Description
WIT Press Mathematical Methods with Applications 2013 Edition by M. Rahman
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.