__MA3351 __

__TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS__

** **

**COURSE 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**

Formation
of partial differential equations –Solutions of standard types of first order
partial differential equations - First order partial differential equations
reducible to standard types- Lagrange’s linear equation - Linear partial
differential equations of second and higher order with constant coefficients of
both homogeneous and non-homogeneous types.

**UNIT - II **

**FOURIER SERIES**

Dirichlet’s
conditions – General Fourier series – Odd and even functions – Half range sine
series and cosine series – Root mean square value – Parseval’s identity –
Harmonic analysis.

**UNIT - III **

**APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS**

Classification
of PDE – Method of separation of variables - Fourier series solutions of
onedimensional wave equation – One dimensional equation of heat conduction –
Steady state solution of two-dimensional equation of heat conduction (Cartesian
coordinates only).

**UNIT - IV **

**FOURIER TRANSFORMS**

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**

Z-transforms
- Elementary properties – Convergence of Z-transforms - – Initial and final
value theorems - Inverse Z-transform using partial fraction and convolution
theorem - Formation of difference equations – Solution of difference equations using
Z - transforms.

**TOTAL:
60 PERIODS**

**OUTCOMES:**

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

1.
Understand how to solve the given standard partial differential equations.

2.
Solve differential equations using Fourier series analysis which plays a vital
role in engineering applications.

3.
Appreciate the physical significance of Fourier series techniques in solving
one- and two dimensional heat flow problems and one-dimensional wave equations.

4.
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.

5.
Use the effective mathematical tools for the solutions of partial differential
equations by using Z transform techniques for discrete time systems

**TEXT BOOKS:**

1.
Grewal B.S., “Higher Engineering Mathematics", 44thEdition, Khanna
Publishers, New Delhi, 2018.

2.
Kreyszig E, "Advanced Engineering Mathematics ", 10th Edition, John
Wiley, New Delhi, India, 2018.

**REFERENCES:**

1.
Andrews. L.C and Shivamoggi. B, "Integral Transforms for Engineers"
SPIE Press, 1999.

2.
Bali. N.P and Manish Goyal, "A Textbook of Engineering Mathematics",
10th Edition, Laxmi Publications Pvt. Ltd, 2021.

3.
James. G., "Advanced Modern Engineering Mathematics", 4thEdition,
Pearson Education, New Delhi, 2016.

4.
Narayanan. S., Manicavachagom Pillay.T.K and Ramanaiah.G "Advanced
Mathematics for Engineering Students", Vol. II & III, S.Viswanathan
Publishers Pvt. Ltd, Chennai, 1998.

5.
Ramana. B.V., "Higher Engineering Mathematics", McGraw Hill Education
Pvt. Ltd, New Delhi, 2018.

6.
Wylie. R.C. and Barrett. L.C., “Advanced Engineering Mathematics “Tata McGraw
Hill Education Pvt. Ltd, 6th Edition, New Delhi, 2012.