Integral Transforms And Their Applications
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 criteriaWeek 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, impulseresponse 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 equationsWeek 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.
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 criteriaWeek 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, impulseresponse 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 equationsWeek 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.
