Page 5 - Essentials of applied mathematics for scientists and engineers

MOBK070-FM MOBKXXX-Sample.cls March 22, 2007 13:6

v

Contents

1. Partial Differential Equations in Engineering . .................................. 1
1.1 Introductory Comments .................................................. 1
1.2 Fundamental Concepts ................................................... 1
Problems......................................................3
1.3 The Heat Conduction (or Diffusion) Equation .............................. 3
1.3.1 Rectangular Cartesian Coordinates . . . .............................. 3
1.3.2 Cylindrical Coordinates . .......................................... 5
1.3.3 Spherical Coordinates . . ........................................... 6
The Laplacian Operator ........................................ 6
1.3.4 Boundary Conditions. .............................................7
1.4 The Vibrating String ..................................................... 7
1.4.1 Boundary Conditions. .............................................8
1.5 Vibrating Membrane ..................................................... 8
1.6 Longitudinal Displacements of an Elastic Bar ............................... 9
Further Reading..........................................................9

2. The Fourier Method: Separation of Variables .................................. 11
2.1 Heat Conduction ....................................................... 12
2.1.1 Scales and Dimensionless Variables. . ..............................12
2.1.2 Separation of Variables . . . . ....................................... 13
2.1.3 Superposition ................................................... 14
2.1.4 Orthogonality . .................................................. 15
2.1.5 Lessons .........................................................15
Problems . . .................................................. 16
2.1.6 Scales and Dimensionless Variables. . ..............................16
2.1.7 Separation of Variables . . . . ....................................... 17
2.1.8 Choosing the Sign of the Separation Constant. . . ...................17
2.1.9 Superposition ................................................... 19
2.1.10 Orthogonality . .................................................. 19
2.1.11 Lessons .........................................................20
2.1.12 Scales and Dimensionless Variables. . ..............................20