The course is designed as an introduction to the theory and applications of integral transforms to problems in linear differential equations, to boundary and initial value problems in partial differential equations and continuum mechanics. Many new applications in applied mathematics, physics, chemistry, biology and engineering are included. This course will serve as a reference for advanced study and research in this subject as well as for its applications in the fields of signal processing, informatics and communications, neuroscience, fluid mechanics, quantum mechanics, computer assisted tomography (CAT). The course is open to all MTech, PhD students, some final year advanced undergraduate and honors students.INTENDED AUDIENCE: Any interested learnersPREREQUISITES: 1) Ordinary Differential Equations (ODEs) ,2) Complex variables (optional)INDUSTRY SUPPORT: (1) Signal Processing and Communications,(2) Data Science,(3) Computational Fluid Dynamics,(4) Software Development
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.