Each of the following problems are related with my recent research. All these problems are the zeroth order steps to reach what I am working on now-a-days.
Basic:
Digitize a traveling wave on a vibrating string [An interactive example!]
What is(are) the normal mode(s) of 4-coupled spring-mass-system arranged in the form of a square?
Solve this functional differential equation: x′(t) = - x(t-τ). Feel free to choose your own initial conditions.
Write down a simple computer program to solve one-dimensional random-walk-problem on a periodically compressed (and stretched) rubber-band.
Develop a baby program (in Fortran or C++) to solve a system of nonlinear equations. [Hint: Use this Algorithm 10.3 @ page 658]
Factorize a number, (for example: {15: 1, 3, 5, 15}), using a computer algorithm that takes polynomial time. [Hint: Use Shor's algorithm and Qiskit architecture]
Intermediate:
If you were a computer, how will you identify, whether a thread/string is 'knotted' or not?
Sketch an algorithm that can distinguish between a 'one-sided-surface' (Möbius strip) and a standard 'two-sided-closed-surface'.
Given a thin current carrying wire of arbitrary shape, can you find the magnetic field at a distance? [Write down a small computer program!]
Can you do the inverse problem, that is, given a prescribed magnetic field profile, can you find the configuration of the current carrying wire? [-do-]
How will you measure the electromagnetic radiation from a current carrying tube of arbitrary shape? [-do-]
Advanced:
If I ask you to write down a simple advection solver using third-order upwind difference scheme, which flux-limiter will you prefer to use and why?
Where is the global-minimum of this function: f(x) = sin(x) - exp(- x^2) [Which optimization method will you choose and why?]
Take a Fourier transform with "pencil-decomposition". [Hint: Use FFTW library]
What is that equation, whose solution is: x(t) = A cos (ω t + φ) [Write down a small computer program in support of your claim!]