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

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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.

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