PH5005: Applicable Mathematics

L-T-P-C: 3-0-2-4

Prerequisites: None

Syllabus

Module 1: Complex variables

( Lecture hours: 9 , Tutorial hours: 1 , Lab hours: 0 )

Analytic functions of a complex variable, differentiation and the Cauchy-Riemann conditions, Contour integrals and Cauchy's theorem, Taylor and Laurent series, Singularities: poles, essential singularities, residue theorem, Analytic continuation, multiple-valued functions, branch points and branch cut integration, Method of steepest descent .

Module 2: Probability theory and random variables

( Lecture hours: 7 , Tutorial hours: 1 , Lab hours: 0 )

Probability distributions and probability densities from a statistical mechanics viewpoint, Dirac delta distributions, Statistical inference: Bayesian statistics, Standard discrete and continuous probability distributions, Moments and generating functions, Central limit theorem (statement and applications) .

Module 3: Numerical methods

( Lecture hours: 29 , Tutorial hours: 0 , Lab hours: 10 )

Basics of computer programming, Modelling data: interpolation and fitting, Lagrange and Newton interpolation methods, divided difference table, Piecewise polynomial interpolation, error in polynomial interpolation, Least squares regression: linear, multiple linear and nonlinear regression, Linear algebra: solving coupled linear equations, eigenvalues and eigenvectors, elimination method and pivoting, LU decomposition, Solutions of nonlinear equations and minimization of functions, Numerical differentiation and integration: divided difference method for differentiation, Newton-Cotes formula, midpoint, trapezoidal, rectangular and Simpson's rules, quadrature methods, Solutions of ordinary differential equations: Euler and predictor-corrector methods, Runge-Kutta method, adaptive step size selection, Solutions of partial differential equations .

Study Materials

Textbooks

  1. Mathematical Methods for Physicists, G. Arfken and H. J. Weber, Academic Press, 6th Edition, Indian Edition (2005), ISBN: 978-9381269558
  2. Mathematical Physics, Vol I, S. D. Joglekar, University Press (2005), ISBN: 978-8173714221
  3. Mathematical Physics, Vol II, S. D. Joglekar, University Press, ISBN: 978-8173715600
  4. Advanced Engineering Mathematics, E. Kreyszig, 9th Edition, ISBN: 978-8126531356
  5. Numerical Methods for Engineers, Steven Chapra, Raymond Canale, 6th Edition, ISBN: 978-0-07-340106-5
  6. Numerical Methods for Scientific and Engineering Computation, M. K. Jain, S. R. K. Iyengar, R. K. Jain, New Age International (2003), ISBN: 978-8122414615

References

  1. Complex Variables and Applications, J. W. Brown, R. V. Churchill, McGraw-Hill Education, ISBN: 978-0073383170
  2. Mathematics for Physicists, P. Dennery and A. Krzywicki, Dover Publications, ISBN: 978-0486691930
  3. Mathematical Physics: With Applications, Problems and Solutions, V. Balakrishnan, Ane Books (2017), ISBN: 9386761114