Mathematical Physics-II

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Sasmita Kumari Pradhan

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Core Courses

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Course Name : Mathematical Physics-II

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

https://www.facebook.com/freemathvideotutorials/videos/339485510121730/

(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

https://www.youtube.com/watch?v=U60jav2afkg

(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 25

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

https://www.youtube.com/watch?v=DITXlRH9YE0

(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

https://www.slideshare.net/muhammadzawawi1/recurrence-relations-45457581

Session 31

Practice-7(2 hours):-

statements, nested loops, logical arrays and vectorization

https://www.youtube.com/watch?v=ctedtxrq8rk

(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/-qt8CPIadWQ

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

https://www.youtube.com/watch?v=SSlZv5AIcgY

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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