# MA8353 Transforms and Partial Differential Equations Syllabus

**OBJECTIVES :**

To introduce the basic concepts of PDE for solving standard partial differential equations.

To introduce Fourier series analysis which is central to many applications in engineering apart

from its use in solving boundary value problems.

To acquaint the student with Fourier series techniques in solving heat flow problems used in various situations.

To acquaint the student with Fourier transform techniques used in wide variety of situations.

To introduce the effective mathematical tools for the solutions of partial differential equations that model several physical processes and to develop Z transform techniques for discrete time systems.

**UNIT I PARTIAL DIFFERENTIAL EQUATIONS 12**

Formation of partial differential equations – Singular integrals – Solutions of standard types of first order partial differential equations – Lagrange’s linear equation – Linear partial differential equations of second and higher order with constant coefficients of both homogeneous and non-omogeneous

types.

**UNIT II FOURIER SERIES 12**

Dirichlet’s conditions – General Fourier series – Odd and even functions – Half range sine series – Half range cosine series – Complex form of Fourier series – Parseval’s identity – Harmonic analysis.

**UNIT III APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS 12**

Classification of PDE – Method of separation of variables – Fourier Series Solutions of one dimensional wave equation – One dimensional equation of heat conduction – Steady state solution of two dimensional equation of heat conduction.** UNIT IV FOURIER TRANSFORMS 12**

Statement of Fourier integral theorem – Fourier transform pair – Fourier sine and cosine transforms – Properties – Transforms of simple functions – Convolution theorem – Parseval’s identity.** UNIT V Z – TRANSFORMS AND DIFFERENCE EQUATIONS 12**

Z-transforms – Elementary properties – Inverse Z-transform (using partial fraction and residues) – Initial and final value theorems – Convolution theorem – Formation of difference equations – Solution of difference equations using Z – transform.

**TOTAL : 60 PERIODS OUTCOMES :**

Upon successful completion of the course, students should be able to:

Understand how to solve the given standard partial differential equations.

Solve differential equations using Fourier series analysis which plays a vital role in engineering

applications.

Appreciate the physical significance of Fourier series techniques in solving one and two

dimensional heat flow problems and one dimensional wave equations.

Understand the mathematical principles on transforms and partial differential equations would

provide them the ability to formulate and solve some of the physical problems of engineering.

Use the effective mathematical tools for the solutions of partial differential equations by using

Z transform techniques for discrete time systems.

**TEXT BOOKS :**

- Grewal B.S., “Higher Engineering Mathematics”, 43rd Edition, Khanna Publishers, New Delhi,

2014. - Narayanan S., Manicavachagom Pillay.T.K and Ramanaiah.G “Advanced Mathematics for Engineering Students”, Vol. II & III, S.Viswanathan Publishers Pvt. Ltd, Chennai, 1998.
**REFERENCES :** - Andrews, L.C and Shivamoggi, B, “Integral Transforms for Engineers” SPIE Press, 1999.
- Bali. N.P and Manish Goyal, “A Textbook of Engineering Mathematics”, 9th Edition, Laxmi Publications Pvt. Ltd, 2014.
- Erwin Kreyszig, “Advanced Engineering Mathematics “, 10th Edition, John Wiley, India, 2016.
- James, G., “Advanced Modern Engineering Mathematics”, 3rd Edition, Pearson Education, 2007.
- Ramana. B.V., “Higher Engineering Mathematics”, McGraw Hill Education Pvt. Ltd, New Delhi, 2016.
- Wylie, R.C. and Barrett, L.C., “Advanced Engineering Mathematics “Tata McGraw Hill Education Pvt. Ltd, 6th Edition, New Delhi, 2012.