Course Hero Logo

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

Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. This preview shows page 1 - 5 out of 20 pages.

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.

Upload your study docs or become a

Course Hero member to access this document

Upload your study docs or become a

Course Hero member to access this document

End of preview. Want to read all 20 pages?

Upload your study docs or become a

Course Hero member to access this document

Term
Fall
Professor
car
Tags
Boundary value problem

Newly uploaded documents

Show More

Newly uploaded documents

Show More

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture