic partial differential equations are discussed.
A gradual and inductive approach to the numerical concepts has been used, such that the presentation of the theory is easily accessible to upper-level undergraduate and graduate students. Special attention has been given to the applications, with many examples and exercises provided along with solutions. For each type of equation, physical models are carefully derived and presented in full details.
Windows programs developed in C++ language have been included in the accompanying CD-ROM. These programs can be easily modified to solve different problems, and the reader is encouraged to take full advantage of the innovative features of this powerful development tool.
Author(s): Berardino D'Acunto
Series: Series on Advances in Mathematics for Applied Sciences 67
Edition: Har/Cdr
Publisher: World Scientific Publishing Co Pte Ltd
Year: 2004
Language: English
Pages: 293
Preface......Page 8
Contents......Page 12
Finite Differences......Page 16
Function discretization......Page 17
Finite-difference approximation of derivatives......Page 18
Approximation for higher-order derivatives......Page 20
Finite-difference operators......Page 22
Fourier model of heat conduction......Page 24
Heat equation......Page 25
Initial and boundary conditions......Page 28
Phase-change problems......Page 29
Heat conduction in a moving medium......Page 32
Fick's law and diffusion......Page 33
An explicit method for the heat equation......Page 36
Non-dimensional form of the heat equation......Page 37
The classical explicit method......Page 38
Matrix form......Page 39
Stability......Page 40
Consistency......Page 43
Convergence......Page 45
Neumann boundary conditions......Page 46
Boundary conditions of the third kind......Page 48
A Windows program......Page 50
Introduction......Page 51
Creating a new project......Page 52
Equation data......Page 55
Initial data......Page 61
Boundary data......Page 65
Numerical results......Page 70
Analysis of data......Page 74
Graphical results. Examples......Page 75
Icons......Page 81
The Heat2 project......Page 82
Introduction......Page 83
The project......Page 84
Equation class......Page 89
Initial and boundary data......Page 95
View class......Page 98
Examples......Page 106
Parabolic equations......Page 108
A simple implicit method......Page 109
Crank-Nicolson method......Page 112
Von Neumann stability......Page 116
Combined scheme......Page 121
An example of unstable method......Page 122
DuFort-Frankel method......Page 123
Matrix stability......Page 127
Stability analysis by the energy method......Page 131
Variable diffusivity coefficient......Page 136
Diffusion-convection equation......Page 138
Nonlinear equation......Page 141
A free boundary problem......Page 143
The Heat3 project......Page 148
Introduction......Page 149
Main menu......Page 152
Implementing the document class......Page 153
Dialog resources......Page 158
Implementing the view class......Page 172
Using the program......Page 186
Wave motions......Page 190
Flexible strings......Page 191
Wave equation......Page 195
Waves in elastic solids......Page 197
Motion of fluids......Page 201
Free piston problem......Page 204
Finite-difference methods for the wave equation......Page 206
Courant-Friederichs-Lewy method......Page 207
Implicit methods......Page 211
Perturbed wave equation......Page 216
Hyperbolic equations......Page 224
First-order equations. Explicit methods......Page 225
Implicit methods......Page 233
Systems of first-order equations......Page 236
Nonlinear systems......Page 241
Elliptic equations......Page 244
Porous media......Page 245
Dirichlet problem......Page 248
Curved boundary......Page 254
Three-dimensional applications......Page 256
Green's identities. Consequences......Page 257
Neumann problem......Page 260
Third boundary value problem......Page 266
Appendix A - Classification of PDEs......Page 270
Appendix B - Elements of linear algebra......Page 278
Bibliography......Page 288
Index......Page 292
