Solution of Stochastic Ordinary Differential Equations Using Explicit Stochastic Rational Runge-Kutta Schemes

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1 Ameca Joual of Computatoal ad Appled Matematc 5, 5(4): 5- DOI:.593/j.ajcam.554. Soluto of Stocatc Oda Dffeetal Equato Ug Explct Stocatc Ratoal Ruge-Kutta Sceme M. R. Odekule, M. O. Egwuube, K. A. Joua,* Depatmet of Matematc, Modbbo Adama Uvet of Tecolog, Yola, gea Depatmet of Matematc, Adamawa tate Uvet, Mub, gea Abtact Ft ode oe-tage explct Stocatc Ratoal Ruge-Kutta metod wee deved fo te oluto of tocatc oda dffeetal equato. Te devato baed o te ue of Talo ee expao fo bot te detemtc ad tocatc pat of te tocatc dffeetal equato. Te tablt ad covegece of te metod, foud to be abolute table. Tee metod wee fute teted o ome umecal poblem. Fom te eult obtaed, t obvou tat te deved metod; pefomed bette ta te oe wt wc we ave aaled ad te wee compaed wt ou eult. Kewod Stocatc dffeetal equato, Ruge-Kutta metod, Explct atoal Ruge-Kutta metod. Itoducto Developmet ecet ea aea of eeace globall, Matematc ad cece elated ubject uc a, Egeeg, Pc, Bolog, Ecolog, Hdolog, Ecoomc, Ivetmet, Face, ad populato damc, jut to meto but a few, ave ealed te mpotace of applcato of tocatc dffeetal equato ad te mpotace of adom effect mot eal poblem. Tee ae dffcult to adle b te detemtc model [, ]. I vew of t, tee ave bee a ceae te eed to cotuct tocatc model to mulate tem tat deal wt eal lfe tuato tat cota ucetate. Ma pcal tem ae modelled b tocatc dffeetal equato wee adom effect (oe) ae beg modelled b a Bowa moto o wat called Wee poce [3]. Suc dffeetal equato ae ael olved aaltcall due to dffuo tem volved. So umecal metod equed ad ould be cotucted le wt te pcple of tocatc pocee, to adle tocatc pocee effect [4]. Tee model ca offe moe ealtc epeetato of te pcal tem. Iteetg eoug, Ruge-Kutta metod pove effectve adlg tocatc dffeetal equato teoe tat ft o adle tocatc pocee, ove ome of te aaltc metod [5, 6]. Teefoe, tee a g eed to develop ad mplemet ome tocatc Ruge-Kutta metod fo olvg tocatc dffeetal equato [7]. I t pape, a explct Stocatc * Coepodg auto: jakwaamu@aoo.com (K. A. Joua) Publed ole at ttp://joual.apub.og/ajcam Copgt 5 Scetfc & Academc Publg. All Rgt Reeved Ratoal Ruge-Kutta metod deved baed o te modfed appoac of tocatc Ruge-Kutta metod fo olvg tocatc oda dffeetal equato. Code oda dffeetal equato of te fom, = f( x, ), a ( ) = η, f=r R R () Te geeal Ratoal Ruge-Kutta ceme fo te oluto of () accodg to [8, 9,,, ] expeed a: + ck + + = () + vh wee K f( x+ c, + aj, Kj), () H = g( x + d, + b H ), () j, j j= = gx (, ) f( x, ) ad = Code te o-autoomou, oe Wee SDE of Statoovc tpe: d() t = f (, t ()) t dt + g(, t ()) t dwt (3) m

2 6 M. R. Odekule et al.: Soluto of Stocatc Oda Dffeetal Equato Ug Explct Stocatc Ratoal Ruge-Kutta Sceme T Y ( T ) = Y () + f (, t ()) t dt + g(, t ()) t dw () t T (4) We all aume a oluto of te fom = + c K + J K + (5) T ow motvate u to fomulate a -tage explct tocatc Ratoal Ruge-Kutta (SRRK) metod fo te oluto of (3) a + ck = + J S K + l + vh (6) t+ t =, a potve tege, Wee J = W = W W l + K = f t + a, + a K ad a = a j j j j= j= H = p( t, b, + b H ) ad b = b j j j j= j= = p ( t, ) f( t, ) ad = K = g t + a, + J b K ad a = b wee l j j j j= j= ( c + v ) = c,, v, a, a, a, b, ad b, j j j fo all, j =,,..., ae cotat to be detemed. We ca claf SRRK metod, a follow: If b = a = b =, < j, te te metod j j j called em-mplct. If b = a = b =, j, te te metod j j j called explct. Otewe t called mplct. Defto : (Stog covegece) We a tat a dcete tme appoxmato Y c covege togl wt ode p > o at tme T f tee ext a potve cotat C, wc doe ot deped o te maxmum tep e, ad δ >, uc tat c E( Y Y ) C T T t old fo eac = (, δ). te umbe of ubteval of te teval, I = [ t, T ], YT te exact T Y te appoxmate oluto at T. We oluto at, c all emplo te tog covegece pcple aalg ou eult.. Devato of te Metod I ode to deve te oe-tage explct SRRK metod, let = (6) to obta wee K = f( t, ) H = pt (, ) K = g( t, ) + c K + = + J K + vh ( c+ v) = p (7) t+ t = a potve tege, J = W = W W + Wt = pt (, ) f( t, ) =, v Expad te R.H.S of (7) bomall, mplf ad eglect tem of ode two ad ge to obta + = + c K v H + J K (8) Expadg K ug Talo ee about ( t, ) we ave: ad K = f( t, ) +... (9) H = pt (, ) +... () K = g( t, ) +... ()

