Numerical Methods for a Class of Hybrid. Weakly Singular Integro-Differential Equations.

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1 Ale Mahemacs h:// ISSN Ole: ISSN Pr: Numercal Mehos for a Class of Hybr Wealy Sgular Iegro-Dffereal Equaos Shhchug Chag Dearme of Face Chug Hua Uversy Hschu Tawa Emal: Chag@chu.eu.w How o ce hs aer: Chag S. (7 Numercal Mehos for a Class of Hybr Wealy Sgular Iegro-Dffereal Equaos. Ale Mahemacs hs://o.org/.436/am Receve: May 5 7 Accee: July 7 Publshe: July 4 7 Coyrgh 7 by auhor a Scefc Research Publshg Ic. Ths wor s lcese uer he Creave Commos Arbuo Ieraoal Lcese (CC BY 4.. h://creavecommos.org/lceses/by/4./ Oe Access Absrac Ths aer rooses umercal mehos for solvg hybr wealy sgular egro-ffereal equaos of he seco. The erms hese equaos are he followg orer: ervave erm of a sae egro-ffereal erm of a sae wh a wealy sgular erel a sae egral erm of a sae wh a smooh erel a force. The orgal class of wealy sgular egro-ffereal equaos of he frs s erve from aeroelascy mahemacal moels. Amog he roose mehos he meho for solvg lear cases s fully base o revously reore aroxmao scheme for equaos of he frs. For olear cases a revse meho s roose. Examles are resee o emosrae he effecveess of he roose mehos a he resuls cae ha he roose mehos faclae achevg sasfacory a accurae aroxmaos. Keywors Hybr Wealy Sgular Iegro-Dffereal Equaos. Irouco The orgal class of egro-ffereal equaos s from a aeroelascy roblem whch he mahemacal moel comrses egh egro-ffereal equaos []. I hs moel he mos eermae equao s a scalar wealy sgular egro-ffereal equao of he frs. For he curre suy a ew equao comrsg aoal ervave erms a egro-ffereal erms wh smooh erel was use. Uer a egrable assumo revous sues hs ew equao ca be rasforme o a Volerra egral equao of he seco [] [3] [4] [5] [6]. The remaer of hs aer s orgaze as follows: Seco reses he equaos. Seco 3 reses he aroach o he umercal mehos from [7] for he lear cases a he revse verso for he olear cases. Seco 4 reses he umercal resuls obae by he me- DOI:.436/am July 4 7

2 S. Chag hos Seco 4. Seco 5 reses he summary.. Problem Descro Coser he class of hybr wealy sgular egro-ffereal equaos x( + Dx = x( + Lx + f ( ( he al coo φ x s = s b s ( where b s a osve cosa. The oeraors D a L are efe as follows: Dx = g s x s s (3 b Lx = c s x s s (4 b where x s = x+ s. (5 s =. Kerel c s smooh o s. The force The weghg erel g s egrable osve oecreasg a wealy sgular a f s assume o be locally egrable for >. Alhough a more geeral erel g s suable hs suy emhass was lace o he Abel ye erel a cosers g( s = s for < <. A secal value of =.5 corresos o he a s [ b] orgal aeroelasc roblem. The al coo φ ( s b s s L g sace whch s a weghe L sace wh wegh g (. The al value roblems ( a ( ca be exresse as ( b + = + φ + ( τ + ( τ + + ( τ τ (6 x Dx Dx x c s x s s f rove ha he fuco Dx = g s x + s s (7 b. The corresog wealy sgular Volerra egral equao of he hybr s s absoluely couous a he fuco g ( φ ( belogs o L [ b ] + s ( s s x c( s x( s s f x s x s s = φ + φ + τ + τ + + τ τ b b b for < b. The roose algorhms use he searag varables meho o recly solve Equaos ( a (. Whou loss of geeraly assumg b = he equao s exresse as x( s x( s s x( c( s x( s s f ( + + = (8 for < wh al aa x s = φ s s (9 957

