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• 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
• 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
THEORY
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.
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.
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.
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.
Some Special Integrals: Beta and Gamma Functions and Relation between them.Expression of Integrals in terms of Gamma Functions. Error Function (Probability Integral).
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.
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.
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).
Orthogonality of sine and cosine functions
https://www.slideserve.com/crete/orthogonal-functions-and-fourier-series
DirichletConditions, Expansion of periodic functions in a series of sine and cosine functions
https://www.slideserve.com/alden-mcmahon/12-1-the-dirichlet-conditions
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
Determination of Fourier coefficients and Complex representation of Fourier series
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 )
Expansion of functions with arbitrary period
https://www.slideserve.com/Jimmy/fourier-series-of-function-with-arbitrary-period-p2l
Practice-2(2 hours):-
Variables and arrays, Initializing variables in Scilab, Multidimensional arrays, Subarray
Using scilab
Expansion of non-periodic functions over an interval
https://www.slideshare.net/hardik6034off/half-range-sine-cosine-fourier-series
Fourier expansions Application. Summing of Infinite Series
https://www.slideserve.com/metea/convergence-of-fourier-series
Assignment-2
Numericals to find Applications of Fourier expansions
Practice-3(2 hours):-
Special values, Displaying output data, data file, Scalar and array operation Hierarchy of operations, Built in Scilab functions
(using scilab)
Term-by-term differentiation and integration of Fourier series
https://www.slideshare.net/DrFereidounDejahang/1531-fourier-series-integrals-and-trans
Singular Points of Second Order Linear Differential Equations and their importance
Assignment-3
Solutions of Second order differential equation for different points.
https://www.slideshare.net/manishreddy27/application-of-differential-equations
Frobenius method and its applications to differential equations
https://www.slideshare.net/kushalpanchal41/series-solution-to-ordinary-differential-equations
Practice-4(2 hours):-
Introduction to plotting, 2D and 3D plotting
https://www.youtube.com/watch?v=U60jav2afkg
(using scilab)
Assignment-4:-
Find the solution of Hermite Differential Equations by Frobenius method.
Practice-5(2 hours):-
Branching Statements and program design, Relational & logical operators
(using scilab)
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
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)
Rodrigues Formula Generating Function
Orthogonality and Simple recurrence relation
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
Practice-7(2 hours):-
statements, nested loops, logical arrays and vectorization
https://www.youtube.com/watch?v=ctedtxrq8rk
(using scilab)
Expansion of function in a series of Legendre Polynomials and Bessel Functions of the First Kind,
Assignment-7, Flipped class-7:-
Generating Function and simple recurrence relations,
https://www.slideshare.net/preethicsekongu/generating-function-121065926
Zeros of Bessel Functions (Jo(x) and J1(x)) and Orthogonality
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)
Beta and Gamma Functions
Properties of Gamma Function and Beta function
Practice-9(2 hours):-
Find the eigen value and eigen vector of the input matrix. Solution Of Radioactive decay
(using scilab)
Assignment-8,Flipped class-8:-
Relation between Beta and Gamma function. Integrals in terms of Gamma Functions
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
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.
https://www.slideshare.net/EdurekaIN/least-squares-regression-method-edureka
Practice-11(2 hours):-
Ohms law to calculate R, Hooke’s law to calculate springconstant
Using scilab
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
Solution of PDEs using separation of variables
Solution of Laplace's Equation in problems of rectangular Coordinate system
https://www.slideshare.net/umeshkmr92/chap6-laplaces-andpoissonsequations
Assignment-10, Flipped class-10:-
Solution of Laplace's Equation in cylindrical Coordinate.
https://www.slideshare.net/abhishekchoksi56/poissons-and-laplaces-equation
Assignment-11, Flipped class-11:-
Wave equation and its solution
https://www.slideshare.net/DhavalJalalpara/the-wave-equation
Practice-12(2 hours):-
Radioactive decay, Current in RC, LC circuits with DC source, Newton’s law of cooling
(using scilab)
vibrational modes of a stretched string
Assignment-12, Flipped class-12:-
Rectangular and circular membranes Diffusion Equation
Assignment-12, Flipped class-12:-
Rectangular and circular membranes Diffusion Equation
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