Computational Methods of Ordinary Differential Equations

Paper Code: 
24MAT323(B)
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. 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. Teach basic scientific computing for solving differential equations.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

 

 

24MAT

323(B)

 

 

 

 

Computational Methods of Ordinary Differential Equations

 (Theory)

 

 

 

 

 

CO89: Analyze the concept of initial value problems (IVPs) for systems of ordinary differential equations (ODEs) and their significance in mathematical modeling.

CO90: Explain the numerical method to apply to solve a given differential equation and quantify the error in the numerical (approximate) solution.

CO91: Examine linear multi-step methods' properties, convergence behavior, stability and accuracy and apply Adams-Moulton and Adams-Bash forth methods to solve initial value problems for ODEs.

CO92: Apply the formulation and significance of boundary value problems (BVPs) for second-order ordinary differential equations (ODEs), particularly those with nonlinearities and mixed boundary conditions.

CO93: Explore the Galerkin method and Ritz method as special cases of finite element methods and understand their theoretical foundations and computational implementation.

CO94: Contribute effectively in course-specific interaction.

Approach in teaching:

Interactive Lectures, Discussion, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions,  Topic  presentation, Assigned tasks

 

 

Quiz, Class Test, Individual projects,

Open Book Test, Continuous Assessment, Semester End Examination

 

 

 

Unit I: 
Initial value problem (IVPS):
15.00

Initial value problem  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: 
Single step method:
15.00

 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 differential equation.

 

Unit 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: 
Boundary value problem (BVP):
15.00

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: 
Finite element methods:
15.00

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.

SUGGESTED READING

  • 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.

e- RESOURCES

 

JOURNALS

 

Academic Year: 
2024-2025
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