# Mathematical Physics-II

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# Code(Credit) : CUTM1487(3-2-1)

## Course Objectives

• Fourier series and its application to the solution of partial differential equations.
• Study of Second order linear differential equations and their importance
• Introduce the concepts of Laplace equation, its application, basic statistical data analysis and curve fitting.

## Learning Outcomes

• Expand a function in Fourier series.
• Find the solution of the wave, diffusion and Laplace equations using the Fourier series.
• Find the mean and the variance of a random variable.
• describe special functions and their recurrence relations

## Course Syllabus

Module-I

Fourier series: Periodic functions. Orthogonality of sine and cosine functions, Dirichlet Conditions (Statement only). Expansion of periodic functions in a series of sine and cosine functions and determination of Fourier coefficients. Complex representation of Fourier series. Expansion of functions with arbitrary period.

Practice-1 Introduction to Scilab, Advantages and disadvantages, Scilab environment, Command window, Figure window, Edit window

Practice-2  Variables and arrays, Initializing variables in Scilab, Multidimensional arrays, Subarray

##### Module-II

Expansion of non-periodic functions over an interval. Even and odd functions and their Fourier expansions. Application. Summing of Infinite Series. Term-by-Term differentiation and integration of Fourier series. Parseval Identity.

Practice-3 Special values, Displaying output data, data file, Scalar and array operations Hierarchy of operations, Built in Scilab functions

Practice-4 Introduction to plotting, 2D and 3D plotting

##### Module-III

Frobenius Method and Special Functions: Singular Points of Second order linear differential Equations and their importance. Frobenius method and its applications to differential equations. Legendre, Bessel, Hermite and Laguerre Differential Equations.

Practice-5 Branching Statements and program design, Relational & logical operators

practice-6 The while loop, for loop, details of loop operations, break & continue statements

##### Module-IV

Properties of Legendre Polynomials: Rodrigues Formula, Generating Function, Orthogonality. Simple recurrence relations. Expansion of function in a series of Legendre Polynomials. Bessel Functions of the First Kind: Generating Function, simple recurrence relations. Zeros of Bessel Functions (Jo(x) and J1(x))and Orthogonality.

Practice-7 Curve fitting for experimental Data. Solution of AX=B Using Gauss Elimination.

Practice-8 Numerical Integration using Trapizoidal Rule and Simpson's Rule

##### Module-V

Some Special Integrals: Beta and Gamma Functions and Relation between them.Expression of Integrals in terms of Gamma Functions. Error Function (Probability Integral).

Practice-9  Find the eigen value and eigen vector of the input matrix. Solution Of Radioactive decay .

##### Module-VI

Theory of Errors: Systematic and Random Errors. Propagation of Errors. Normal Law of Errors. Standard and Probable Error. Least-squares fit. Error on the slope and intercept of a fitted line.

Practice-10 An introduction to Scilab file processing, file opening and closing, Binary I/o functions, comparing binary and formatted functions, Numerical methods and developing the skills of writing a program

Practice-11 Ohms law to calculate R, Hooke’s law to calculate spring constant.

##### Module-VII

Partial Differential Equations: Solutions to partial differential equations, using separation of variables: Laplace's Equation in problems of rectangular, cylindrical and spherical symmetry. Wave equation and its solution for vibrational modes of a stretched string, rectangular and circular membranes. Diffusion Equation.

Practice-12 Solution of simple pendulum  and Double Pendulum using Euler and RK4 method

##### Text Book:
1. Mathematical Methods for Physicists: Arfken, Weber, 2005, Harris, Elsevier.
##### Reference Books:
1. Mathematical Physics by B.S.Rajput, APragati
2. Mathematical Physics by Satyaprakash, Chand&Co.
3. Introduction to Mathematical Physics by Charlie
4. Mathematical Physics by H.K.Dass and Dr. Rama Verma, S.Chand
5. Mathematical Physics by D.Gupta.
6. Fourier analysis by M.R. Spiegel, 2004, Tata McGraw-Hill.
7. Mathematics for Physicists, Susan M. Lea, 2004, Thomson Brooks/Cole.
8. Differential Equations, George F. Simmons, 2006, Tata McGraw-Hill.
9. Partial Differential Equations for Scientists & Engineers, S.J. Farlow, 1993, Dover
10. Engineering Mathematics, S.Pal and S.C. Bhunia, 2015, Oxford University Press
11. Mathematical methods for Scientists & Engineers, D.A. McQuarrie, 2003, Viva Books

## Session 4

DirichletConditions, Expansion of periodic functions in a series of sine and cosine functions

https://youtu.be/Xdn5rsisF6E

https://www.slideserve.com/alden-mcmahon/12-1-the-dirichlet-conditions

## Session 5

Assignment-1-

Numerical problems to find Fourier series and Fourier's coefficient

https://youtu.be/OjHgoZOdRKM

https://www.slideserve.com/kolton/chapter-6-periodic-functions-6-1-the-sine-and-cosine-functions

## Session 6

Determination of Fourier coefficients and  Complex representation of Fourier series

https://youtu.be/oXrbcRyXj84

https://youtu.be/k8FXF1KjzY0

## Session 7

Practice-1(2 hours):-

Introduction to Scilab, Advantages and disadvantages, Scilab environment, Command window, Figure window, Edit window

https://youtu.be/VfNV7wr4338

(Using Scilab )