3 S. Chag where f ( > s a locally egrable fuco. By he form of he sae he hybr egro-ffereal Equao (8 we oba he followg resul: = s > x + s x + s s a Equao (8 ca be rewre as x s x s s x c s x s s f s ( + ( + = + ( + +. ( 3. Numercal Meho 3.. Lear Problems The roose meho eals screzg Equao (. The sace mesh os (corresog o he s varable are screze as = τ < τ < < τ < τ = a a ew varable ξ s efe as follows: ξ s = x+ s s <. ( Equao ( ca he be reformulae as a frs-orer hyerbolc equao wh he coo ξ( s = ξ( s s s ξ s ξ ξ ξ (3 ( + s ( s s= ( + cs ( s s+ f(. (4 Nex assume ha he soluo o Equaos (3 a (4 has he form ξ = (5 ( s ( B( s = where he bass B ( s = s gve by ( sτ+ : s [ τ+ τ] τ τ + B( s = ( τ s : s [ τ τ ] ττ : oherwse where B ( s = s a ecewse lear fuco. Afer subsug he secal form of ξ exresse Equao (5 o Equaos (3 a (4 he goverg equaos for j ( j = ca be exresse as follows: ( ( ( ( = δ = (6 B s B s s + = = s ( B c( s ( B ( s s f( = + + = where δ = τ τ > for =. By efg cosas c a (7 for 958

4 S. Chag = a alyg he roery of B ( s Equao (7 ca be wre as ( + c ( = ( + ( + f (. (8 = = Noe ha Equaos (6 a (8 ca form a sysem of frs orer orary m ffereal equaos. For me he screzao coas T T T for m = T < T < < T =. Defe + = T T for = m. Assume α = for = a = m. Wh frs erm of Equao (8 relace by he frs orer fe fferece Equaos (6 a (8 ca ow be exresse as follows: ( α + α + cα + = α + + α + + f ( T + (9 = = for = m. Furhermore assume a uform mesh for boh sace a me varables he mesh os are τ = a T = m. Secf cally τ = a T = for some osve egers a m. The corres m og ffereces are efe as + = T T = m for he me varable a δ = τ τ = for he sace varable. Thus = m a δ = for = m a =. Seg m= leas o he relao = δ = for = m a = a Equaos (9 a (8 lea o he followg sysem: α + = α ( a α α + δ + ( c α ( c + α ( c f + + α = + + ( for =. Afer efg he corresog cosas e e e Equao ( ca be smlfe as follows: α α α α ( + e + + e+ + + e + e f T = for =. x a α s s as follows: Because ξ ( s = x ( + s for s a > a ξ s B s. x ca be obae for > he followg The coeco of he soluo = case: = l l xt = T B = T = α for j =. (3 j j j j l= Equaos above wh he al coo ca be se u as [ A][ x] = [ b] where j he vecor [ x ] comrses he uows α. j = The srucure of ma- 959

5 S. Chag rx [ A ] s a ha of vecor [ b ] s e e e e e e f T αeαe α e f T αe αe3 αe f αe. 3.. Nolear Problems The seco roose meho coas ar of he frs meho. By assumg ξ x+ s = s he he roery of Equao (3: ξ s ( s = ξ( s for s sll hols. The screze Equao ( follows he suy [7]: x ( + τ j x ( + τ j+ τ j x( + s s τ j+ s τ j ( τ j+ = x+ csx+ s+ f τ j+ for s eve. s = τ τ for j = s cosa for uform mesh. j+ j By alyg he roery of τ = Equao (4 ca be wre as s( ( ( ξ( + ξ( s = ξ( τ + s Nex assume ha ( τ j ( τ j+ ( j ( j+ s c( j+ ( j+ f ( ξ τ ξ τ + τ ξ τ +. ( s ( B( s = (4 (5 ξ = (6 where he bass B ( s = are he same as above c(s = Equao (5 becomes 96

6 S. Chag Seg T s( ( ( ( + ( s = ( + s ( τ j ( τ j+ ( ( + s + ( f ( + +. j j j = a assumg ( = s + s s φ = + φτ ( s s + s ( τ j ( τ j+ j j+ s ( j f he φτ φτ + φτ +. (7 (8 Smlarly seg = T he ( τ s s s s s( s( = φ + τ τ ( j ( j+ j j s ( j f φτ φτ + φτ +. For 3 a = T ( τ ( + + s s s s + + τ 4 τ + τ s = φτ s( + s ( τ j ( τ j+ φτ ( j + φτ ( j + 3 ( j + f + s φτ +. (9 (3 For = T 96

7 S. Chag ( ( τ s [ s] s s s + + = φ s( (3 + j j+ φτ ( j φτ ( j + s ( j + f + s φτ + he for ( = T ( τ + + s s s s + + T τ4 τ + τ s (3 = φτ s( + j j+ φτ ( j + φτ ( j + 3 s + sφ ( + f a for = T + + [ s] s s ( τ [ s] (33 s s( = φ( ( τ + f. s( By collecg Equaos (8 (9 (3 (3 (3 (33 a assumg α = T for A x = b s cosruce where [ A] a ha of vecor [ b ] s [ b] = he sysem [ ][ ] [ ] s s = * * * * * * f + φ + + s φτ ( j s f + φ + + s φτ ( j = s( τ τ f T φ s(. 96

