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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