Mathematical Methods with Applications

In this textbook the author applies differential equations to physical problems, and highlights solution techniques with practical examples. Emphasis is placed on: operator methods; the convolution integral; periodic signals; the energy and power spectra; Frobenius method; Laplace, Fourier, Hankel and Z-transforms; Green's Function method; the similarity technique; the method of characteristics; the separation of variable method; and Bessel functions and Legendre polynomials. Four tables of integral transforms are also included, while exercises and answers are given on an accompanying CD-ROM.

Table of Contents:
Ordinary differential equations - Classification of first order differential equations; First order nonlinear differential equations; Singular solutions of differential equations; Orthogonal trajectories; Higher order linear differential equations; The solution of the nonhomogeneous equations; The method of variation of parameters; The method of differential operator; Euler-Cauchy differential equations; Applications to practical problems. Fourier series and Fourier transform - Introduction; Definition of a periodic function; Fourier series and Fourier coefficients; Complex form of Fourier series; Half-range Fourier sine and cosine series; Parseval's theorem; Gibbs' phenomenon; Development of Fourier integral and transform; Relationship of Fourier and Laplace transforms; Applications of Fourier transforms; Parseval's theorem for energy signals; Heaviside unit step function and Dirac delta function; Some Fourier transforms involving impulse functions; Properties of the Fourier transform; The frequency transfer function. Laplace transforms - Introduction; Definition of Laplace transform; Laplace transform properties; Laplace transforms of special functions; Some important theorems; The unit step function and the Dirac delta function; The Heaviside expansion theorems to find inverses; The method of residues to find inverses; The Laplace transform of a periodic function; Convolution. Series solution: method of Frobenius - Introduction; Definition of ordinary and singular points; Series expansion about an ordinary point; Series expansion about a regular singular point. Partial differential equations - Introduction; Mathematical formulation of equations; Classification of PDE: Method of characteristics; The D'Alembert solution of the wave equation; The method of separation of variables; Laplace and Fourier transform methods; Similarity technique; Applications to miscellaneous problems; Sturm-Liouville problems. Bessel functions and Legendre polynomials - Introduction; Series solution of Bessel's equation; Modified Bessel functions; Ber, Bei, Ker and Kei functions; Equations solvable in terms of Bessel functions; Recurrence relations of Bessel functions; Orthogonality of Bessel functions; Legendre polynomials; Applications. Applications - Applications of Fourier series; Applications of Fourier integrals; Applications of Laplace transforms; Applications with PDE; Transmission lines; The heat conduction problem; The chemical diffusion problem; Vibration of beams; The hydrodynamics of waves and tides. Green's function - One-dimensional Green's function; Green's function using variation of parameters; Developments of Green's function in 2D; Development of Green's function in 3D; Numerical formulation. Integral transforms - Introduction; The Hankel transform; The Mellin transform; The Z-transform.