Transform Calculus and Its Applications in Differential Equations

COURSE OUTLINE: For undergraduate students in the discipline of Mathematics, the course on Transform Calculus has become an integral part. This course is designed to train students with the basic Integral Transform techniques. Application of these transforms techniques in solving ordinary differential equations and partial differential equations will be discussed. We will also introduce some higher level concepts that will prepare them for future research and development projects. The course outline is given for each week. We will introduce each topic and give an overview of the topic and underlying theory. This will be followed by some solved numerical examples on each topic for their better understanding. Weekly assignments will be provided and graded.

Transform Calculus and its applications in Differential Equations.
Lecture 01: Introduction to Integral Transform and Laplace Transform.
Lecture 02: Existence of Laplace Transform.
Lecture 03: Shifting properties of Laplace Transform.
Lecture 04: Laplace Transform of Derivative and Integration of a Function - I.
Lecture 05: Laplace Transform of Derivative and Integration of a Function - II.
Lecture 06: Explanation of properties of Laplace Transform using Examples.
Lecture 07: Laplace Transform of Periodic Function.
Lecture 08: Laplace Transform of some special Functions.
Lecture 09: Error Function, Dirac Delta Function and their Laplace Transform.
Lecture 10: Bessel Function and its Laplace Transform.
Lecture 11: Introduction to Inverse Laplace Transform.
Lecture 12: Properties of Inverse Laplace Transform.
Lecture 13: Convolution and its Applications.
Lecture 14: Evaluation of Integrals using Laplace Transform.
Lecture 15.
Lecture 16.
Lecture 17: Solution of Simultaneous Ordinary Differential Equations using Laplace Transform.
Lecture 18: Introduction to Integral Equation and its Solution Process.
Lecture 19: Introduction to Fourier Series.
Lecture 20: Fourier Series for Even and Odd Functions.
Lecture 21: Fourier Series of Functions having arbitrary period - I.
Lecture 22: Fourier Series of Functions having arbitrary period - II.
Lecture 23: Half Range Fourier Series.
Lecture 24: Parseval's Theorem and its Applications.
Lecture 25: Complex form of Fourier Series.
Lecture 26: Fourier Integral Representation.
Lecture 27: Introduction to Fourier Transform.
Lecture 28: Derivation of Fourier Cosine Transform and Fourier Sine Transform of Functions.
Lecture 29: Evaluation of Fourier Transform of various functions.
Lecture 30: Linearity Property and Shifting Properties of Fourier Transform.
Lecture 31: Change of Scale and Modulation Properties of Fourier Transform.
Lecture 32: Fourier Transform of Derivative and Integral of a Function.
Lecture 33: Applications of Properties of Fourier Transform - I.
Lecture 34: Applications of Properties of Fourier Transform - II.
Lecture 35: Fourier Transform of Convolution of two functions.
Lecture 36: Parseval's Identity and its Application.
Lecture 37: Evaluation of Definite Integrals using Properties of Fourier Transform.
Lecture 38: Fourier Transform of Dirac Delta Function.
Lecture 39: Representation of a function as Fourier Integral.
Lecture 40: Applications of Fourier Transform to Ordinary Differential Equations - I.
Lecture 41: Applications of Fourier Transform to Ordinary Differential Equations - II.
Lecture 42: Solution of Integral Equations using Fourier Transform.
Lecture 43: Introduction to Partial Differential Equations.
Lecture 44: Solution of Partial Differential Equations using Laplace Transform.
Lecture 45: Solution of Heat Equation and Wave Equation using Laplace Transform.
Lecture 46:.
Lecture 47:.
Lecture 48: Solution of Partial Differential Equations using Fourier Transform - I.
Lecture 49: Solution of Partial Differential Equations using Fourier Transform - II.
Lecture 50: Solving problems on Partial Differential Equations using Transform Techniques.
Lecture 51: Introduction to Finite Fourier Transform.
Lecture 52: Solution of Boundary Value Problems using Finite Fourier Transform - I.
Lecture 53: Solution of Boundary Value Problems using Finite Fourier Transform - II.
Lecture 54: Introduction to Mellin Transform.
Lecture 55: Properties of Mellin Transform.
Lecture 56: Examples of Mellin Transform - I.
Lecture 57: Examples of Mellin Transform - II.
Lecture 58: Introduction to Z-Transform.
Lecture 59: Properties of Z-Transform.
Lecture 60: Evaluation of Z-Transform of some functions.