Mathematical Physics-II
Teacher
Sasmita Kumari Pradhan
Category
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
| COs | Course outcomes | Mapping COs with POs (High-3, Medium-2, Low-1) |
| CO1 | Understand Fourier series and its application to the solution of partial differential equations | PO1-3, PO2-3 |
| CO2 | Solve different types of second order linear differential equations and special integrals. | PO1-3, PO2-22, PO5-2 |
| CO5 | Able to solve basic statistical data analysis and curve fitting. | PO5-3 |
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:
- Mathematical Methods for Physicists: Arfken, Weber, 2005, Harris, Elsevier.
Reference Books:
- Mathematical Physics by B.S.Rajput, APragati
- Mathematical Physics by Satyaprakash, Chand&Co.
- Introduction to Mathematical Physics by Charlie
- Mathematical Physics by H.K.Dass and Dr. Rama Verma, S.Chand
- Mathematical Physics by D.Gupta.
- Fourier analysis by M.R. Spiegel, 2004, Tata McGraw-Hill.
- Mathematics for Physicists, Susan M. Lea, 2004, Thomson Brooks/Cole.
- Differential Equations, George F. Simmons, 2006, Tata McGraw-Hill.
- Partial Differential Equations for Scientists & Engineers, S.J. Farlow, 1993, Dover
- Engineering Mathematics, S.Pal and S.C. Bhunia, 2015, Oxford University Press
- Mathematical methods for Scientists & Engineers, D.A. McQuarrie, 2003, Viva Books
Session Plan
Session 1
Session 3
Orthogonality of sine and cosine functions
https://www.slideserve.com/crete/orthogonal-functions-and-fourier-series
Session 4
DirichletConditions, Expansion of periodic functions in a series of sine and cosine functions
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://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
Session 7
Practice-1(2 hours):-
Introduction to Scilab, Advantages and disadvantages, Scilab environment, Command window, Figure window, Edit window
https://www.facebook.com/freemathvideotutorials/videos/339485510121730/
(Using Scilab )
Session 8
Expansion of functions with arbitrary period
https://www.slideserve.com/Jimmy/fourier-series-of-function-with-arbitrary-period-p2l
Session 9
Practice-2(2 hours):-
Variables and arrays, Initializing variables in Scilab, Multidimensional arrays, Subarray
Using scilab
Session 10
Expansion of non-periodic functions over an interval
https://www.slideshare.net/hardik6034off/half-range-sine-cosine-fourier-series
Session 11
Session 12
Fourier expansions Application. Summing of Infinite Series
https://www.slideserve.com/metea/convergence-of-fourier-series
Session 13
Assignment-2
Numericals to find Applications of Fourier expansions
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 15
Term-by-term differentiation and integration of Fourier series
https://www.slideshare.net/DrFereidounDejahang/1531-fourier-series-integrals-and-trans
Session 17
Singular Points of Second Order Linear Differential Equations and their importance
Session 18
Assignment-3
Solutions of Second order differential equation for different points.
https://www.slideshare.net/manishreddy27/application-of-differential-equations
Session 19
Frobenius method and its applications to differential equations
https://www.slideshare.net/kushalpanchal41/series-solution-to-ordinary-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 21
Session 23
Assignment-4:-
Find the solution of Hermite Differential Equations by Frobenius method.
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://www.slideshare.net/sohag0331/applications-of-differential-equations-78670756
Session 27
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
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,
Session 33
Assignment-7, Flipped class-7:-
Generating Function and simple recurrence relations,
https://www.slideshare.net/preethicsekongu/generating-function-121065926
Session 34
Zeros of Bessel Functions (Jo(x) and J1(x)) and Orthogonality
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 36
Beta and Gamma Functions
Session 37
Properties of Gamma Function and Beta function
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
Session 40
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
Session 43
Normal Law of Errors Standard and Probable Error
Session 44
Least-squares fit ,Error on the slope and intercept of a fitted line.
https://www.slideshare.net/EdurekaIN/least-squares-regression-method-edureka
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://www.slideshare.net/aman1894/partial-differential-equations-42054204
Session 47
Solution of PDEs using separation of variables
Session 48
Solution of Laplace's Equation in problems of rectangular Coordinate system
https://www.slideshare.net/umeshkmr92/chap6-laplaces-andpoissonsequations
Session 49
Assignment-10, Flipped class-10:-
Solution of Laplace's Equation in cylindrical Coordinate.
https://www.slideshare.net/abhishekchoksi56/poissons-and-laplaces-equation
Session 51
Assignment-11, Flipped class-11:-
Wave equation and its solution
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 53
vibrational modes of a stretched string
Session 54
Assignment-12, Flipped class-12:-
Rectangular and circular membranes Diffusion Equation
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