Mathematical Physics-II

Teacher

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 

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

THEORY

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.

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.

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.

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.

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).

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.

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.

 

 

PRACTICES

PRACTICE-1

Introduction to Scilab: command window, variable definition, basic operations, creating vectors and matrices.

PRACTICE-2

Plotting basics: plot simple functions (sin(x), cos(x), exp(x)), label axes, and add title and legend.

PRACTICE-3

Matrix operations: transpose, determinant, inverse, solving AX=B using \ operator.

PRACTICE-4

Writing and running scripts: create a .sce file, use loops (for, while), and use if statements.

PRACTICE-5

Using built-in functions: sum, mean, sqrt, exp, log, abs, etc.

PRACTICE-6

Plot a basic periodic function (square wave or sawtooth wave).

PRACTICE-7

Write a Scilab script to compute and plot the Fourier series approximation of a square wave (say, up to N=10 terms).

PRACTICE-8

Plot complex exponential representation of Fourier series.

PRACTICE-9

Compute and plot Fourier series for a function with an arbitrary period (e.g., period=4).

PRACTICE-10

Plot even and odd extensions of a given function.

PRACTICE-11

Use Parseval’s identity numerically to verify energy equality.

PRACTICE-12

Use Scilab to solve a second-order ODE near a singular point using the Frobenius method (simple DE, e.g., x2y′′+xy′+y=0x^2 y''+xy'+y=0x2y′′+xy′+y=0).

PRACTICE-13

Plot Legendre polynomials P0​ to P4over [−1,1].

PRACTICE-14

Plot Bessel functions J0(x) and J1(x) over [0,20].

PRACTICE-15

Plot Hermite polynomials H0 to H4​.

PRACTICE-16

Plot Laguerre polynomials L0 to L4.

PRACTICE-17

Use Rodrigues' formula in Scilab to compute Pn(x) and plot

PRACTICE-18

Verify orthogonality of Legendre polynomials numerically (compute integral).

PRACTICE-19

Compute and plot the beta function B(x,y) for some x, y.

PRACTICE-20

Compute the Gamma function Γ(z)\Gamma(z)Γ(z) numerically and plot over z.

PRACTICE-21

Plot the error function erf(x)\mathrm{erf}(x)erf(x) over suitable interval

PRACTICE-22

Generate noisy data, fit a straight line using least squares, and plot data and fitted line.

PRACTICE-23

Compute and display standard and probable errors of slope and intercept.

PRACTICE-24

Solve and plot the solution of the 1D wave equation or Laplace’s equation in a rectangle using separation of variables (numerical approximation).

 

Project List

  1. Fourier Series Analysis of Real-World Signals (e.g., ECG, Sound Waves).
  2. Visualization and Computation of Even and Odd Function Expansions.
  3. Frobenius Method for Solving Differential Equations with Singular Points.
  4. Numerical Computation of Legendre and Bessel Functions.
  5. Evaluation of Improper Integrals Using Beta and Gamma Functions.
  6. Error Propagation and Least Squares Fit in Experimental Physics Data
  7. Simulation of Vibrating String Using the Wave Equation.
  8. Laplace Equation Solutions in Cylindrical and Spherical Symmetry.
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 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):-

PRACTICE-1

Introduction to Scilab: command window, variable definition, basic operations, creating vectors and matrices.

https://youtu.be/VfNV7wr4338

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

(Using Scilab )

Session 9

Practice-2(2 hours):-

PRACTICE-2

Plotting basics: plot simple functions (sin(x), cos(x), exp(x)), label axes, and add title and legend.

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

Practice-3(2 hrs)

Matrix operations: transpose, determinant, inverse, solving AX=B using \ operator.

Session 14

Numericals  to find Applications of Fourier expansions

https://youtu.be/GbH8lPNwNtw

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

Session 15

Practice-4(2 hours):-

PRACTICE-4

Writing and running scripts: create a .sce file, use loops (for, while), and use if statements.

Session 18

Singular Points of Second Order Linear Differential Equations and their importance

https://youtu.be/oWuFqbNun7c

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

Solutions of Second order differential equation for different points.

https://youtu.be/UFWAu8Ptth0

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

Session 20

Session 19

Session 21

Practice-5(2 hours):-

PRACTICE-5

Using built-in functions: sum, mean, sqrt, exp, log, abs, etc.

