NPTEL Phase II :: Mathematics - Numerical Solution of Ordinary and Partial Differential Equations

Untitled Document

COURSE OUTLINE

A. Numerical Solution of Ordinary Differential Equations

Numerical solution of first order ordinary differential equations: Piccard’s method Taylor series method, Euler and modified Euler method, Runge Kutta methods.
Multi-step methods: Predictor corrector methods
Systems of equations and higher order equations.
Linear Boundary value problems: Shooting methods, Finite Difference Methods
Convergence criteria, Errors and error propagation, Stiff equations.
Nonlinear Boundary Value Problems

B. Numerical Solution of Partial Differential Equations

Classification, Finite Difference representation
Parabolic PDE: Explicit and implicit schemes. Compatibility, Stability and Convergence
Elliptic PDE: Solution of Laplace/Poisson PDE ADI and SOR schemes,
Hyperbolic equations: Finite difference schemes, Method of characteristics

COURSE DETAIL

Particulars	Hours
Numerical solution of order Ordinary Differential Equations
Initial value problems: definition, existence of solution, need for numerical solutions, Finite difference equation, truncation error	1
Piccard’s method of successive approximation. Taylor series method, Euler and modified Euler method	4
Runge Kutta methods, Stability Analysis	2
Multi-step methods: Predictor corrector methods Milne’s method, Adams-Moulton method, Adams Bashforth method	4
Systems of equations and higher order equations	2
Linear Boundary value problems: Shooting methods, Finite Difference Methods	4
Convergence criteria, Errors and error propagation, Stiff equations	2
Nonlinear Boundary Value Problems	1
Numerical solution of Partial Differential Equations
Introduction to well posed PDE, Classification, various types of governing conditions, Finite Difference representation of derivatives	3
Parabolic PDE: Solution for one Dimensional equation, explicit and various implicit schemes	3
Discussion on compatibility, stability and convergence of above schemes, extension to 2d Heat Conduction equation	3
Elliptic PDE:, Solution of Laplace/ Poisson PDE in Cartesian and Polar system	2
ADI and SOR schemes	2
Methods for solving diagonal systems, Treatment of irregular boundaries	2
Hyperbolic equations – wave equation, Finite difference explicit and implicit schemes, stability analysis	3
Method of characteristics and their significance	2

PREREQUISITES

A sufficient knowledge of Differential equations and Numerical methods

REFERENCES

G.D. Smith, "Numerical Solution of Partial Differential Equations : Finite Difference Methods" (Oxford Applied Mathematics & Computing Science Series).
R K Jain , "Numerical Methods for Scientific and Engineering Computations": M K Jain, S R K Iyengar.
John Wiley, "Finite Difference methods for partial Differential equations": Forsythe G.E. & Wasow, WR.
Gerald, C.F. & Wheatley P.O. "Applied Numerical Analysis", Pearson Education Asia.

ADDITIONAL READINGS

Fox, L. Numerical Solution of Ordinary & Partial Differential Equation, Pergamon Press