# hmk#11_s.pdf - ENM510 - Advanced Engineering Mathematics I...

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ENM510 - Advanced Engineering Mathematics I (Practice)Fall Semester, 2018M. Carchidi––––––––––––––––––––––––––––––––––––Problem #1Determine the specific solution to the heat equation2u(x, t)x26x=u(x, t)tfor a one-dimensional rod betweenx= 0andx= 1for0< t, assuming theinsulated boundary conditions:u(x, t)x¯¯¯¯¯x=0= 0andu(x, t)x¯¯¯¯¯x=1= 0for0t, and the initial conditionu(x,0) = 0for0x1.Is yoursolution unique? Explain.
Problem #2Solve completely foru(x, t)in the regionR={(x, t)|0x1,0t}given the heat equation,2u(x, t)x2=u(x, t)talong with the boundary conditions:u(0, t) = 0andu(1, t) =t2, and theinitial condition,u(x,0) =x.––––––––––––––––––––––––––––––––––––
––––––––––––––––––––––––––––––––––––Problem #3Consider the following boundary-value, initial-value problem in the regionR={(x, t)|0x1,0t}, having partial differential equation2u(x, t)x2=u(x, t)t,boundary conditions:u(0, t) = 0andu(1, t)x+u(1, t)t= 0,and initial condition:u(x,0) =f(x).a.)(10 points)Show that the separation of variables method applied to thisproblem does not lead to a regular Sturm-Liouville problem.b.)(10 points)Continue with solving this problem using the method of sepa-ration of variables anyway, and explain why the hard part of this problemisfitting the initial condition.
Problem #4Solve foru(x, t)in the regionR={(x, t)|0x1,0t}given the partial differential equation,x2u(x, t)x2u(x, t)x=x3u(x, t)t,the boundary conditions:u(0, t) = 1,u(1, t) = 0and the initial condition,u(x,0) = 0.––––––––––––––––––––––––––––––––––––2
––––––––––––––––––––––––––––––––––––Problem #5Solve completely foru(x, t)in the regionR={(x, t)|0x1,0t}given the equation,x22u(x, t)x22xu(x, t)x+ 2u(x, t) =x22u(x, t)t2+ 2x3along with the boundary conditions:u(0, t)x= 1andu(1, t) = 3and the initial conditions:u(x,0) =x3andu(x,0)t=x2.––––––––––––––––––––––––––––––––––––3
––––––––––––––––––––––––––––––––––––Solution to Problem #1Using the approach offinding aue(x)so thatu00e(x)6x= 0oru00e(x) = 6xyieldingu0e(x) = 3x2+Aandue(x) =x3+Ax+B.

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Term
Fall
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car
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Boundary value problem