Paper 6 Practical

Paper 6 Laboratory

Laboratory : Credit-1 Scilab based simulations (XCos) experiments based on Mathematical Physics problems like. * Solve simple differential equations like: - $\frac{dy}{dx} = e^{-x}$ with $y(x = 0) = 0$ - $\frac{dy}{dx} + e^{-x}=x^2$ with $y(x = 0) = 0$ - $\frac{d^2y}{dx^2}+2\frac{dy}{dx}=-y$ with $y(x=0)=0$, $y'(x=0)=1$ - $\frac{d^2y}{dx^2}+e^{-x}\frac{dy}{dx}=-y$ with $y(x=0)=0$, $y'(x=0)=1$

  • Dirac Delta Function: Evaluate $\int_{-3}^3 dx \frac{x=3}{\sqrt{2\pi a^2}} e^{-\frac{(x-2)^2}{2\sigma^2}}$, for $\sigma = 0.1, 0.01, 0.001$ and show that it tends to 5.

  • Fourier Series:
    Program to sum; evaluate the Fourier Coefficients of a given periodic function (Square Wave)

  • Frobenius Method and Special Functions: $$\int_{-1}^1 d\mu P_n\mu P_m \mu = \frac{2}{2n+1}\delta_{mn}$$ Plot $P_n(x)$, Legendre polynomial of degree n, and $J_n (x)$, Bessel function of first kind. Show Recursion relation.

  • Calculation of error for each data point of observations recorded in experiments done in previous semesters (choose any two)

  • Calculation of least square fitting manually without giving weightage to error. Confirmation of least square fitting of data through computer Programme.
  • Evaluation of trigonometric functions e.g. SinĪ˜, Given Bessel’s function at N points, find its value at an intermediate point.
  • Complex analysis: Calculate $\int \frac{dx}{x^2+2}$ and check it with computer integration.
  • Integral transform: FFT of $e^{-x^2}$
Reference Books:
  • Mathematical Methods for Physics and Engineers, K. F. Riley, M. P. Hobson and S. J. Bence, 3rd ed., 2006, Cambridge University Press.
  • Mathematics for Physicists, P. Dennery and A. Krzywicki, 1967, Dover Publications.
  • Simulation of ODE/PDE Models with MATLAB, OCTAVE and SCILAB: Scientific and Engineering Applications: A. Vande Wouwer, P. Saucez, C.V. Fernndez. 2014 Springer ISBN: 978-3319067896
  • Scilab by example: M. Affouf, 2012. ISBN: 978-1479203444
  • Silab (A freesoftware to Matlab): H. Ramchandran, A. S. Nair. 2011 S. Chand And Company,
  • Scilab Image Processing: Lambert M. Surhone. 2010 Betas cript Publishing.

Questions

2025 | +3-IV-S-CBCS(MS)-Sc(H)-Core-VIII-Phy-Pract. | Full Marks : 25

Answer any one question using Scilab, Record-5 marks, Viva-5 marks, Experiment- 15 marks

  1. Solve the differential equations.
    a. $\quad \frac{d^{2} y}{d x^{2}}+e^{-x} \frac{d y}{d x}=y$, with $y(0)=0, y^{\prime}(0)=1$.
    b. $\quad \frac{\mathrm{dy}}{\mathrm{dx}}=\ln x$, with $y(0)=0$.

  2. a. Plot Dirac delta function about $x=2$.
    b. Evaluate $\int_{-3}^{3} \mathrm{dx} \frac{(\mathrm{x}-2)}{\sqrt{2} \pi \sigma^{2}} \mathrm{e} \frac{-(\mathrm{x}+3)^{2}}{2 \sigma^{2}}$ for $\sigma=0.1$.

  3. Reproduce the square wave given by $\mathrm{f(x)}=\begin{cases}0, & -\pi<x<0 \ 2, &0<x<\pi \end{cases}$ by application of Fourier series.

  4. a. Plot Legendre polynomials $P_{1}(x)$ and $P_{2}(x)$.
    b. Verify the orthogonality condition for Legendre polynomials.
  5. Verify the recursion relations for Legendre polynomials.
  6. Calculate $\int_0^\infty \frac{dx}{x^2+2}$ using Complex analysis and check it with computer programming.
  7. Plot Bessel function of first kind for n=0 ans n=1. Show value of their first roots respectively.

2024 | +3-IV-S-CBCS(MS)-Sc.(H)-Core-VIII-Phy-Pract | Full Marks: 25

Ditto in 2025

Comments

Post a Comment