INTEGRAL EQUATIONS (Optional Paper)
- Understand the concept of the relationship between the integral equations and ordinary differential equations.
- Understand the linear and nonlinear integral equations by different methods with some problems which give rise to integral equations.
- 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):
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Course |
Learning outcomes (at course level) |
Learning and teaching strategies |
Assessment Strategies |
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Paper Code |
Paper Title |
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MAT 424A
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Integral Equations (Theory)
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The students will be able to –
CO137: Acquire knowledge of different types of Integral equations: Fredholm and Volterra integral equations. CO138: 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. CO139: Demonstrate a depth of understanding in advanced mathematical topics in relation to geometry of curves and surfaces. CO140: Think logically and mathematically and apply the knowledge of transforms to solve complex problems CO141: Construct Green functions in solving boundary value problem by converting it to an 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
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Linear integral equations: Definition and classification, Conversion of initial and boundary value problems to an integral equation, Eigenvalues and Eigenfunctions, Solution of homogeneous and general Fredholm integral equations of second kind with separable kernels.
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.
Integral equations with symmetric kernels: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.
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.
Green’s function: Definition, Construction, Properties, Green’s function approach for integral equation formulation of ordinary differential equation of any order, Laplace and Poisson's equations.
- 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.
