Syllabus (MATHEMATICS)

Course Type: MAJ-15

Semester: 8

Course Code: BMTMMAJ15T

Course Title: Integral Transforms & Integral Equations

(L-P-Tu): 3-0-1

Credit: 4

Practical/Theory: Theory

Course Objective: To be done.

Learning Outcome: Course Outcomes (CO): The whole course will have the following outcomes: Upon successful completion, the students will be able to CO1: understand the ideas and importance of the Integral transforms that are used to map one domain into another in which th

Syllabus:

Unit -1: Laplace Transform [Credit: 1, 15L]

Definition and properties of Laplace transforms, Sufficient conditions for the existence of Laplace Transform (proof not required), Laplace Transform of some elementary functions, Laplace Transforms of derivatives and integrals. Derivatives and integrals of Laplace transform. Initial and final value theorems, Convolution theorems, Inverse of Laplace Transform, Application to ordinary and partial differential equations.

Unit-2: Fourier Transform [Credit: 1, 15L]

Fourier integral theorem, Definition and Properties, Fourier transform of some elementary functions, Fourier transform of the derivative, Derivative of Fourier transform, Fourier sine and cosine transforms, Inverse of Fourier transform, Convolution, Properties of convolution functions, Convolution theorem and applications. Application to ordinary and partial differential equations.

Unit -2: Integral Equations [Credit: 1, 15L]

Linear Integral Equations of 1^st and 2^nd kinds- Fredholm and Volterra types, Relation between integral equations and initial boundary value problems, Existence and uniqueness of continuous solutions of Fredholm and Volterra’s integral equations of 2^nd kind, Iterated Kernels, Reciprocal Kernels, Solution by the method of successive approximations, Solution by Resolvent kernel method and iterated kernel method. Volterra’s solution of Fredholm’s integral equations. Reduction of boundary value problem of an ordinary differential equation to an integral equation and vice-versa.

Reading References:

1. Sneddon, I. N., Fourier Transform, McGraw Hill.
2. Miller, F. H., Partial Differential Equations, John Wiley and Sons
3. Sneddon, I.N., Use of Integral Transforms, McGraw-Hill Pub.
4. Tranter, C. J., Integral Transforms
5. Tricomi, Integral Equations
6. Andrews, L.C., Shivamoggi, B., Integral Transforms for Engineers, PHI.
7. Debnath, L., Bhatta,D., Integral Transforms and Their Applications, CRC Press, 2007.
8. G. C. Gorain, Laplace Transformations, Narosa Publishing House, New Delhi.
9. Lovitt, Linear Integral Equations