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#### Title-Special Functions And Integral Transforms (Code-09050402)

 S.No. Contents of the topics Learning   Objectives Teaching Guidelines Methodology Time (Hour) 1 ·     Series solution of Des-Power series method ·     Definitions of Beta and Gamma functions ·     Bessel equation and its solution ·     Bessel functions and their properties-Convergence, recurrence Relations ·     Orthogonality of Bessel functions Students will be able to understand Study of series solution of DEs and Bessel’s equations with its properties Lecture should be effective so that student can grasp the topics easily Assignments/ seminars/ Class tests/ Presentations 12 2 ·      Legendre and Hermite DEs and their solutions ·      Legendre and Hermite functions and their properties ·      Recurrence relations and generating functions ·      Orthogonality of Legendre and hermite polynomials ·      Rodrigues’ formula for Legendre and hermite polynomials ·      Lapalace integral representation of Legendre polynomial Students will be able to understand Study of Legendre and Hermite DEs and polynomials Lecture should be effective so that student can grasp the topics easily Assignments/ seminars/ Class tests/ Presentations 15 3 ·      Laplace transforms-Existence theorem for Laplace transforms ·      Linearity of the Laplace transforms, Shifting theorems ·      Laplace transforms of derivatives and integrals ·      Differentiation and integration of Laplace transforms, Convolution theorem ·      Inverse Laplace transforms of derivatives and integrals ·      Solution of ODEs using Laplace transform Students will be able to understand Study of Laplace Transforms with its properties Lecture should be effective so that student can grasp the topics easily Assignments/ seminars/ Class tests/ Presentations 15 4 ·     Fourier transforms-Linearity property, Shifting, Modulation ·     Convolution Theorem ·     Fourier transform of derivatives ·     Relations between Fourier transform and Laplace transform ·     Parseval’s Identity for Fourier transforms ·     Solution of DEs using Fourier Transforms Students will be able to understand Study of Fourier Transforms with its properties Lecture should be effective so that student can grasp the topics easily Assignments/ seminars/ Class tests/ Presentations 10

Books Recommended:

1. Erwin Kreyszing : Advanced Engineering Mathematics, John Wiley & Sons, Inc., New York, 1999
2. R. Forsyth: A Treatise on Differential Equations, Macmillan and Co. Ltd.
3. N. Sneddon: Special Functions on mathematics, Physics & Chemistry.
4. W. Bell: Special Functions for Scientists & Engineers.
5. Murray R. Spiegel: Laplace transform, Schaum’s Series