Mathematical Physics III (Practical)

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### 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.