Computational Methods of Ordinary Differential Equations

Paper Code: 
MAT323 B
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. To enable students to design and analyze numerical methods to approximate solutions to differential equations for which finding an analytic (closed-form) solution is not possible.
  2. To teach basic scientific computing for solving differential equations.

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

MAT 323B

 

 

 

 

Computational Methods of Ordinary Differential Equations

 (Theory)

 

 

 

 

 

The students will be able to –

 

CO87: Understand the key ideas, concepts and definitions of the computational algorithms, sources of errors, convergence theorems.

CO88: Implement a given algorithm in Matlab (or related software package) and test and validate codes to solve a given differential equation numerically.

CO89: Choose the best numerical method to apply to solve a given differential equation and quantify the error in the numerical (approximate) solution.

CO90: Analyze an algorithm’s accuracy, efficiency and convergence properties

CO91: Create own algorithms to solve the differential equations numerically.

CO92: Apply the techniques to solve differential equations representing the models of anf specific problems.

 

Approach in teaching:

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Field trips

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

Unit I: 
I
15.00
Initial value problem (IVPS) for the system of ordinary differential equation (ODEs) difference equations numerical methods, Local truncation error, Stability analysis, Interval of absolute stability, Convergence and consistency.
 
Unit II: 
II
Single-step method: Taylor series method, Explicit and implicit Runga-Kutta method and their stability and convergence analysis, Extrapolation method, Runga–Kutta method for first-order initial value problems, Runga-Kutta method for the second-order initial value problems and their stability analysis, Stiff system of the differential equation.
 
Unit III: 
III
Multi-step methods: Explicit and implicit multi-step methods, General linear multi-step method and their convergence analysis, Adams-Moulton method, Adams-Bashforth method, Nystorm- method, Multi-step methods for the second order IVPS.
 
Unit IV: 
IV
15.00
Boundary value problem (BVP): Two point nonlinear BVPs for second order ordinary differential equation, Shooting method, Finite difference methods, Convergence analysis, Difference scheme based on quadrature formula, Difference scheme for linear eigen value problems, Mixed boundary condition.
 
Unit V: 
V
15.00
Finite element methods: Assemble of element equations, Variational formulation of BVPs and their solutions, Galerikin method, Ritz method, Finite element solution of BVPs.
 
Essential Readings: 
  • J.C. Butcher, Numerical Method for Ordinary Differential Equations, John Wiley & Sons, New York, 2003.
  • J.D. Lambert, Numerical Method for Ordinary Differential Systems: The initial Value Problem, John Wiley & Sons, New York, 1991.
  • M. K. Jain, S.R.K. Iyenger and R. K. Jain, Numerical methods and Solution, New Age Publications, 2004.
 
References: 
  • K. Atkinson, W. Han, and D.E. Stewart, Numerical Solution of Ordinary Differential Equations,  John Wiley & Sons, New York, 2009.
  • C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, Addison-Wesley, 2003.
  • H.T.H. Piaggio, Elementary Treatise on Differential Equations and Their Applications, C.B.S. Publisher & Distributors, Delhi, 2019.
  • M.K. Jain, Numerical Solution of Differential Equations: Finite difference and Finite Element Approach, New Age Publications, 2018.
  • E.A. Codington, An Introduction to Ordinary Differential Equation, Prentice Hall of India, 1961. 
 
Academic Year: 
2022-2023
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