Numrical Eampl In thi final chaptr, w tart b illutrating om known rult in th thor and thn procd to giv a fw novl ampl. All ampl conidr th quation F(u) = u f(u) = g, (-) with Dirichlt boundar condition on th rctangl Ω = [, ] [, ]. Hr, λ = π., λ = π 9.7, λ = 7 π.9. Rcall that olving (-) boil down to prforming two tp. Firt, w mov along th pac of fibr to idntif a point in th fibr α g. Thn w mov along thi fibr to obtain, in principl, all olution of th quation. In th firt ampl, dcribd in Sction., w choo a right-hand id g and comput an lmnt in th fibr α g. Th ubqunt ampl, in Sction., corrpond to th cond tp in th algorithm. W uppo that a fibr ha bn idntifid and mov along th fibr. W firt conidr cnario whr th fibr i on dimnional.. J =, in th pirit of Hammrtin and Dolph.. A tpical fibr in th Ambrotti-Prodi ca (J = {} and f conv), which i th α g obtaind in th prviou ubction. In particular, w find th olution of -.. Non conv f with J = {}.. J = {}, with conv and non conv f. In Sction., w conclud with a ca in which J = {, }, for which w prnt four olution for a particular g. Th conv nonlinariti f() ar contructd a follow. W choo contant α and β o that f () = α arctan() + β ha prcribd amptotic bhavior.
Numrical Anali of Ambrotti-Prodi Tp Oprator. Moving Horiontall Th firt ampl i a gnuin Ambrotti-Prodi ituation: ( Ran(f λ λ ) =, λ ) + λ >. Th right-hand id i chon to rmbl a vr ngativ multipl of th ground tat, g() = ( )( ). W tak a initial gu th ro function, u (). Uuall on or two itration of th horiontal tp lad to an rror which can onl dcra b chooing a finr mh. An m-triangulation T i th on obtaind b plitting [, ] and [, ] ach in m qual intrval on th right of Figur A., w hav a -triangulation. W prnt th normalid projctd rror for triangulation with m =,, and 6: for an approimation u n, w how n = P ξ n Q ξ n, whr ξ n = g F(u n ) and th norm ar in L. m.69.689.688.6.7.7.8.8.8 6.7.. In Figur. w how g and th function u...... Figur.: A right-hand id and a function on it fibr
Numrical Anali of Ambrotti-Prodi Tp Oprator. Moving along a fibr Unl othrwi tatd, w conidr fibr through th point u () = ϕ () + ϕ (), and th vrtical ubpac i V = V = Span{ϕ } (w u H -normalid ignfunction)... Th Ca J = Lt u tart with th implt of ca, naml b conidring linar function f, who drivativ ar not qual to ignvalu of Dir. In thi ca, b th linar thor, thr i actl on olution for ach right-hand id g. Figur. i th graph of f : in thi ca, it li alwa blow th firt f () F u Figur.: f < λ ɛ f () F u Figur.: λ + ɛ < f < λ ɛ ignvalu. Th ignvalu ar markd a dottd lin. Th lin on th right i
Numrical Anali of Ambrotti-Prodi Tp Oprator 6 th graph of an incraing function, rprnting th fact that a w mov up along th fibr u = u +tϕ, th corrponding point in th rang F = F(u ) alo mov up. Similarl, in Figur., th drivativ of f li trictl btwn λ and th cond ignvalu. In thi ca, moving up in th fibr, corrpond to moving down in th rang. W now conidr th hight variation with rpct to th cond ignvctor. W t th vrtical ubpac V = V = Span{ϕ } and conidr th fibr through u () = ϕ () + ϕ (). Thr i a ubtantial diffrnc btwn both ca: Whil on th lft of Figur., w th am pictur a in Figur.. Yt on it right w hav an incraing lin. All i till wll, inc w ar now projcting in ϕ. What w till i th up-down bhavior in th imag a w go from bing blow λ to bing abov it, a Figur. confirm. f () F u Figur.: λ + ɛ < f < λ ɛ f () F u Figur.: λ + ɛ < f < λ ɛ
Numrical Anali of Ambrotti-Prodi Tp Oprator 7.. Th Ambrotti-Prodi Ca W now rturn to th ampl of Sction.. Hr w ar actl in th ca conidrd b Ambrotti and Prodi, of an incraing f intracting onl with th firt ignvalu. A naïv analog with Figur. and. uggt that w tart b going up in th rang a w mov up in th fibr, until w rach a turning point, tarting at which w onl go downward. Thi i actl f () F 6 8 u Figur.6: Ranf σ( ) = {λ } what w in Figur.6. A prdictd b th thor, w that th horiontal lin corrponding to th hight of th right-hand id g := F(u ) i crod twic, indicating that g ha two primag in th fibr, which w prnt in Figur.7. 6 8.. Figur.7: Ambrotti-Prodi Solution W combind th corrcting algorithm dcribd in Sction.. with rgula fali.