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

(using scilab)

Session 24

Practice-6(2hrs)

PRACTICE-6

Plot a basic periodic function (square wave or sawtooth wave).

https://youtu.be/DotPCf6srlk

https://www.slideshare.net/Pokar/bessels-equation-4619114

Session 25

Find the solution of Hermite Differential Equations by Frobenius method.

https://youtu.be/1ZrJGJ93yV8

https://youtu.be/LoaagZPWvpM

Session 26

Practice-7(2 hours):-

PRACTICE-7

Write a Scilab script to compute and plot the Fourier series approximation of a square wave (say, up to N=10 terms).

(using scilab)

Session 27

Session 28

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 29

Properties of Legendre Polynomials

https://youtu.be/piYY_U_01js

https://youtu.be/ll7xNsbI37A

Session 30

Practice-8(2 hours):-

PRACTICE-8

Plot complex exponential representation of Fourier series.

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

(Using Scilab)

Session 31

Rodrigues Formula Generating Function

Orthogonality and Simple recurrence relation

https://youtu.be/Zm3iVO2d_2c

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

Session 32

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 33

Practice-9(2 hours):-

PRACTICE-9

Compute and plot Fourier series for a function with an arbitrary period (e.g., period=4).

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

(using scilab)

Session 34

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 35

Practice-14(2hrs)

PRACTICE-14

Plot Bessel functions J0(x) and J1(x) over [0,20].

Session 36

Session 37

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

https://youtu.be/TW_20TBF0XY

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

Session 38

Practice-15(2 hours):-

PRACTICE-15

Plot Hermite polynomials H0 to H4​.

(using scilab)

Session 41

Practice-16(2 hours):-

PRACTICE-16

Plot Laguerre polynomials L0 to L4.

(using scilab)

Session 42

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

https://youtu.be/8WeDgw7p_Q4

https://youtu.be/jJRQkyaE75w

Session 43

Error Function (Probability Integral)

https://youtu.be/-qt8CPIadWQ

https://youtu.be/CcFUQhorgdc

Session 44

Error Function (Probability Integral)

https://youtu.be/-qt8CPIadWQ

https://youtu.be/CcFUQhorgdc

Session 45

Practice-17(2 hours):-

PRACTICE-17

Use Rodrigues' formula in Scilab to compute Pn(x) and plot

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

Session 46

Systematic and Random Errors Propagation of Errors

https://youtu.be/WwXGwgDEbE0

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

Session 47

Practice-18(2hrs)

PRACTICE-18

Verify orthogonality of Legendre polynomials numerically (compute integral).

Session 48

Normal Law of Errors Standard and Probable Error

https://youtu.be/RKdB1d5-OE0

https://youtu.be/iYiOVISWXS4

Session 50

Practice-19(2 hours):-

PRACTICE-19

Compute and plot the beta function B(x,y) for some x, y.

Using scilab

Session 51

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 52

Practice-20(2hrs)

PRACTICE-20

Compute the Gamma function Γ(z)\Gamma(z)Γ(z) numerically and plot over z.

Session 53

Solution of PDEs using separation of variables

https://youtu.be/Ol1BPCRlTvw

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

Session 54

Practice-21(2hrs)

PRACTICE-21

Plot the error function erf(x)\mathrm{erf}(x)erf(x) over suitable interval

Session 55

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 56

Session 58

Practice-22(2hrs)

PRACTICE-22

Generate noisy data, fit a straight line using least squares, and plot data and fitted line

https://youtu.be/4p83iaUeTro

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

Session 60

Practice-23 (2 hours):

PRACTICE-23

Compute and display standard and probable errors of slope and intercept.

(using scilab)

Session 62

Rectangular and circular membranes Diffusion Equation

https://youtu.be/ISMPMj0QpAc

https://youtu.be/P9qar8mv3Tk

Session 63

Practice-24(2hrs)

PRACTICE-24

Solve and plot the solution of the 1D wave equation or Laplace’s equation in a rectangle using separation of variables (numerical approximation).

https://youtu.be/ISMPMj0QpAc

https://youtu.be/P9qar8mv3Tk

Session 64

Rectangular and circular membranes Diffusion Equation

https://youtu.be/ISMPMj0QpAc

https://youtu.be/P9qar8mv3Tk

Case Studies

Case Studies

Our Main Teachers

Sasmita Kumari Pradhan

Assistant Professor, CUTM Bolangir Campus
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