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Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0...

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Advance Math

Consider the following wave equation for u(t, x) with boundaryand initial conditions defined for 0 ? x ? 2 and t ? 0.

? 2u ?t2 = 0.01 ? 2u ?x2 (0 ? x ? 2, t ? 0) (1)

?u ?x(t, 0) = 0 and ?u ?x(t, 2) = 0 (2)

u(0, x) = f(x) = x if 0 ? x ? 1 1 if 1 ? x ? 2.

(a) Compute the coefficients a0, a1, a2, . . . of the Fouriercosine series of f(x) in the interval 0 ? x ? 2. For yourcalculation, recall that sin(?n) = 0 for all integers n.

(b) Find all the possible values of ?, µ ? 0 such that thefunction v(t, x) = cos(?x) cos(µt) solves the wave equation (1) andthe boundary condition (2).

(c) Use the superposition principle, the Fourier seriescoefficeints a0, a1, . . . from part (a) and the functions v(t, x)from part (b) to write down an expression for the solution u(t, x)of the boundary and initial condition problem (1)–(3).

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