Numrical Anali of Ambrotti-Prodi Tp Oprator 8 f () F 9 9 u Figur.8: Non-conv f.. A Non-conv Nonlinarit with J = {} Thing gt mor intrting if w go bond th Ambrotti-Prodi ca and rla th condition that f b conv. In Figur.8 w anal th ituation in which w till intract onl with λ, but altrnating btwn th bhavior n in Figur. and.. Now w hav thr ditinct olution, η = ϕ + ϕ, η and η. W cho g = F(η ). Th olution ar diplad in Figur.9.... Figur.9: Solution of non-conv ca Th qunc in Figur. how that th action of F on fibr i not homognou. Th plot ar for fibr α gi with g i = F( ϕ + c i ϕ ), whr c = (am a Fig..8), c = and c =. F F F u u u Figur.: Fibr Gtting Mappd Non-uniforml
Numrical Anali of Ambrotti-Prodi Tp Oprator 9.. Th Ca J = {}, for both Conv and Non-conv f f () F. u Figur.: Ranf σ( ) = {λ } For an ampl of a conv f intracting with λ, w tak f () = α arctan()+β o that Ranf = ( ) λ λ λ, λ + λ λ. W hav now thr primag β = ϕ + ϕ, β and β in th fibr. Again, g = F(β ). Figur. how th bhavior of th projction along th fibr and Figur. th thr olution.. 8 6.. Figur.: Conv Nonlinarit, J = {} In Figur. w chang ϕ and ϕ and prnt a qunc imilar to th on in Figur.. Thi tim, howvr, th action of F m uniform F u F u F u Figur.: Fibr Gtting Mappd Uniforml in Conv Ca acro fibr.
Numrical Anali of Ambrotti-Prodi Tp Oprator f () F 7... u Figur.: Non-conv f W now conidr a non-conv nonlinarit givn in Figur.. Thing hr ar omwhat imilar. Thr ar again thr olution γ = β, γ and γ for th right-hand g = F(γ ), hown in Figur.. In thi ca, thr i. 8 6.. Figur.: Solution of non-conv ca nonuniformit acro fibr, a hown in Figur.6. F F F u u u Figur.6: Non-Uniform Bhavior Acro Fibr. Intracting With Two Eignvalu In thi concluding ction, w attmpt to pictur th action of F on a two-dimnional fibr. Mor pcificall, w conidr th fibr α through th ro function, for which F() =. Conidr th curv C α α which projct bijctivl
Numrical Anali of Ambrotti-Prodi Tp Oprator f () undr Q to th unit circl C in th vrtical plan pannd b th firt two ignfunction ϕ and ϕ. Notic that, givn a point in C, on ma idntif th corrponding point in C α b moving horiontall with th firt tp of th algorithm. U 6 7 u F 6 7 6 D u 8 6 F Figur.7: Solution U and D on th circl C and imag. Nt, w comput F(C α ), and projct it b Q to th vrtical plan in th imag. Th rult i th fih-hapd curv on th right of Figur.7. Notic that vn point wr labld, to indicat thir poition in th domain and thir imag, giving an ida of how th curv i bing travrd. In particular, it i clar that thr hould b point U and V btwn point and and 6 and 7, rpctivl, with a common imag g markd with a bullt on th right of Figur.7. Notic alo that th origin i takn to outid of F(C α ), to it right. Now radial lin in th domain from th origin to point in C giv ri to lin from F() = to point in F(C α ), a in.8. From thi pictur, w wr abl to obtain two additional approimation L and R along th horiontal ai for primag of g.
Numrical Anali of Ambrotti-Prodi Tp Oprator Th four approimat primag wr thn takn a initial gu for Nwton Mthod and th four computd olution ar illutratd in Figur.9. n 6 n n w w w L n w n 6 u F Figur.8: Solution L and R on th u ai...9(a): U and D PUC-Rio - Crtificação Digital Nº 86/CA w w n w R F u w n n...9(b): L and R Figur.9: Computd Solution, -D ca