3 Ameca Joual of Computatoal ad Appled Matematc 5, 5(4): 5-7 Subttutg(9), () ad () te R.H.S of (8) we get But + = + c f ( t, ) v p( t, ) + Jgt (, ) = pt (, ) f( t, ) Te () become =, v () + = + ( c + v) f J g +... (3) Fom Talo ee expao of + about 3 J 3 x we get + = J! 3! J ! 3! If we ue te otato = f = Df (4) (5) ce = f( t, ), ad let = g(, t ) be te oluto of te tocatc pat deoted b ad we: = f + ff, = g + gg Teefoe, (4) become t t + = + f + ( ft ff )...! + + Fo cotec, J + Jg + ( gt + gg ) +...! c v (6) + = (7) = Wc te multaeou equato to be olved tee ukow v, c,, wee c a fee paamete. To deteme a patcula ceme of oe-tage explct Stocatc Ratoal Ruge-Kutta Metod (SRRK): Cae, let T gve c =, te, v =, ad = wee K = f( t, ) H = pt (, ) K = g( t, ) ( c+ v) = + K = + J K + H + Pt (, ) = Z ( t, ), Z = ad t+ t =, J = W = W W Wee J + (8) Gaua adom omal dtbuto wt mea eo ad tadad devato oe ( (,) ) Cae, f 3 c =, v =, =, we obta K = 4 + J K 3 + H 4 + (9) wee K, H, K, P( t, ),, ad J ae a defed (8) Cae 3, f c =, v =, =, te, + = JK H + () + wee K, H, K, P( t, ),, ad J ae a defed (8) Cae 4, f c =, v =, = K 3 + = + JK () + H 3 wee K, H, K, P( t, ),, ad J ae a

4 8 M. R. Odekule et al.: Soluto of Stocatc Oda Dffeetal Equato Ug Explct Stocatc Ratoal Ruge-Kutta Sceme defed (8) Tu, te ceme (8), (9), () ad () ae te oe-tage SRRK metod. 3. Stablt Aal Teoem : (Covegece, [3]) () Let te fucto ϕ ( x,, ) be cotuoul jotl a a fucto of t tee agumet, te X ab,, (, ), ego F defed b [ ] () Let ϕ (,, ) atf a Lpct codto of te fom all pot * * ϕ( x,, ) ϕ( x,, ) M fo * ( x,, ),( x,, ) F te te metod + = ϕ ( x,, ) coveget f ad ol f t cotet. Teoem [8]: Fo te tet equato d = λdt + σdw ( t), () =, λσ, C, Rλ < te Eule-Mauama metod umecal table mea f + λ <. [ ], > ote tat te tablt codto + λ < te ame a Eule metod fo ODE accodg to Lambet (99). I geeal te umecal tablt codto fo te addtve cae ae cocdet wt tem fo ODE (Hemade ad Spgle, 99). Teoem 3 [8]: If te Eule-Mauama metod atfe te umecal table codto mea,.e. + λ <, te Eule-Mauama metod amptotcall cotet mea quae. Fo te tablt aal of te deved ceme, we all ue te Su metod ad utle te pcple of [4, 5, 6, 7, 8, 9], tat tate, te tablt aal of tocatc metod coepod wt t detemtc coutepat. Ug te tet equato = λ () Fom (8) + ( K)( H) Expadg ad tucatg tem of ecod ad ge ode, we obta + K H (3) But = λ, ad K = f( t, ) = f ce = f = λ H = p( t, ) = ubttutg tee (3), we get f( t, ), = + λ ( + λ) te caactetc polomal ς = + λ, λ < (4) Teefoe, (-., ) te teval of te abolute tablt of te oe-tage ceme (8). Smlal, we ca ee tat (8), (9), () ad () all ave te ame teval of abolute tablt. Wc called umecal table mea tocatc ee. 4. umecal Poblem ad Reult Poblem Code te SDE [6] Wt te exact oluto gve b: Poblem Code te SDE [] d( t) = ( +. )( ) dt +.( ) dw ( t) () = ( t+. W( t)) ( + ()) e + () t () = ( t+. W( t)) ( + ()) e () + [ ] d = a ( ) dt + a( ) dw ( t), () =, t, wt exact oluto ( t) = ta( aw ( t)) + acta ( ); a =, ε =.

