力学中的偏微分方程（第1卷）

1. Mathematical preliminaries
1.1 Components of a vector
1.2 Dot or scalar product
1.3 Cross or vector product
1.4 Derivative of a vector
1.5 Results involving derivatives
1.6 Partial derivatives of vectors
1.6.1 The gradient of a scalar field
1.6.2 The divergence of a vector field
1.6.3 The Laplacian of a scalar or vector field
1.6.4 The curl of a vector field
1.6.5 Other formulae involving
1.7 Divergence of a vector field: an application
1.8 Divergence or Green''s theorem
1.9 Green''s theorem in two dimensions
1.10 Orthogonal curvilinear coordinates
1.11 Gradient and Laplacian in orthogonal curvilinear coordinates
1.12 Integral transforms
1.12.1 Laplace transform
1.12.2 Fourier transforms
1.12.3 Hankel transforms
1.13 PROBLEM SET 1
2. General concepts in partial differential equations
2.1 Fundamental concepts
2.1.1 The order of a partial differential equation
2.1.2 The linearity of a partial differential equation
2.1.3 Homogeneity of a partial differential equation
2.2 Well-posed problems
2.2.1 Boundary conditions
2.2.2 Initial conditions
2.2.3 Well-posed problems
2.3 PROBLEM SET 2
3. Partial differential equations of the first-order
3.1 General concepts
3.2 Examples involving first-order equations
3.3 Advective transport in reactor column
3.3.1 Governing equation - one dimensional case
3.3.2 Governing equation - generalized formulation
3.4 A heat exchanger problem
3.5 PROBLEM SET 3
4. Partial differential equations of the second-order
4.1 Classification of second-order partial differential equations
4.2 Reduction to canonical forms
4.3 Applications of the procedures
4.4 Classification of second-order pdes
for n independent variables
4.5 PROBLEM SET 4
5. Laplace''s equation
5.1 Derivation of Laplace''s equation
5.1.1 Irrotational flow in fluid mechanics
5.1.2 Flow of fluids in porous media
5.2 Boundary conditions
5.2.1 Boundary conditions for fluid flow
5.2.2 Boundary conditions for porous media flow
5.2.3 Boundary conditions for heat conduction
5.3 Generalized results
5.4 Methods of solution of Laplace''s equation
5.4.1 Direct solution procedure
5.4.2 Separation of variables method Cartesian coordinates
5.4.3 Separation of variables method plane polar coordinates
5.5 Integral transform solution of Laplace''s equation
5.6 Line source within a half-plane region
5.7 Uniqueness theorem
5.8 A maximum principle
5.9 PROBLEM SET 5
6. The diffusion equation
6.1 Derivation of the diffusion equation
6.1.1 Heat conduction in solids
6.1.2 Pressure transients in porous media
6.1.3 Chemical mass transport in porous media
6.1.4 Drying of porous solids
6.1.5 Thermal oxidation of silicon
6.1.6 Motion of a plate on a viscous fluid
6.2 Initial conditions and boundary conditions
6.2.1 Dirichlet-type boundary condition
6.2.2 Neumann-type boundary conditions
6.2.3 Combined boundary conditions
6.2.4 Mixed boundary conditions
6.2.5 Initial conditions
6.2.6 Change in dependent variable for homogeneous initial conditions
6.3 Methods of solution of the diffusion equation
6.3.1 Direct solution procedure
6.3.2 Trial function approach
6.3.3 Separation of variables method Cartesian coordinates
6.3.4 Separation of variables method plane polar coordinates
6.4 Some generalized results associated with the diffusion equation
6.4.1 Reduction to Helmholtz equation
6.4.2 Product solutions for the diffusion equation
6.4.3 Sturm-Liouville problems
6.5 Separation of variables method for spatially two-dimensional problems
6.5.1 Spatially two-dimensional problems Cartesian coordinates
6.5.2 Spatially two-dimensional problems plane polar coordinates
6.5.3 Product solutions and solutions for infinite domains
6.6 Uniqueness theorem
6.7 A maximum principle
6.8 PROBLEM SET 6
7. The wave equation
7.1 Wave motion in strings
7.1.1 Harmonic waves
7.1.2 d''Alembert''s solution
7.1.3 Fourier analysis of the stretched string
7.1.4 Reflection and transmission at boundaries
7.1.5 Energy in a string
7.1.6 Forced motion of a semi-infinite string
7.1.7 Forced motion of an infinite string
7.2 Wave motion in stretched finite strings
7.2.1 Waves in a stretched finite string
7.2.2 Vibrations of a stretched finite string:trial function approach
7.2.3 Vibrations of a stretched finite string variables separable solution
7.2.4 Vibrations of a stretched string: variable boundary conditions
7.2.5 Forced vibration of a stretched finite string
7.3 Wave motion in stretched strings: non-classical effects
7.3.1 Elastically supported string
7.3.2 Energy dissipation and damping in a stretched string
7.4 Waves and vibrations in stretched membranes
7.4.1 Equation of motion for a stretched membrane
7.4.2 Plane wave motion in a stretched infinite membrane
7.4.3 Free vibrations of a stretched membrane of infinite extent
7.4.4 Symmetric free vibrations of the stretched membrane
7.4.5 Green''s function for the vibration of a stretched membrane
7.5 Vibrations of stretched finite membranes
7.5.1 Vibrations of a stretched square membrane
7.5.2 Free vibrations of a stretched rectangular membrane
7.5.3 Forced vibrations of a stretched rectangular membrane
7.5.4 Free vibrations of a stretched circular membrane
7.5.5 Hankel transform analysis of free vibrations of a stretched circular membrane
7.5.6 Hankel transform analysis of forced vibrations of a stretched circular membrane
7.5.7 Vibrations of a circular membrane-general formulation
7.6 Wave motion and vibrations in membranes: non-classical effects
7.7 Wave equation for problems in solid mechanics
7.7.1 Longitudinal wave motion in a slender elastic rod
7.7.2 Torsional waves in a slender circular elastic rod
7.8 Shallow water waves
7.9 Uniqueness theorem
7.10 PROBLEM SET 7
Bibliography
Index