The material presented in these monographs is the outcome of the author's long-standing interest in the analytical modelling of problems in mechanics by appeal to the theory of partial differential equations. The impetus for writing these volumes was the opportunity to teach the subject matter to both undergraduate and graduate students in engineering at several universities. The approach is distinctly different to that which would adopted should such a course be given to students in pure mathematics; in this sense, the teaching of partial differential equations within an engineering curriculum should be viewed in the broader perspective of "The Modelling of Problems in Engineering" . An engineering student should be given the opportunity to appreciate how the various combination of balance laws, conservation equations, kinematic constraints, constitutive responses, thermodynamic restrictions, etc., culminates in the development of a partial differential equation, or sets of partial differential equations, with potential for applications to engineering problems. This ability to distill all the diverse information about a physical or mechanical process into partial differential equations is a particular attraction of the subject area.
目录
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 1.12.3ccHankelctransforms 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 mechani s 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 7.The wave equation Bibliography Index