## Session 9

Practice-2(2 hours):-

Variables and arrays, Initializing variables in Scilab, Multidimensional arrays, Subarray

https://youtu.be/FxtcyHW9mP4

https://youtu.be/39RCsMbiseg

Using scilab

## Session 12

Fourier expansions Application. Summing of Infinite Series

https://youtu.be/jPx2S4mSZWc

https://www.slideserve.com/metea/convergence-of-fourier-series

## Session 13

Assignment-2

Numericals  to find Applications of Fourier expansions

https://youtu.be/GbH8lPNwNtw

https://slideplayer.com/slide/4923187/

## Session 14

Practice-3(2 hours):-

Special values, Displaying output data, data file, Scalar and array operation Hierarchy of operations, Built in Scilab functions

(using scilab)

## Session 17

Singular Points of Second Order Linear Differential Equations and their importance

https://youtu.be/oWuFqbNun7c

https://slideplayer.com/slide/4918704/

## Session 18

Assignment-3

Solutions of Second order differential equation for different points.

https://youtu.be/UFWAu8Ptth0

https://www.slideshare.net/manishreddy27/application-of-differential-equations

## Session 20

Practice-4(2 hours):-

Introduction to plotting, 2D and 3D plotting

(using scilab)

## Session 23

Assignment-4:-

Find the solution of Hermite Differential Equations by Frobenius method.

https://youtu.be/1ZrJGJ93yV8

https://youtu.be/LoaagZPWvpM

## Session 24

Practice-5(2 hours):-

Branching Statements and program design, Relational & logical operators

(using scilab)

## Session 26

Assignment-5 , flipped class-5:-

Solutions of second order differential equation by using Frobenius method.

https://youtu.be/6o7b9yyhH7k

https://www.slideshare.net/sohag0331/applications-of-differential-equations-78670756

## Session 27

Properties of Legendre Polynomials

https://youtu.be/piYY_U_01js

https://youtu.be/ll7xNsbI37A

## Session 28

Practice-6(2 hours):-

The while loop, for loop, details of loop operations, break & continue statements

(Using Scilab)

## Session 29

Rodrigues Formula Generating Function

Orthogonality and Simple recurrence relation

https://youtu.be/Zm3iVO2d_2c

https://slideplayer.com/slide/10002071/

## Session 30

Assignment-6

Numericals to find the properties and problems of Legendre's Polynomials

https://youtu.be/qeOkxZek_VsU

## Session 31

Practice-7(2 hours):-

statements, nested loops, logical arrays and vectorization

(using scilab)

## Session 32

Expansion of function in a series of Legendre Polynomials and Bessel Functions of the First Kind,

https://youtu.be/lf8Nzb622s4

https://youtu.be/f86K0QerUug

https://slideplayer.com/slide/11575472/

## Session 33

Assignment-7, Flipped class-7:-

Generating Function and simple recurrence relations,

https://youtu.be/Pp4PWCPzeQs

https://www.slideshare.net/preethicsekongu/generating-function-121065926

## Session 34

Zeros of Bessel Functions (Jo(x) and J1(x)) and Orthogonality

https://youtu.be/TW_20TBF0XY

https://www.slideshare.net/rendyrobert7/ch05-8

## Session 35

Practice-8(2 hours):-

User defined functions, Introduction to Scilab functions, Variable passing in Scilab, optional arguments, preserving data between calls to a function,

(using scilab)

## Session 38

Practice-9(2 hours):-

Find the eigen value and eigen vector of the input matrix. Solution Of Radioactive decay

(using scilab)

## Session 39

Assignment-8,Flipped class-8:-

Relation between Beta and Gamma function. Integrals in terms of Gamma Functions

https://youtu.be/8WeDgw7p_Q4

https://youtu.be/jJRQkyaE75w

## Session 40

Error Function (Probability Integral)

https://youtu.be/CcFUQhorgdc

## Session 41

Practice-10(2 hours):-

An introduction to Scilab file processing, file opening and closing, Binary I/o functions, comparing binary and formatted functions, Numerical methods and developing the skills of writing a program

## Session 42

Systematic and Random Errors Propagation of Errors

https://youtu.be/WwXGwgDEbE0

https://nptel.ac.in/courses/122107035/

## Session 43

Normal Law of Errors Standard and Probable Error

https://youtu.be/RKdB1d5-OE0

https://youtu.be/iYiOVISWXS4

## Session 45

Practice-11(2 hours):-

Ohms law to calculate R, Hooke’s law to calculate springconstant

Using scilab

## Session 46

Assignment-9, Flipped class-9 :-

Solutions to partial differential equations by using Complimentary function and particular integral

https://youtu.be/Gxe7-wGwqFk

https://www.slideshare.net/aman1894/partial-differential-equations-42054204

## Session 47

Solution of PDEs using separation of variables

https://youtu.be/Ol1BPCRlTvw

https://slideplayer.com/slide/9505473/

## Session 48

Solution of Laplace's Equation in problems of rectangular Coordinate system

https://youtu.be/TOpxK_eMR8Y

https://www.slideshare.net/umeshkmr92/chap6-laplaces-andpoissonsequations

## Session 49

Assignment-10, Flipped class-10:-

Solution of Laplace's Equation in cylindrical Coordinate.

https://youtu.be/54XDoIyZOXY

https://www.slideshare.net/abhishekchoksi56/poissons-and-laplaces-equation

## Session 51

Assignment-11, Flipped class-11:-

Wave equation and its solution

https://youtu.be/4p83iaUeTro

https://www.slideshare.net/DhavalJalalpara/the-wave-equation

## Session 52

Practice-12(2 hours):-

Radioactive decay,  Current in RC, LC circuits with DC source, Newton’s law of cooling

(using scilab)

## Session 54

Assignment-12, Flipped class-12:-

Rectangular and circular membranes Diffusion Equation

https://youtu.be/ISMPMj0QpAc

https://youtu.be/P9qar8mv3Tk

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