Integral Equations

Paper Code: 
24MAT424(A)
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: 

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

24MAT

424(A)

Integral Equations (Theory)

 

 

 

CO172: Acquire knowledge of different types of Integral equations: Fredholm and Volterra integral equations and Obtain solution of integral equations with eigenvalues and eigen function.

CO173: Demonstrate linear integral equations using methods such as t6he method of successive approximations, resolvent kernel methods and the eigenfunction expansion method.

CO174: Demonstrate  application of the Hilbert-Schmidt theorem to analyze and solve integral equations

CO175: Explore volterra integral equation by using transform method.

CO176: Construct Green functions in solving boundary value problems by converting it to an integral equation.

CO177: 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: 
Linear integral equations:
15.00

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: 
Solution of Fredholm and Volterra integral equations II kind via successive method:
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: 
Integral equations by using Hilbert-Schmidt theorem:
15.00

Orthogonal system of functions, Fundamental properties of eigenvalues and eigenfunctions for symmetric kernels, Hilbert-Schmidt theorem, Solution of Fredholm integral equations of second kind by using Hilbert-Schmidt theorem.

 

Unit IV: 
Solution of Fredholm and Volterra integral equations II kind via different methods:
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: 
Green’s function:
15.00

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, 2014.
  • 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.

SUGGESTED READING

  • 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, 2009. 
  • A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations. CRC Press, Boca Raton, 2008.
  • W. V. Lovitt, Linear Integral Equations, Dover Publication, 2005.

e- RESOURCES

 

JOURNALS

 

Academic Year: 
2024-2025
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