Basic concepts of integral transforms.
Fourier transforms: Introduction, basic properties, applications to solutions of Ordinary Differential Equations (ODE),
Partial Differential Equations (PDE).
Applications of Fourier Transforms to solutions of ODEs, PDEs and Integral Equations,
evaluation of definite integrals.
Laplace transforms: Introduction, existence criteria
Laplace transforms: Convolution, differentiation, integration, inverse transform,
Tauberian Theorems, Watson’s Lemma, solutions to ODE, PDE including Initial Value Problems (IVP) and Boundary Value Problems (BVP).
Applications of joint Fourier-Laplace transform, definite integrals, summation of infinite series, transfer functions, impulse response function of linear systems.
Hankel Transforms: Introduction,
properties and applications to PDE Mellin transforms: Introduction, properties, applications;
Generalized Mellin transforms.
Hilbert Transforms: Introduction, definition, basic properties, Hilbert transforms in complex plane, applications;
asymptotic expansions of 1-sided Hilbert transforms.
Stieltjes Transform: definition, properties, applications, inversion theorems, properties of generalized Stieltjes transform.
Legendre transforms: Intro, definition, properties, applications.
Z Transforms: Introduction, definition, properties; dynamic linear system and impulse response, inverse Z transforms,
summation of infinite series, applications to finite differential equations
Week 9: Radon transforms: Introduction, properties, derivatives, convolution theorem, applications, inverse radon transform.
Week 10: Fractional Calculus and its applications: Intro, fractional derivatives, integrals, Laplace transform of fractional integrals and derivatives.
Integral transforms in fractional equations: fractional ODE, integral equations,
IVP for fractional Differential Equations (DE), fractional PDE, green’s function for fractional DE.
Week 12: Wavelet Transform: Discussion on continuous and discrete, Haar, Shannon and Daubechie Wavelets.