1Frobenius method Applied to Frobenius method Applied to BesselBessel’’s Equations EquationLarry CarettoMechanical Engineering 501ABSeminar in Engineering AnalysisOctober 14, 20042Outline• Review last week– Power series solutions– Frobenius Method• Apply Frobenius method to Bessel’s equation– Obtained indicial equation last week– Get first solution– Differences in second solution– Definition of Bessel Functions3Review Power Series Solutions• Look at following equation and proposed power series solution• Requires p(x), q(x) and r(x) that can be expanded in power series about x = x0)()()(22xryxqdxdyxpdxyd=++∑∞==00)-()(nnnxxaxy∑∞=−=010)-(nnnxxnadxdy∑∞=−−=02022)-()1(nnnxxanndxyd4Review Getting the Solutions• Manipulate series to get single summation with common power of x and common limits– Use substitution of exponents to get common exponents– Remove terms from summations, giving individual terms, plus a common sum)()-()()-()()-()1(00010020xrxxaxqxxnaxpxxannnnnnnnnnn=++−∑∑∑∞=∞=−∞=−5Review Getting the Solutions II• Result of manipulating sum is series that has form Σmcmxm= 0– Can only satisfy this equation if all cm= 0– The cmusually involve combinations of the original anterms– This gives equations between anand a coefficients with subscripts n-1, n-2, etc.– Initial few coefficients unknown, used to match boundary conditions– Can get all original anin terms of these original coefficients6Review Frobenius Method• Applied to differential equation below• Usual power series method inapplicable• Solution similar to previous power series (with x0= 0) except for xrfactor0)()()()(222=++ yxxcdxxdyxxbdxxyd∑∑∞=+∞===00)(nrnnnnnrxaxaxxy27Review Frobenius Method II• Differentiate proposed solution two times• Get power series for b(x) and c(x)• Substitute into original equation• Set coefficient of lowest term, xr, to zero• This gives indicial equation, a quadratic equation with two roots for r, r1and r2• Need two solutions but have different second solution depending on r1and r2– Same, differ by integer, differ by noninteger8Review Frobenius Method III• First and second solutions y1(x) and y2(x) • Double root∑∞==011)(nnnrxaxxy∑∞==022)(nnnrxAxxy• First solution, all cases• Root difference r1–r2not an integer∑∞=+=112)ln()()(nnnxAxxyxy• Roots differ by integer (k may be 0)∑∞=+=0122)ln()()(nnnrxAxxxkyxy9Bessel’s Equation• Arises in mechanical and thermal problems in circular geometries• The value of ν is a known parameter• Solve by Frobenius method 0)(1)(22222=−++ yxxdxxdyxdxxydν∑∑∞=+∞===00)(nrnnnnnrxaxaxxy∑∑∞=−+∞=−+−++=+=022201)1)(()(nrnnnrnnxarnrndxydxarndxdy10Bessel’s Equation II• Plug solution and derivatives into Bessel’s equation and rearrange()0)()1)((02200=−+++−++∑∑∑∞=+∞=+∞=+nrnnnrnnnrnnxaxxarnxarnrnν[]0)()1)((0202=+−++−++∑∑∞=++∞=+nrnnnrnnxaxarnrnrnν[][]0)()(2202202022=+−+=+−+∑∑∑∑∞=+−∞=+∞=++∞=+nrnnnrnnnrnnnrnnxaxarnxaxarnννL+++=+++423120rrrxaxaxaBoth11Bessel’s Equation III• Final arrangement gets