Course Hero Logo

hmk#8_s (1).pdf - MATH 241 - Partial Differential Equations...

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 - 3 out of 15 pages.

MATH 241-Partial Differential Equations(Homework#8)Fall Semester, 2020M. Carchidi———————————————————————————————————————Problem#1(25 Points) -Why Not A Regular Sturm-Liouville Problema.) (5 points) While solving a regular Sturm-Liouville Problem a student found eigenvaluesnsatisfying the equation 100nsin1/n1. What is wrong with this result?b.) (5 points) While solving a regular Sturm-Liouville Problem another student foundeigenvalues given bynn2and corresponding eigenfunctions given byunxsinnx3forn1,2,3,, and 0x1. What is wrong with this result?c.) (5 points) While solving a regular Sturm-Liouville Problem another student foundeigenvalues given bynnand corresponding eigenfunctions given byunxsinnxforn1,2,3,, and 0x1. What is wrong with this result?d.) (5 points) While solving a regular Sturm-Liouville Problem another student foundeigenvaluesnto satisfy the equationnsin1/ncos1/n. What is wrong with thisresult?e.) (5 points) While solving a Regular Sturm-Liouville Problem on the interval 0x1, astudent claims to have determinednnfor the eigenvalues, with correspondingeigenfunctionsnxsin2nxforn1,2,3,....Explain why the student must be mistaken.———————————————————————————————————————Problem#2(20 Points) -Orthogonal Seriesa.) (10 Points) Show that the functionsnxsinnx2, forn1,2,3,..., form areorthogonalset with respect to the “dot” productfg01fxgxxdx.b.) (10 Points) Use this property to then determine an expression forcnif a functionhxisexpanded ashxn1cnnxfor 0x1 and compute thecn’s for the special case whenhxx2. Then make plotsofhxagainsthmxnmcnnxfor 0x1 and form1,2,3,4,5,10 and 20 and comment on your results.1———————————————————————————————————————
———————————————————————————————————————Problem#3(15 points) -Does This Contradict Sturm-Liouville TheoryConsider the boundary-value problem:u′′xux0, for 0x1, withu00andu1u10.a.) (10 points) Show that the eigenvalues to this boundary-value problem are the positivesolutions to the equation cotnn, and the corresponding eigenfunctions areunxsinxn, forn1,2,3,.b.) (5 points) By direct calculation, show that01unxumxdx0fornm. Is this a contradiction to Sturm-Liouville Theory? Explain.———————————————————————————————————————Problem#4(20 points) -A Regular Sturm-Liouville Problema.) (10 points) Determine the eigenvalues (n) and eigenfunctions (nx) for the differentialequation′′x2xxx2x0for 0x1, along with the boundary conditions,010.Hint: See Problem#4 of Homework #4.

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 15 pages?

Upload your study docs or become a

Course Hero member to access this document

Term
Fall
Professor
RIMMER
Tags
lim, Boundary value problem, Partial differential equation, 2 w, Sturm Liouville theory, Orthogonal Series

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