# MA8353 Tranforms and Partial Differential Equations Lecture Notes

0

Anna University, Chennai
Department of Electrical and Electronics Engineering

Subject Code: MA8353
Subject Name: Transforms and Partial Differential Equations

Lecture Notes – All Units

PDF Format
Number of Pages: 61

Unit I: Fourier Series

Introduction to Fourier Series:

Definition and significance.
Periodic functions and their properties.
Fourier Series Representation:

Fourier series expansion for periodic functions.
Trigonometric Fourier series and exponential Fourier series.
Properties of Fourier Series:

Linearity property.
Symmetry property.
Differentiation property.
Parseval's theorem.
Convergence of Fourier Series:

Pointwise convergence.
Uniform convergence.
Gibbs phenomenon.
Unit II: Fourier Transforms

Introduction to Fourier Transforms:

Definition and applications.
Fourier transform pair.
Properties of Fourier Transforms:

Linearity property.
Time shifting property.
Frequency shifting property.
Time scaling property.
Frequency scaling property.
Convolution theorem.
Applications of Fourier Transforms:

Signal processing.
Filtering.
Communication systems.
Unit III: Laplace Transforms

Introduction to Laplace Transforms:

Definition and properties.
Region of convergence (ROC).
Laplace Transform of Elementary Functions:

Exponential functions.
Sinusoidal functions.
Step functions.
Properties of Laplace Transforms:

Linearity property.
Time shifting property.
Time scaling property.
Differentiation property.
Integration property.
Inverse Laplace Transform:

Partial fraction expansion method.
Complex inversion integral method.
Unit IV: Z-Transforms

Introduction to Z-Transforms:

Definition and properties.
Region of convergence (ROC).
Z-Transform of Discrete-Time Signals:

Definition and examples.
Z-transform pair.
Properties of Z-Transforms:

Linearity property.
Time shifting property.
Time scaling property.
Convolution property.
Differentiation property.
Inverse Z-Transform:

Partial fraction expansion method.
Inverse Z-transform table.
Unit V: Partial Differential Equations

Introduction to Partial Differential Equations (PDEs):

Definition and classification.
Examples of PDEs in engineering.
Solution Methods for First-Order PDEs:

Method of characteristics.
Separation of variables method.
Solution Methods for Second-Order PDEs:

Classification of second-order PDEs.
Method of separation of variables.
Fourier series method.
Laplace transform method.
Applications of PDEs in Engineering:

Heat conduction.
Wave propagation.
Fluid dynamics.
These lecture notes provide a comprehensive overview of the subject "Transforms and Partial Differential Equations" as per the curriculum of Anna University, Chennai. Each unit covers the theoretical concepts, properties, solution methods, and applications in engineering. These notes serve as a valuable resource for students to understand the fundamentals of the subject and excel in their academic pursuits.