MA4157 Mathematical Modeling and Simulation Syllabus:

MA4157 Mathematical Modeling and Simulation Syllabus – Anna University PG Syllabus Regulation 2021

COURSE OBJECTIVES:

This course will help the students to
 acquire the knowledge of solving system of linear equations using an appropriate numerical methods.
 approximate the functions using polynomial interpolation numerical differentiation and integration using interpolating polynomials.
 acquire the knowledge of numerical solution of ordinary differential equation by single and multi step0 methods.
 obtain the solution of boundary value problems in partial differential equations using finite differences.
 study simulation and Monte-Carlo methods and their applications.

UNIT I MATRICES AND LINEAR SYSTEMS OF EQUATIONS

Solution of Linear Systems : Cramer’s Rule – Gaussian elimination and Gauss Jordon methods – Cholesky decomposition method – Gauss Seidel iteration method – Eigenvalue problems : Power method with deflation for both symmetric and non symmetric matrices and Jacobi method for symmetric matrices.

UNIT II INTERPOLATION, DIFFERENTIATION AND INTEGRATION

Lagrange’s interpolation – Newton’s divided differences – Hermite’s interpolation – Newton’s forward and backward differences – Numerical differentiation – Numerical integration : Trapezoidal and Simpson’s rules – Gaussian quadrature : 2 and 3 point rules.

UNIT III DIFFERENTIAL EQUATIONS

Initial value problems for first and second order ODEs : Single step methods – Taylor’s series method – Euler’s and modified Euler’s methods – Runge – Kutta method of fourth order – Multi step methods : Milne’s and Adam Bashforth methods – Boundary value problems : Finite difference approximations to derivatives – Finite difference method of solving second order ODEs .

UNIT IV PARTIAL DIFFERENTIAL EQUATIONS

Classification of second order PDE’s – Finite difference approximations to partial derivatives – Elliptic equations : Solution of Laplace and Poisson equations – One dimensional parabolic equation – Bender Schmidt method – Hyperbolic equation : One dimensional wave equation.

UNIT V SIMULATION AND MONTE CARLO METHODS

Random numbers : Random number algorithms and generators – Estimation of areas and volumes by Monte Carlo techniques – Numerical integration – Computing volumes – Simulation : Loaded Die Problem – Birthday problem – Buffon’s needle problem – Two dice problem and Neutron shielding problem.

TOTAL: 60 PERIODS

COURSE OUTCOMES :

At the end of the course, students will be able to
 solve an algebraic or transcendental equation and linear system of equations using an appropriate numerical method.
 approximation of functions using polynomial interpolation, numerical differentiation and integration using interpolating polynomials.
 numerical solution of differential equations by single and multistep methods.
 solution of boundary value problems and initial boundary value problems in partial differential equations using finite differences.
 simulation and Monte-Carlo methods and their applications.

REFERENCES :

1. Burden, R.L. and Faires, J.D. “Numerical Analysis”, 9th Edition, Cengage Learning, Delhi, 2016.
2. Cheney, W and Kincaid D., “Numerical Mathematics and Computing”, 7th Edition, Cengage Learning , Delhi, 2014.
3. Jain, M.K., Iyengar, S.R.K. and Jain R.K. “Numerical Methods for Scientific and Engineering Computation”, 6th Edition, New Age International Pvt. Ltd., Delhi, 2014.
4. Landau, D.P. and Binder, K., “A Guide to Monte – Carlo Simulations in Statistical Physics”, 3rd Edition, Cambridge University Press, Cambridge, 2009.
5. Maki, D P and Thompson, M., “Mathematical Modelling with Computer Simulation”, Cengage Learning, Delhi , 2011.
6. Sastry, S.S., “Introductory Methods of Numerical Analysis”, 5th Edition, PHI Learning Pvt. Ltd., Delhi, 2012.
7. Taha, H.A. “Operations Research”, 10th Edition, Pearson Education India, Delhi, 2018.