indicial equation[] [][][ ]0)()1()0()(0222112202222022=+−++−++−+=+−+∑∑∑∞=+−+∞=+−∞=+nrnnnnrrnrnnnrnnxaaarnxarxarxaxarnνννν• Indicial equation (r2– ν2= 0) roots ±ν– Solution gives double root if ν = 0– Roots differ (do not differ) by an integer for integer (non-integer) ν12Bessel’s Equation IV•With r = ν, we must have a1= 0[][]0)()1(22221122=+−++−+∑∞=+−+nnnnnxaaanxaνννννν•With a1= 0, all anwith n odd vanish• Unknown coefficient a0from initial conditions on the differential equation• For coefficients of xn+νto vanish )2()(2222ννν+−=−+−=−−nnanaannn313Bessel’s Equation V• Get new subscript, m = n/2 (n = 2m)• Test general result proposed below• Get even coefficients, a2m, in terms of a0)2(2ν+−=−nnaann)(2)22(2222222νν+−=+−=−−mmammaammm)1(2202ν+−=aa)1)(2)(2(2)2)(2(240224ννν++=+−=aaa)1)(2()1)((!2)1(202νννν+++−+−=Lmmmaammm14Bessel’s Equation VI• Compute a2m/a2m-2from general equation• Result matches equation from last chart• Now have general result for first root of indicial equation, r = ν)(21)(!2)!1)(1(22222νν+−=+−−=−mmmmmaamm)1)(2()2)(1()!1(2)1()1)(2()1)((!2)1(220120)1(22222νννννννν+++−+−−−+++−+−=−−−−LLmmmammmaaaaammmmmmmm15Bessel’s Equation VII• For integer ν = n, multiply a2mby n!/n!• Pick a0= 1/(2nn!) to give convenient functions for tabulation• Use gamma functions to get similar result for non-integer ν)!(!2!)1(!!)1)(2()1)((!2)1(20202nmmnannnnnmnmmaammmmm+−=+++−+−=L)!(!2)1()!(!2!)1(!212222nmmnmmnnanmmmmm+−=+−=+16Gamma Functions• Function Γ(x) generalizes factorials to non-integer arguments (Appendix C)∫∞−−=Γ01)( dttexxt)()1( xxxΓ=+Γ• Definition• Analog of (n+1)! = (n+1)n!• For integer x, Γ(n+1) = n! = nΓ(n)• Application to Bessel coefficients below)1(!2)1()1()1)(2()1)((!2)1(20202++Γ+Γ−=+++−+−=ννννννmmammmaammmmmL17Bessel Functions• Solutions use specific definition of a0= 1/[2νΓ(n+1)] for tables giving)1(!2)1()1(!2)1()1()1(1222++Γ−=++Γ+Γ−+Γ=ννννmmmmammmmm• Substitute into original solution for r = ν∑∑∑∞=++∞=+∞=+++Γ−===0220220)1(!2)1()(mmmmmmmnnnmmxxaxaxyννννν• Look at integer and non-integer ν18Bessel Functions II• Use n for integer values of ν• For integer x, Γ(x + 1) = x!• Bessel function, first kind, integer order• First few terms (we chose n ≥ 0)∑∑∞=++∞=+++−=⇒++Γ−=022022)!(!2)1()()1(!2)1()(mnmnmmnmmmmnmmxxJmmxxJνννν• Plots for n = 0,1, and 4 on next chart⎥⎥⎦⎤⎢⎢⎣⎡+⎟⎠⎞⎜⎝⎛++⎟⎠⎞⎜⎝⎛+−⎟⎠⎞⎜⎝⎛= L422)!2(!212)!1(!11!12)(xnxnnxxJnn4Bessel Functions IIIBessel Functions of the First Kind for Integer Orders-0.4-0.3-0.2-0.100.10.20.30.40.50.60.70.80.910123456789101112131415xJn(x)n = 0n = 1n = 420Bessel Functions IV• Back to Frobenius method for second solutions in three cases–n = ν = 0, the double root– Integer ν = n ≠ 0, roots differ by an integer, J-n(x) = (-1)nJn(x)– Non-integer ν, easiest case, Jνand J-νare two linearly independent solutions• General case for second solution∑∞=−+=]1,0[2)ln()()(mnmmnxAxxkJxy• For n = 0, k = mfirst= 121Bessel Functions V• Substitute proposed second solution into