# hmk#10_s.pdf - MATH 241 - Partial Differential Equations...

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MATH 241-Partial Differential Equations(Homework#10)Fall Semester, 2020M. Carchidi———————————————————————————————————————A Useful Result Involving The Rayleigh QuotientGiven the Regular Sturm-Liouville Problem consisting of the ODEddxsxdxdxqxxwxx0forxalong with the BCsc1c20andd1d20,The Rayleigh Quotient is given byRQxsxx2qxx2dxsxxx|wxx2dxand it can be shown that the minimum value ofRQxfor all continuous functionsxthat satisfy the BCs (but not necessarily the ODE) will yield the lowest eigenvalue for theRSLP,i.e.,1minover allxsatisfying the BCssxx2qxx2dxsxxx|wxx2dx.You shall use this property in some of the problems in the following Homework set.———————————————————————————————————————
———————————————————————————————————————Problem#1(25 Points) -Approximating EigenvaluesConsider the RSLP having ODE′′xx0for 0x1 with BCs000 and10. In Problem #1 of Homework #9,we found the eigenvaluesnin which10 andsinnncosn0forn2,3,4,..., and this lead to220.191,359.680and4118.8998.a.) (5 points) By using a trial function of the form2xx2AxBxadetermineAandBin terms ofa.b.) (5 points) Next use this, along with the Rayleigh Quotient to approximate2as a functionofa, call it2a.c.) (10 points) Next plot2aas a function ofaand note that the statement given at thebeginning of this homework does not hold for2since there are values ofafor which theRayleigh Quotient for2ais less that the exact result of 20.191.d.) (5 points) Finally, determine the estimated value of2usinga1/3 and see how itcompares to220.191.———————————————————————————————————————Problem#2(25 Points) -Approximating EigenvaluesIn Problem #4 of Homework #9, you found the eigenvaluesand corresponding non-zeroeigenfunctionsxto the RSLP consisting of the ODE′′x4xx4x22x0for 0x1 with the BCs00 and10 and specifically you found that thesmallesteigenvalue was1≃ −4.2655 with eigenfunction

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