# hmk#6_s (1).pdf - MATH 241 - Partial Differential Equations...

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MATH 241-Partial Differential Equations(Homework#6)Fall Semester, 2020M. Carchidi———————————————————————————————————————Problem#1(20 Points) -Convergence of a Fourier SeriesConsider the functionfxx992x86x7215x52x3310xover the range 0x1 and periodic there after.a.) (10 points) Without computing the coefficienta0,anandbn, first explain whya0an0.Hint: How doesfx1compare withfx?b.) (10 points) Then determine how fastbnshould converge to zero. In other words,determinemso thatbnO1/nm.———————————————————————————————————————Problem#2(25 Points) -A Wave EquationDetermine a general solution to the wave equationProblem#3(20 points) -A Wave Equation
Starting with your result in Problem #2, determine the specific solution to the waveequation2ux,tx22ux,tt2for 0x1 and 0t, given the boundary conditionsux,txx00andux,txx10for 0tand the initial conditionsux,0x2andux,ttt0x2for 0x1.———————————————————————————————————————
———————————————————————————————————————Problem#4(25 points) -A Time-Independent Solution to a Wave EquationConstruct the time-independent solution to the PDE———————————————————————————————————————
Problem#5(10 points) -Complex Form of the Fourier SeriesWe had seen in class that the complex form of a Fourier Series is given byfxn−cne2inx/Pwithcn1PaaPfxe2inx/Pdxfor any constanta. Determine the coefficientscnfor the functionfx|sinx|.———————————————————————————————————————2
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Term
Spring
Professor
Kobylinski
Tags
Partial differential equation, 1 L, 3 k, x, 2020 M, u x