Integral Equations (Optional Paper)

Paper Code: 
MAT 424A
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
Objective: 
This course will enable the students to -
  1. Understand the concept of the relationship between the integral equations and ordinary differential equations.
  2. Understand the linear and nonlinear integral equations by different methods with some problems which give rise to integral equations. 
  3. Learn different types of solution methods like successive approximation, resolvent kernel and iteration method, integral transform method and which method is applicable for which type of integral equation. 

Course Outcomes (COs):

                 Course                

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Paper Code

Paper Title

MAT 424A

 

 

Integral Equations (Theory)

 

 

 

The students will be able to –

 

CO170: Acquire knowledge of different types of Integral equations: Fredholm and Volterra integral equations.

CO171: Obtain integral equations from ODE and PDE arising in applied mathematics and different engineering branches and solve accordingly using various method of solving integral equation.

CO172: Demonstrate a depth of understanding in advanced mathematical topics in relation to geometry of curves and surfaces.

CO173: Think logically and mathematically and apply the knowledge of transforms to solve complex problems

CO174: Construct Green functions in solving boundary value problem by converting it to an integral equation.

CO175: Apply various techniques to solve integral equation.

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
Linear integral equations: Definition and classification, Conversion of initial and boundary value problems to an integral equation, Eigen values and Eigen functions, Solution of homogeneous and general Fredholm integral equations of second kind with separable kernels.
 
Unit II: 
II
15.00
Solution of Fredholm and Volterra integral equations of second kind by methods of successive substitutions and successive approximations, Resolvent kernel and its result, Conditions of uniform convergence and uniqueness of series solution.
 
Unit III: 
III
15.00
Integral equations with symmetric kernels: Orthogonal system of functions, Fundamental properties of eigen values and eigen functions for symmetric kernels, Hilbert-Schmidt theorem, Solution of Fredholm integral equations of second kind by using Hilbert-Schmidt theorem.
 
Unit IV: 
IV
15.00
Solution of Fredholm integral equation of second kind by using Fredholm first theorem, Solution of Volterra integral equations of second kind with convolution type kernels by Laplace transform, Solution of singular integral equations by Fourier transform.
 
Unit V: 
V
15.00
Green’s  function: Definition, Construction, Properties, Green’s function approach for integral equation formulation of ordinary differential equation of any order, Laplace and Poission’s  equations.
 
Essential Readings: 
  • Shanti Swaroop, Integral Equations, Krishna Publication, Meerut, 2014.
  • S.P. Goyal and A.K. Goyal, Integral Equations, Jaipur publishing House, Jaipur, 2013.
  • R. P. Kanwal, Linear Integral Equations, Academic Press, 1974.
  • J. L. Bansal and H.S. Dhami, Differential Equations, JPH Vol. I & II, 2014.
  • M.D. Raisinghania, Advanced Differential Equation, S. Chand and Company ltd., 2012.
  • S. K.  Pundir and R. Pundir, Integral Equations and Boundary Value Problems, Pragati Prakashan, 2014.
  • K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge Monographs on Applied and Computational Mathematics, 1997.  
  • A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations. CRC Press, Boca Raton, 1998. 
  • W. V. Lovitt, Linear Integral Equations, Dover Publication, 2005.
 
Academic Year: 
2021-2022
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