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Spectral methods in Matlab - 2 small problem

I have attached two problems. Fairly simple. It require Fourier Spectral method to solve.

I would need them by 2 PM (CST) May 3rd.

Although sort time but I will also provide a starting point and a pseudo code for each problem.

reply as soon as possible and only if you can finish before the deadline.

2 problems to solve: problem # 2 n # 4


[1] To make the Problem #4 simpler, please do the following change:

Domain: Both intervals for x and y are (-4, 4);

Boundary condition: Set u(-4,y,t) = u(4,y,t) = u(x, -4,t) = u(x,4,t) = -1

Initial condition: u(x,y, 0) = 1, if x^2 + y^2 < 1; u(x,y,0) = -1, otherwise.

Thus, you may use FDM or Fourier spectral method to discretize the space. In addition, the periodic Dirichelet boundary conditions would make the calculation much simpler.

[2] If forward Euler method is used for time discretization, then in part (c), in general, the smaller \epsilon, the smaller time step demanded for stability issue.

So one way without significantly increasing computational cost is to use semi-implicit discretization for time.

[3] for problem 2(a) use following instructions instead what is printed: construct a scheme with semi-implicit discretization for time and Fourier Spectral method for space to numerically solve the problem.

Kỹ năng: Toán học, Matlab and Mathematica

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