1st Edition

Advanced Engineering Mathematics with Modeling Applications

By S. Graham Kelly Copyright 2008
    538 Pages 159 B/W Illustrations
    by CRC Press

    Engineers require a solid knowledge of the relationship between engineering applications and underlying mathematical theory. However, most books do not present sufficient theory, or they do not fully explain its importance and relevance in understanding those applications.

    Advanced Engineering Mathematics with Modeling Applications employs a balanced approach to address this informational void, providing a solid comprehension of mathematical theory that will enhance understanding of applications – and vice versa. With a focus on modeling, this book illustrates why mathematical methods work, when they apply, and what their limitations are. Designed specifically for use in graduate-level courses, this book:

    • Emphasizes mathematical modeling, dimensional analysis, scaling, and their application to macroscale and nanoscale problems
    • Explores eigenvalue problems for discrete and continuous systems and many applications
    • Develops and applies approximate methods, such as Rayleigh-Ritz and finite element methods
    • Presents applications that use contemporary research in areas such as nanotechnology

    Apply the Same Theory to Vastly Different Physical Problems
    Presenting mathematical theory at an understandable level, this text explores topics from real and functional analysis, such as vector spaces, inner products, norms, and linear operators, to formulate mathematical models of engineering problems for both discrete and continuous systems. The author presents theorems and proofs, but without the full detail found in mathematical books, so that development of the theory does not obscure its application to engineering problems. He applies principles and theorems of linear algebra to derive solutions, including proofs of theorems when they are instructive. Tying mathematical theory to applications, this book provides engineering students with a strong foundation in mathematical terminology and methods.

    Foundations of mathematical modeling

    Engineering analysis

    Conservation laws and mathematical modeling

    Problem formulation



    Linear algebra


    Three-dimensional space

    Vector spaces

    Linear independence

    Basis and dimension

    Inner products


    Gram-Schmidt orthonormalization

    Orthogonal expansions

    Linear operators

    Adjoint operators

    Positive definite operators

    Energy inner products

    Ordinary differential equations

    Linear differential equations

    General theory for second-order differential equations

    Differential equations with constant coefficients

    Differential equations with variable coefficients

    Singular points of second-order equations

    Bessel functions

    Differential equations whose solutions are expressible in terms of Bessel functions

    Legendre functions

    Variational methods


    The general variational problem

    Variational solutions of operator equations

    Finite-element method

    Galerkin’s method

    Eigenvalue problems

    Eigenvalue and eigenvector problems

    Eigenvalues of adjoint operators

    Eigenvalues of positive definite operators

    Eigenvalue problems for operators in finite-dimensional vector spaces

    Second-order differential operators

    Eigenvector expansions

    Fourth-order differential operators

    Differential operators with eigenvalues in boundary conditions

    Eigenvalue problems involving Bessel functions

    Eigenvalue problems in other infinite-dimensional vector spaces

    Solvability conditions

    Asymptotic approximations to solutions of eigenvalue problems

    Rayleigh’s quotient

    Rayleigh-Ritz method

    Green’s functions

    Partial differential equations

    Homogeneous partial differential equations

    Second-order steady-state problems, Laplace’s equation

    Time-dependent problems: Initial value problems

    Nonhomogeneous partial differential equations

    Problems in cylindrical coordinates

    Problems in spherical coordinates



    S. Graham Kelly