Mathematical methods with applications

Abstract

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.