HW5, due Wednesday, Aug. 11
数学作业代写 Determine graphically (from the error plot) how many terms in the partial sum of the Fourier series of the odd extension
Part I. 数学作业代写
(See accompanying Mathematica file)
1.Determine graphically (from the error plot) how many terms in the partial sum of the Fourier series of the odd extension of the functions f(x)=exp(-x^2) and f(x)=x^3 given on interval [0, Pi] are needed to preserve the energy in the original signal to 10^(-16)? (or, equivalently, to satisfy the Parceval’s identity within 10^(-16)).
Part II. 数学作业代写
- Convert to a non-dimensional form the following equation, u_t + a u_x = b u^2, where aand b are constants.
- Use Fourier series to solve the following initial boundary value problem for the heat equation u_t = u_xx, hat function as an initial condition, i.e. u(x,0) = 1, on (0,Pi) and zero boundary conditions. Plot resulting solution at t=0.1 and t=4 with partial sum approximation forN1=5.
- Apply the separation of variables to the heat equation with the source (denoted as Method I in class notes), u_t=u_xx +h(x), for x in (0, Pi), assuming that both the source and the initial condition can be expanded in a sine Fourier series on interval [0, Pi]. You do not have to solve the resulting equation, but identify the method from HW3 that can be used to solve it.
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Part III.
1.Calculate and plot the Fourier transforms of the following fourfunctions:
Plot real and/or imaginary parts as appropriate, see Mathematica file illustration for f(x) =1/x.
a)exp( -pix^2),
b)xrect(x),
c)exp(- pi a^2 x^2) cos( 2 pi b x) for a=2;b=3,
d)x exp(-x)Heaviside(x),
e)1/x, 数学作业代写
f)Heaviside(x), (is called UnitStep[x] inMathematica),
g)Haar function, h(x) = {{ 1, 0<x<1/2 }, {-1, 1/2x<1}} and zero outside interval[0,1].
2.Prove statement (5.20) of the Shift Theorem 5.7 in sec. 5.3, p. 285 (FTHansen1d.pdf)
3.Following six problems from FTHansen1D.pdf
p.351, 5.1 (c), (d); 5. 2(b);5.3(a),(b) (You may use any software for plotting.)
p.352,5.13 (You may assume a & b to be positive anda<b)
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