Topics

• 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 = √^π⁄^aexp(^b2⁄_4a+c).
  • Verifying that the convolution of two Gaussian function is a Gaussian.
  • Verifying that ∫_a-x1^a+x2 δ(x-a)f(x)dx = f(a) using different limiting representation of δ(x).
• Solution of Differential Equation:
  • 1^st order & 2^nd order Ordinary Differential Equation (ODE) by scipy.integrate.odeint().
• 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,
    • ∫_-∞^∞ P_n(μ)P_m(μ)dμ = δ_nm ²⁄_2n+1.
    • (1-x²)P_n′(x)+(n+1)xP_n(x)=(n+1)P_n+1(x).
    • z²J_ν′(z)+νJ_ν(z)=zJ_ν-1(z).
• Solution of some basic PDEs:
  • Boundary value problems. 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.
• 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.

Study Materials

• Scientific Computing in python - Avijit Kar Gupta.
• Computational Physics problem solving with Computers - Landau, Paez, Bordeianu.
• Programming for Computation-Python - Svein Linge & Hans Petter Lantangen.
• Numerical Solution of Partial Differential Equations in Science and Engineering - L. Lapidus & G.F. Pinder.
• Spectral Methods in MATLAB - L.N.Trefethen.
• Limit Representation of Dirac Delta.