Mathematical Physics III (Practical)



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Lectures

Codes

Data

Plots

Topics

  • Solution of Differential Equation:
    • 1st order & 2nd order Ordinary Differential Equation (ODE) by scipy.integrate.odeint().
    • Initial Value Problem (IVP): Modified-Euler and Runge-Kutta second order and fourth order methods.
    • Boundary Value Problem (BVP): Finite discrete method with fixed step sizes. Idea of stability. Application to simple physical problems.
      • Laplace equation 2φ=0 on a square grid with specified potential at the boundaries.
      • Wave equation in 1+1 dimension: t2φ=λ∂x2φ Vibration of a string with ends fixed with given initial configurations: φ(x,0) and ∂tφ(x,0).
      • Heat equation in 1+1 dimension, tφ=α∂x2φ with specified value of temperature at the boundaries with given initial temperature at the boundaries with given initial temperature profile.
  • Dirac-delta function:
    • Numerically handling improper integrals over infinite intervals.
    • Evaluate 1⁄√2πσ2∫e-(x-2)2⁄2σ2 (x+3) dx for x = 1, 0.1, 0.01 and show that it tends to 5.
    • -∞ exp[-(ax2+bx+c)]dx = √πaexp(b24a+c).
    • Verifying that the convolution of two Gaussian function is a Gaussian.
    • Verifying that a-x1a+x2 δ(x-a)f(x)dx = f(a) using different limiting representation of δ(x).
  • Fourier Series:
      Evaluate the Fourier coefficients of a given periodic function using scipy.integrate.quad(). Examples: square wave, triangular wave, saw-tooth wave. Plot to see a wave form from scipy.signal and the constructed series along with.
  • Special Functions:
    • Use of special functions taken from scipy.special. Plotting and verification of the properties of special functions. Orthogonality relations and recursion relations. Examples,
      • -∞ Pn(μ)Pm(μ)dμ = δnm 22n+1.
      • (1-x2)Pn′(x)+(n+1)xPn(x)=(n+1)Pn+1(x).
      • z2Jν′(z)+νJν(z)=zJν-1(z).
  • Complex analysis:
    • Integrate 0 sinx/x dx numerically and check with computer integration.
    • Root finding:
      • Compute the nth roots of unity for n=2,3,and 4.
      • Find the two square roots of −5 + 12i.

Remarks

  • Will be updated from time to time.

Study Materials

  • Numerical Analysis, Mathematics of Scientific Computing - David Kincaid, Ward Cheney.
  • Numerical Methods for Engineers - D.V. Griffiths and I.M. Smith.
  • An Introduction to computational Physics - T.Pang.
  • Scientific Computing in python - Avijit Kar Gupta.
  • Computational Physics problem solving with Computers - Landau, Paez, Bordeianu.
  • Computational Methods for Physics - Joel Franklin.
  • Programming for Computation-Python - Svein Linge & Hans Petter Lantangen.
  • Numerical Python - Robert Johansson.