Week 1:
Basic concepts of integral transforms.
Fourier transforms: Introduction, basic properties, applications to solutions of Ordinary Differential Equations (ODE),
Partial Differential Equations (PDE).
Week 2:
Applications of Fourier Transforms to solutions of ODEs, PDEs and Integral Equations,
evaluation of definite integrals.
Laplace transforms: Introduction, existence criteria
Week 3:
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).
Week 4:
Applications of joint Fourier-Laplace transform, definite integrals, summation of infinite series, transfer functions, impulse response function of linear systems.
Week 5:
Hankel Transforms: Introduction,
properties and applications to PDE Mellin transforms: Introduction, properties, applications;
Generalized Mellin transforms.
Week 6:
Hilbert Transforms: Introduction, definition, basic properties, Hilbert transforms in complex plane, applications;
asymptotic expansions of 1-sided Hilbert transforms.
Week 7:
Stieltjes Transform: definition, properties, applications, inversion theorems, properties of generalized Stieltjes transform.
Legendre transforms: Intro, definition, properties, applications.
Week 8:
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.
Week 11:
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.
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