Differential equations Syllabus S2 2015-16

Course No.
Course Name
L-T-P-Credits
Year of Introduction
MA102
DIFFERENTIAL EQUATIONS
3-1-0-4
2015
Course Objectives
This course introduces basic ideas of differential equations, both ordinary and partial, which are widely used in the modeling and analysis of a wide range of physical phenomena and has got applications across all branches of engineering.  The course also introduces Fourier series which is used by engineers to represent and analyze periodic functions in terms of their frequency components.  
Syllabus
Homogeneous linear ordinary differential equations, non-homogeneous linear ordinary differential equations, numerical solutions of ordinary differential equations, Fourier series, partial differential equations, applications of partial differential equations.  
Expected outcome
At the end of the course students will have acquired basic knowledge of differential equations and methods of solving them and their use in analyzing typical mechanical or electrical systems. The included set of assignments will familiarize the students with the use of software packages for analyzing systems modeled by differential equations.
Text Books:
      Kreyszig, E., Advanced Engineering Mathematics, Wiley
      Srivastava, A. C. and Srivasthava, P. K., Engineering Mathematics, Vol 2. PHI Learning Pvt.
Ltd.
References Books:
      Bali, N. P. and Goyal, M., Engineering Mathematics, Lakshmy Publications
      Datta, Mathematical Methods for Science and Engineering. Cengage Learning
      Edwards, C. H. and Penney, D. E., Differential Equations and Boundary Value Problems.
Computing and Modelling, Pearson.
      Grewal, B. S., Higher Engineering Mathematics, Khanna Publishers, New Delhi.
      Jordan, D. W. and Smith, P., Mathematical Techniques, Oxford University Press 
      Pal, S and Bhunia, S. C., Engineering Mathematics, Oxford, 2015
      Ross, S. L., Differential Equations, Wiley
Course Plan
Module
Contents
Hours
Sem.
Exam
Marks
I
HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS
(Text Book 1: Sections: 1.7, 2.1, 2.2,2.4,2.6, 3.1, 3.2)
Existence and Uniqueness theorem for solutions of initial value problems (without proof).  Basic theory of solutions of homogeneous differential equations (superposition principle, basis of solutions, general and
5
15%



particular solutions).


Methods of solving homogeneous linear differential equations with constant coefficients of orders two or higher. Modelling of free oscillations of a mass-spring system.

(For practice and submission as assignment only:
Solutions of separable, exact and first order linear differential equations  and orthogonal trajectories )
4
II
NON-HOMOGENEOUS    LINEAR    ORDINARY       DIFFERENTIAL
EQUATIONS
(Text Book 1: Sections: 2.7—2.10, 3.3)
Basic theory of non-homogeneous linear differential equations. Methods of solving non-homogeneous linear differential equations with constant coefficients
4
15%
Method of undetermined coefficients and method of variation of parameters.  
4
Legendre and Cauchy’s differential equations.
Modelling of forced oscillations of mass-spring system and electric circuits.

(For practice and submission as assignment only:
Sketching, plotting and interpretation  of solutions of differential equations using suitable software packages)
2
FIRST INTERNAL EXAM
III
NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS 
(Text Book 1: sections 21.1, 21.2)

Basic idea of numerical solutions of differential equations.  Euler-method, improved Euler method, Runge-kutta method of fourth order (without proof)
6
15%
Predictor-corrector method of Adams-Moulton (without proof).  Error bounds of these methods.

(For practice and submission as assignment only:
Implementation of the above numerical methods in any programming language or using software packages)
2
IV
FOURIER SERIES 
(Text Book 1: Sections: 11.1-11.2 )
Periodic Functions- Orthogonality of Sine and Cosine functions-Fourier series of periodic functions, Euler’s formula, Condition for Convergence of Fourier series (without proof)
3
15%
Fourier series for even and odd functions, Half range expansion

(For practice and submission as assignment only:
Plots of partial sums of Fourier series and demonstration of convergence using plotting software)
6

SECOND INTERNAL EXAM


V
PARTIAL DIFFERENTIAL EQUATION 
(Text Book 2: Section: 5.1.1, 5.1.2, 5.1.3, 5.1.4, 5.1.5, 5.1.9, 5.1.10, 5.2.6,
5.2.7, 5.2.8, 5.2.9, 5.2.10)
Formation of PDEs, solutions of  first order PDEs, 
General integral, complete integral, Lagrange’s linear equation, 
5
20%
Higher order PDE-Solution of Linear Homogeneous PDE with Constant Coefficients.
5
VI
APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS
(Text Book 2: Section: 6.1, 6.2, 6.3, 6.4, 6.7, 6. 8, 6. 9, 6.9.1, 6.9.2)
Method of Separation of Variables
2
20%
Modelling Vibrations of a Stretched sting-One dimensional wave equation and its Solution by Fourier series.
4
Heat transfer through an insulated rod-one dimensional heat equation. 


Solution of heat equation by Fourier series in special cases– insulated rod with ends at zero temperatures, insulated rod with ends at non-zero temperatures.

(For practice and submission as assignment only:
Plots of partial sums of Fourier series solutions of PDEs and
demonstration of convergence using plotting software)
4


END SEMESTER EXAM



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