Video Lecture Series on Mathematics - III by Dr.P.N.Agrawal, Department of Mathematics, IIT Roorkee.


1 - Solution of ODE of First Order and First Degree [53:50]
2 - Linear Differential Equations of the First Order [50:03]
3 - Approximate Solution of An Initial Value [49:50]
4 - Series Solution of Homogeneous Linear [1:01:15]
5 - Series Solution of Homogeneous Linear [50:56]
6 - Bessel Functions and Their Properties [51:10]
7 - Bessel Functions And Their Properties(Contd..) [52:31]
8 - Laplace Transformation [53:26]
9 - Laplace Transformation(contd..) [55:58]
10 - Applications Of Laplace Transformation [59:53]
11 - Applications Of Laplace Transformation(Contd.) [59:23]
12 - One Dimensional Wave Equation [54:08]
13 - One Dimensional Heat Equation [55:28]


These video lectures are delivered by Prof P.N.Agrawal, Department of Mathematics, IIT Roorkee as a part of NPTEL project. After listening to these videos courses kindly post your comments and doubts.

Detailed Syllabus

COURSE OUTLINE

Sl no.
Topics and Contents
No of lectures
No of Modules
1
Complex Numbers and Complex Algebra:
Geometry of complex numbers, Polar form, Powers and roots of complex numbers.
1
1
2
Complex Functions:
Limits of Functions, Continuity, Differentiability, Analytic functions, Cauchy-Riemann Equations, Necessary and Sufficient condition for analyticity, Properties of Analytic Functions, Laplace Equation, Harmonic Functions, Finding Harmonic Conjugate functions
5
1
3
Elementary Analytic Functions:
Exponential, Trigonometric, Hyperbolic functions and its properties. Multiple valued function and its branches - Logarithmic function and Complex Exponent function.
4
1
4
Complex Integration:
Curves, Line Integrals (contour integral) and its properties. Line integrals of single valued functions, Line integrals of multiple valued functions (by choosing suitable branches). Cauchy-Goursat Theorem, Cauchy Integral Formula, Liouville, FTA, Max/Min Modulus Theorems.
5
1
5
Power Series:
Convergence (Ordinary, Uniform, Absoulte) of power series, Taylor and Laurent Theorems, Finding Laurent series expansions.
2
1
6
Zeros, Singularities, Residues:
Zeros of analytic functions, Singularities and its properties, Residues, Residue Theorem, Rouche’s Theorem, Argument Principle.
2
1
7
Applications of Contour Integration:
Evaluating various type of indefinite real integrals using contour integration method.
4
1
8
Conformal Mapping and its applications:
Mappings by elementary functions, Mobius transformations, Schwarz-Christofel transformation, Poisson formula, Dirichlet and Neumann Problems.
5
1
9
Solution in Series:
Second order linear equations with ordinary points, Legendre equation, Second order equations with regular singular points, The method of Frobenius, Bessel equation.
4
1
10
Properties of Legendre Polynomials and Bessel Functions

2
1
11
Fourier Series:
Orthogonal Family, Fourier Series of 2? periodic functions, Formula for Fourier Coefficients, Fourier series of Odd and Even functions, Half-range series, Fourier series of a T-periodic function, Convergence of Fourier Series, Gibb’s Phenomenon, Differentiation and Integration of Fourier series, Complex form of Fourier series.
4
1
12
Fourier Transforms:
Fourier Integral Theorem, Fourier Transforms, Properties of Fourier Transform, Convolution and its physical interpretation, Statement of Fubini’s theorem, Convolution theorems, Inversion theorem, Laplace Transform.
4
1
13
Second order PDE:
Second order PDE and classification of 2nd order quasi-linear PDE (canonical form)
1
1
14
Wave Equation:
Modeling a vibrating string, D’Alembert’s solution, Duhamel’s principle for one-dimensional wave equation.
2
1
15
Heat Equation:
Heat equation, Solution by separation of variables.
2
1
16
Laplace Equation:
Laplace Equation in Cartesian, Cylindrical polar and Spherical polar coordinates, Solution by separation of variables.
3
1
17
Solution by Transform Methods:
Solutions of PDEs by Fourier and Laplace Transform methods.
2
1

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