RG Tutorial xlc3.doc 1/10. To apply the R-G method, the differential equation must be represented in the form:
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1 G Tuorial xlc3.oc / iear roblem i e C i e C ( ie ( Differeial equaio for C (3 Thi fir orer iffereial equaio ca eaily be ole bu he uroe of hi uorial i o how how o ue he iz-galerki meho o fi ou he oluio. iz Galerki Meho (wih weihe reiual To aly he -G meho, he iffereial equaio mu be rereee i he form:, /, / ] x (4 where x i a liear or oliear oeraor The aume aroximae oluio mu be rereee i he form below y N ( Y λ ( λ λϕ (5 where he ϕ λ are N liear ieee fucio a he Y λ are N ajuable coa coefficie. Sice he aume oluio (5 oe o i eeral exacly aify he iffereial equaio (4, he exreio obaie by ubiui (5 io (4 i o loer ieically equal o zero:, /, / ] ε ( x (6 where ε( i he reiual ha meaure he error.
2 G Tuorial xlc3.oc / Thi error i miimize if he followi yem of N weihe reiual, calle he iz coiio, i aifie. x, /, / ] ϕ ( λ,,,... N λ Thi roceure ie a yem of N alebraic equaio i N ukow. (7 For circui how aboe, if.co( he he ouu olae a he fuameal frequecy require wo arameer o be efie. Thee wo arameer are he maiue a he hae or he maiue of he i-hae a he i-quaraure comoe. Coieri he i-hae a i-quaraure comoe, he oluio for ca be aroximae by he followi relaio: ( i( ( co (8 We hae wo ukow & meai ha wo equaio are eee o fi ou & The iz coiio will ie u hoe wo eee equaio. By coieri reiual ol he ( Yλϕ λ ( ϕ ϕ ( co( ( i( (9 ( π ξ [,,, /, / ] ϕ ( ( λ, Wih x efie i (3 λ ( co λ ϕ efie i (9 & ( ( ( ( efie i (8 ( From (8 ( i ( co( (3 G meho lea o ole he wo followi equaio or iz coiio From ( [ co(, co( i(,, /, / ] co( ( λ ξ From ( (4 [ co(, co( i(,, /, / ] i( ( λ ξ (5
3 G Tuorial xlc3.oc 3/ Fir iz Coiio From (4 C co( ( (6 (8, (, (3 i (6 C i ( C co ( co( i( co( ( co( From (7 C co From (8 π i( i( co( ( ( co( C co( [ C i( ] i( co( ( π [ C ] co ( ( (7 (8 (9 From (9 [ C ] co ( ( From ( ( co [ ] ( C From ( ( ( [ C ] ( From ( ( [ C ] π Fially from (3 (3 C (4
4 G Tuorial xlc3.oc 4/ Seco iz Coiio From (5 C i( ( (5 (8, (, (3 i (5 C i ( C co ( co( i( i( ( co( From (6 C co From (7 i( i( i( ( ( co( C co( π [ C ] i ( ( π [ C ] co( i( ( (6 (7 (8 From (8 [ C ] i ( ( From (9 (9 co [ ] ( C From (3 ( (3 [ C ] ( From (3 (3 [ C ] π (3 Fially from (3 C (33
5 G Tuorial xlc3.oc 5/ From (33 (34 C (34 i (4 ( ( C C (35 Fially from (35 C ( C (36 (36 i (33 C C (37 From (37 C Fially from (38 ( C C (38 ( C ( C (39 From (39 C ( co( i( (4 ( C ( C From (4 ( [ co( C i( ] (4 ( C e efie φ a follow a( φ C (4 (4 i (4 ( [ co( a( φi( ] (43 ( C From (43 co ( C co( φ i( φ ( i( ( From (44 ( ( C co( φ co ( i( φi( co( φ (44 (45
6 G Tuorial xlc3.oc 6/ From (45 ( ( C ( φ co co( φ (46 From (4 ( a ( φ C co ( φ From (47 co( φ (47 ( (48 C Fially we fi ou he well kow relaio for he C circui. ( co ( C ( φ (49 Wih φ efie a below ( C φ arca (5
7 G Tuorial xlc3.oc 7/ Noliear roblem i e e C (5 ie C ie Ex (5 i (53 e Differeial equaio for (5 i (53 C (54 (54 i (5 i Ex C e (55 (55 i (5 i e C. Ex C (56 Ex C C (57 Thi iffereial equaio cao be eaily ole, bu he iz-galerki meho ca be ue o fi ou a aroximae oluio. iz Galerki Meho (wih weihe reiual To aly he -G meho, he iffereial equaio mu be rereee i he form:
8 G Tuorial xlc3.oc 8/, /, / ] x (58 where x i a liear or oliear oeraor f oly oe reiual i coiere he aume aroximae oluio mu be rereee i he form below y( Yϕ ( (59 where ϕ i a fucio a he Y i a ajuable coa coefficie. Sice he aume oluio (59 oe o i eeral exacly aify he iffereial equaio (58, he exreio obaie by ubiui (59 io (59 i o loer ieically equal o zero:, /, / ] ε ( x (6 where ε( i he reiual ha meaure he error. Thi error i miimize if he followi reiual, calle he iz coiio, i aifie., /, / ] ϕ ( x (6 Thi roceure ie oe alebraic equaio i oe ukow. For circui how aboe, if.co( a rile elece, he he oluio for ca be aroximae by a ure DC olae wih he relaio below. ( (6 We hae oe ukow meai ha oe equaio i eee o fi ou The iz coiio will ie u hi eee equaio. By coieri reiual ol he ( ( λ ϕ λ λ (63 ( λ ϕ (64 [,,, /, / ] ϕ λ ( ( π ξ (65 Wih x efie i (57 ϕ efie i (64 λ ( ( co (66 efie i (6 ( G meho lea o ole he uique followi equaio From (65 [ co(,,, /, / ] ( λ ξ (67
9 G Tuorial xlc3.oc 9/ Fir a uique iz Coiio From (67 ( C C Exπ (68 (63, (66 i (68, a hai i mi ha rile i elece, he ( ( co Exπ (69 From (69 ( ( ( π π co Exπ (7 From (7 ( ( Ex co (7 From (7 ( ( Ex co (7 From (7 ( ( [ ] ( Ex co (73 From (73 ( ( Ex co (74 From (74 ( ( Ex Ex co (75 From (75 ( ( Ex Ex co (76 From (76
10 G Tuorial xlc3.oc / ( ( co Ex Ex (77 From Abramowiz & Seu (Habook of Mahemaical Fucio ( [ ] ( ( co z z Ex (78 i he zero orer moifie Beel fucio of he fir ki From (76 wih (78 ( ( ( co Ex Ex (79 Fially ( Ex (8
( ) ( t) ( 0) ( ) dw w. = = β. Then the solution of (1.1) is easily found to. wt = t+ t. We generalize this to the following nonlinear differential
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