Advanced Differential Equations

Paper Code: 
24MAT123
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Understand the fundamentals of ordinary and partial differential equations and its applications to calculating boundary value problems.
  2. Aware about the concept of heat and wave equations, conditions at the boundary of the spatial domain and initial conditions at time zero.
  3. Learn technique of separation of variables to solve PDE’s and analyze the behavior of solutions in terms of eigen function expansions.

 

Course Outcomes: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

24MAT

123

Advanced Differential Equations

(Theory)

 

CO13: Get the Competence in solving applied problems of linear and nonlinear forms like Riccati etc.

CO14: Explore the problems choosing the most suitable methods for partial differential equations like Monge’s method, Canonical form etc.

CO15: Demonstrate boundary value problems for linear second-order differential equations using methods such as eigen function expansions and Sturm-Liouville theory.

CO16: Apply the Euler-Lagrange equation to find extremals for single and multiple integral variational problems.

CO17: Demonstrate isoperimetric problems and understand their geometric significance.

CO18: 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: 
Non-linear ordinary differential equations of particular forms, Riccati's equation:
15.00

 General solution and the solution when one, two or three particular solutions are known, Total differential equations.

 

Unit II: 
Second order partial differential equations:
15.00

 Formulation and classification of second order partial differential equations, Monge’s methods:  Canonical forms, classification of second order partial differential equations of the type Rr+Ss+Tt+ f (x,y,z,p,q)=0 and second order partial differential equations in more than two independent variables, Method of separation of variables, Laplace, Wave and diffusion equations.

 

Unit III: 
Boundary value problems and series solution:
15.00

Linear homogeneous boundary value problems, Eigen values and eigen functions, Strum-Liouville boundary value problems, Orthogonality of eigen functions, Reality of eigen values, Series solution (all four cases).

 

Unit IV: 
Calculus of variation in general forms:
15.00

 Functionals, Variation of a functional and its properties, Variational problems with fixed boundaries, Euler's equation and it’s alternative forms, Extremals, Functionals dependent on several unknown functions and their first order derivatives, Functionals dependent on higher order derivatives, Functionals dependent on the function of more than one independent variable.

 

Unit V: 
Calculus of variation with special cases:
15.00

Variational problem in parametric forms, Isoperimetric problem and conditions, Geodesic problems, Variational problems with moving (or free) boundaries: One sided variations only for a functional dependent in one or two functions.

Essential Readings: 
  • Z. Ahsan, Differential Equations & Their Applications, PHI, New Delhi, 2016.
  • J. L. Bansal and H. S. Dhami, Differential Equations, Jaipur Publishing House, Jaipur, 2014.
  • M. D. Raisinghania, Advanced differential equation, S. Chand and Co. Ltd., 2012.

SUGGESTED READING

  • R. Forsyth, A Treatise on Differential Equations, Macmillan and Co. Ltd, London, 1956.
  • Frank Ayres, Schaum’s Theory and Problems of Differential Equations, McGraw Hill, 2012.
  • D.A. Murray, Introductory Course in Differential Equations, University of   Michigan Library, 1902.
  • W.E. Boyce and P.C. Diprima, Elementary Differential Equations and Boundary Value Problems, John Wiley, 2009.
  • E. A. Coddington, An Introduction to Ordinary Differential Equations, PHI New Delhi, 2003.
  • G. F. Simmons, Differential Equations with Applications and Historical Notes, McGraw Hill, New York, 2017.
  • I. N. Sneddon, Elements of Partial Differential Equations, McGraw Hill, 2006.

e- RESOURCES

JOURNALS

 

Academic Year: 
2024-2025
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