8 S. Chag 3... c(s = s I hs case Equao (5 becomes Seg T Smlarly seg ( ( ( + ( s = ( s( + s ( τ j ( τ j+ ( ( + s + + ( f ( + τ + j j j j = a assumg ( = s + s s φ = + φτ ( s s( + s ( τ j ( τ j+ he j j+ s j+ ( j f φτ φτ + τ φτ +. = T he ( τ s( s( s s s s = φ + τ τ ( j ( j+ j j s j+ ( j f φτ φτ + τ φτ +. For 3 a For = T ( τ + + s s s τ s + + τ 4 τ + τ s = φτ s( + φτ φτ s ( j+ j + j j+ j + j+ 3 f + s τ φτ +. = T (34 (35 (36 (37 963

9 S. Chag ( ( τ s s s + + T + T sτ + + T sτ j+ j + [ ] = φ s( + j j+ φτ ( j φτ ( j + s + s τ φτ + f T (38 he for = T ( τ + + s s s τ s + + τ 4 τ + τ s = φτ ( τ + sτ φ( + f s( (39 a for = T + + T sτ s s ( τ + 4 [ ] [ sτ ] (4 s s = φ( ( τ + f. s( By collecg Equaos (35 (36 (37 (38 (39 (4 a assumg α = T for A x = b s cosruce where [ A] a ha of vecor [ b ] s [ b] = he sysem [ ][ ] [ ] s s = * * * * * * f + φ + + s τ j+ φτ ( j s f + φ + + s τ φτ ( j = s( τ τ f T φ s( j+. 964

10 S. Chag 4. Numercal Examles A eso comuer (Iel Peum 4 mcrorocessor.8 GHz CPU a 4 MB RAM was use for esg he examles. Examle. φ c s = s = s f = 3.5 x =. meho Max error rec e 3 rec.7453e revse 6.6e 5 revse 6.56e 4.5 Examle. φ c s = s = f = + x =. meho Max error rec 4.99e 3 rec.6994e revse.4 revse e 5 Examle 3. φ c s = s = s f = + x = meho Max error rec rec 4.38 revse.3 revse.6 Examle 4. φ c s = s s = s f = 8 3 x =. meho Max error rec 6.47e rec 7.968e revse.755e 5 revse.758e Examle 5. φ c s = s s = f = x =. meho Max error rec 6.783e rec 7.738e revse.5 revse 5.354e 5 965

11 S. Chag Examle 6. φ c s = s s = s f = + x = meho Max error rec.7 rec.5983 revse.7 revse Summary I hs suy we rese umercal mehos for solvg a class of hybr egro-ffereal equaos; he equao of he frs orgaes from a aeroelascy moel. The rec meho from revous suy roves sasfacory resuls esseally for he lear cases. For he olear cases a revse meho s roose o oba more accurae resuls. Refereces [] Burs J.A. Clff E.M. a Herma T.L. (983 A Sae-Sace Moel for a Aeroelasc Sysem. IEEE Coferece o Decso a Corol hs://o.org/.9/cdc [] Burs J.A. Herma T.L. a Sech H.W. (983 Lear Fucoal Dffereal Equaos as Semgrous o Prouc Saces. SIAM Joural o Mahemacal Aalyss hs://o.org/.37/547 [3] Che H.-H. a Chag S. (9 A Secfc Proceure for Aalyc Soluos o a Class of Sgular Iegral Equaos. Chug Hua Joural of Comuaoal Scece [4] Chag S. (5 Noes o he Soluo of a Class of Sgular Iegral Equaos. Chug Hua Joural of Scece a Egeerg [5] Chag S. (6 O he Numercal Soluo of a Class of Sgular Iegro-Dffereal Equaos. Chug Hua Joural of Scece a Egeerg [6] Kael F. a Zhag K.P. (986 Equvalece of Fucoal Equaos of Neural Tye a Absrac Cauchy Problems. Moashefe für Mahema hs://o.org/.7/bf9895 [7] Chag S. (6 Numercal Algorhm a Is Revso for Solvg a Class of Nosgular Iegro-Dffereal Equaos of he Frs K. Chug Hua Joural of Comuaoal Scece

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