This learning solution provides a clear exposition of essential tools of Series solutions and special functions, Fourier Series, Partial Differential Equations, Fourier Transforms and Z Transforms.
Series solutions and special functions - Motivation for power series solutions of Differential Equation - Power Series - Analytic Function - Ordinary Point - Singular Point - Power Series Solution of the Differential equation when x = 0 is an ordinary point i.e., when P does not vanish for x = 0 - Example Problems. Extended power series method-indicial equation - Extended power series method-indicial equation - Series solution about regular point x - Frobenius method - Example Problems. Beta function - Beta function - Beta function expressed as improper integral - Beta function in terms of trigonometric functions - Example Problems. Gamma function - Gamma function - Value of Г(n) in terms of factorial - Properties of Gamma Function - Example Problems. Relation between Beta and Gamma functions - Relation between beta and gamma functions - Example Problems. Lengendre polynomials - Lengendre polynomials - Rodrique's Formula - Express Legendre’s polynomial in terms of algebraic polynomial - Express algebraic polynomials in terms of Legendre polynomials - Example Problems. Rodrigue's formula - Example Problems. Generating function for Pn(x) - Example Problems. Orthogonal Properties of Legendre’s polynomial - Laplace’s first and second integral of Pn(x). Recurrence Relations - Recurrence Relations - Example Problems - Beltrami's result - Example Problems - Fourier - Legendre expansion of f(x) - Example Problem. Bessel’s Functions - Bessel’s Functions - Solution of Bessel’s Equations - Expansions for J0 and J1 - Recurrence formulae for Jn(x) - Formula Two - Formula Three - Formula Four - Formula five - Formula six - Example Problems. Generating Function for Jn(x) - Generating Function for Jn(x) - Example Problems - Orthogonality of Bessel functions.
Fourier series - History of Fourier Series - Periodic Function - Example Problem - Introduction of Fourier Series - Dirichlet Conditions - Determination of Fourier Coefficients - Euler’s Formula - Example Problems. Function having points of discontinuity - Function having points of discontinuity - Example Problems. Even and odd functions - Even and odd functions - Fourier series for Even and Odd Function - Example Problems. Half range Fourier series - Introduction to Half–Range Fourier Series - Example Problems. Complex form - Complex form of Fourier series - Example Problems. Parseval’s Theorem - Parseval’s Theorem - Example Problems. Harmonic analysis - Practical harmonic analysis - Example Problems.
Introduction to formation of Partial Differential Equations - Introduction to formation of Partial Differential Equations - Formation of Partial Differential Equations - Example Problems. Elimination of Arbitrary Functions - Introduction - Example Problems. Solutions of standard types of first order partial differential equations - Singular integral - Type - I f(p, q) = 0 - Example Problems - Type II: Clairaut’s form - Example Problems - Type : 3(a) - Example Problems - Type : 3(b) - Example Problems - Type : 3(c) - Example Problems - Type IV: Equation containing x, y, p, q - Example Problems - Type V: - Example Problems - Type VI: - Example Problems. First order linear (Lagrange) equation - Solution of a Partial Differential Equation - Linear Partial differential equations of the first order - Example Problems. Higher order partial differential equations - Higher order partial differential equations - Type - I(Homogeneous Equations ) R.H.S = 0 - Example Problem - Type - II R.H.S = eax + by - Example Problems - Type - III - Example Problems - Type - IV R.H.S = eax + by - Example Problem - Complementary function for a non-homogeneous linear equation - Complementary function for a non-homogeneous linear equation - Example Problems. Classification of second order PDE - Classification of partial differential equations - Example Problems. Method of separation of Variables - Example Problems. Solution of One dimensional Wave - Applications of partial differential equations - Transverse vibrations of a stretched string – one dimensional wave equation - Transmission line equations - Variable separable solutions of the wave - Choice of proper solution - Solution of a damped vibrating string equation - Example Problems. Problems on vibrating string with non-zero initial velocity - Example Problems. Solution of One dimensional Heat - One - dimensional heat flow - Equation of variable heat flow in one dimension - Variable separable solutions of the heat equation - Choice of proper solution - One dimensional heat equation - Example Problems. Solution of Two-dimensional Laplace equation - Steady state heat flow two dimensions [Cartesian Coordinates] - Equation of variable heat flow in two dimensions in Cartesian coordinates - Variable separable solutions of Laplace equation - Choice of proper solution - Two dimensional heat equation - Example Problems. Two Dimensional wave equation - Solution of Two-dimensional Wave Equation - Solution of Two-dimensional Wave Equation - Two-dimensional Wave Equation of a Rectangular Membrane - Example Problems.
Statement of Fourier integral theorem - Fourier integral theorem - Integral transforms - Fourier integral theorem - Example Problems. Fourier sine and cosine integral - Fourier sine and cosine integral - Example Problems. Fourier transform pair - Complex Fourier transforms and its inversion formula - Properties - Example Problems. Fourier sine transform - Fourier sine transform - Example Problems. Fourier cosine transform - Fourier cosine transform - Example Problems. Properties - Example Problems.
Z-Transform - Introduction - Definition of Z-Transform - Example Problems - Problems based on bilateral Z-transform - Example Problems. Linear Property - Linear Property - Example Problems. Shifting and Damping rule - Shifting and Damping rule - Example Problems. Differentiation in the z-domain - Differentiation in the z-domain - Example Problems. Second shifting theorem - Second shifting theorem - Example Problems. Unit sample sequence and unit step sequence - Unit sample sequence and unit step sequence - Example Problems. Initial and final value theorems - Initial and final value theorems - Example Problems - Differentiation - Example Problem. Inverse Z-Transform - Inverse Z-Transform - Partial Fractions Method - Example Problems - Inverse of Z-transform by inverse integral method (Cauchy’s residue theorem) - Example Problems. Convolution Theorem - Convolution Theorem - Z-transform of f(x)*g(x) type - Example Problems. Formation of difference equations - Formation of difference equations - Example Problems. Solution of difference equations using z-transform - Solution of difference equations using z-transform - Example Problems.
