Advanced Differential Equations

Paper Code: 
MAT123
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 their applications to calculating boundary value problems.
  2. Aware of the concept of heat and wave equations, conditions at the boundary of the spatial domain, and initial conditions at time zero.
  3. Learn the technique of separation of variables to solve PDEs and analyze the behavior of solutions in terms of Eigenfunction expansions.

 

Course Outcomes (COs):

course learning

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

MAT 123

Advanced Differential Equations

(Theory)

 

The students will be able to –

 

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

CO14: Solve the problems by choosing the most suitable methods.

CO15: Determine the solutions of differential equations with initial and boundary conditions

CO16: Enhance and develop the ability to use the language of mathematics in analyzing the real-world problems of sciences and engineering.

CO17: Techniques to predict the behavior of certain phenomena.

CO18: Demonstrate ability to cover a topic independently related to the rate of change.

 

The approach to teaching:

Discussion, Demonstration,

Team teaching, Presentation

 

Learning activities for the students:

Self-learning,

Presentation,

Effective questions,

Giving task observation

Observation, Presentation, Report writing,

 

Unit I: 
I
15.00
Non-linear ordinary differential equations of particular forms, Riccati's equation: General solution and the solution when one, two, or three particular solutions are known, Total differential equations.
 
Unit II: 
II
15.00
Second-order partial differential equations: 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: 
III
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: 
IV
15.00
Calculus of variation: 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: 
V
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.
References: 
  • 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.
  • E. D. Ranville, Elementary Differential Equations, Macmillan Company New York, 1988.
  • N. Sneddon, Elements of Partial Differential Equations, McGraw Hill, 2006.
Academic Year: 
2022-2023
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