5 Ameca Joual of Computatoal ad Appled Matematc 5, 5(4): 5-9 Teefoe, te umecal oluto ug te explct SRRK metod fo te oe-tage ceme a obtaed t wok wt abolute eo ae gve te Table ad a ee below. Te followg otato wll be ued to epeet eult te table. PL, RAe-3: Reult fom [6], Soel, Reult fom [] ad JAk, Reult obtaed b ou ew metod. Table. umecal eult of oe-tage JAk explct SRRK compao wt [6] fo Poblem t W Exact Soluto PL Abolute Eo RAe (Pa) Abolute eo RAe Abolute Eo RAE3 Abolute Eo JAk Abolute Eo Table. umecal eult of oe-tage JAk explct SRRK compao wt [] fo Poblem Soel JAK abolute eo abolute eo e e e Dcuo of Reult Wt te oe- tage explct tocatc atoal Ruge-Kutta ceme (SRRK) deoted JAk te umecal eult table. We deved faml of fou ceme of oe-tage SRRK metod Te faml ceme fo te oetage wee teted to olve umecal Poblem ad fom [6] ad [] epectvel fo ome of te deved ceme a peeted table ad. Matlab oftwae (veo ) wa emploed to u te mulato, baed o omal dtbuted adom umbe wt mea eo ad vaace (tadad devato) oe,.e (,). Fom Table ad we ca ee te pefomace of ou oe tage ceme wt te extg ceme [6] ad []. Alo tablt aal of te oe-tage developed wee caed out ug Scu metod, le wt wat we call mea ad mea quae tablt pcple tocatc tablt aal dcued b ome auto ecto 3. ad te tablt aal of te deved ceme owed te ae bouded b te teval (-., ), jut a tat of detemtc oe-tage explct Ruge-Kutta metod. 6. Cocluo Cleal ou oe- tage ceme pefom bette tem of covegece ad accuac, alo te deved oe -tage ceme wa able to ecove te eult of te two tage ceme bette. T mea tat ou explct SRRK metod a alteatve ceme to be ued olve uc poblem. Alo oe left wt a opto of coog a ceme to wok wt ou cae, ulke Eule Ruge-Kutta metod tat a jut a gle ceme. REFERCES [] Kloede, P.E. & Plate, E. (999). umecal oluto of tocatc dffeetal equato (3d.ed). Applcato of Matematc, ewyok: Spge-Velag, Bel. [] Buage, K., & Buage, P. M. Ad Ta, T. (3). umecal metod fo tog oluto of tocatc dffeetal equato: a ovevew. Reteved fom tt:/cteex.t.pu.edu/vewdoc/umma. [3] Sobek, K. (99) Stocatc dffeetal equato wt applcato to pc ad egeeg, Kluwe Academc Puble, Dodect. SIAM Joual of umecal Aal 9(3), [4] Soel, R. A. (8). Stocatc Ruge-Kutta Metod wt weak ad tog covegec. Iteatoal Joual of Cotempoa Matematcal Scece, 3(9), [5] Hgam, J. D. (). A Algotmc Itoducto of umecal mulato of tocatc dffeetal equato. SIAM Revew, 43(3) [6] Fadel, S. F. & Abdulamea, A. A. (). Explct Ruge-Kutta Metod fo olvg tocatc dffeetal equato. Joual of Reeace Baa ((cece)),37(4) [7] Seve, E. S. (4). Stocatc calculu fo Face II. Cotuou Tme Model Spge Face text book,usa. [8] Hog, Y. F. (98). A cla of A-table o A(α) table explct

6 M. R. Odekule et al.: Soluto of Stocatc Oda Dffeetal Equato Ug Explct Stocatc Ratoal Ruge-Kutta Sceme ceme computatoal ad amptotc metod fo Bouda ad teo lae poc. Of BAIL cofeece. Tt college, Dubl [9] Okubo D.I. (987). Explct atoal Ruge-Kutta ceme fo tff tem of oda dffeetal equato. M.Sc. Te. Uvet of Be, Be ct, gea. [] Odekule, M. R. (). Some em-mplct atoal Ruge-Kutta ceme. Bagale Joual of pue ad Appled cece, () -4. [] Ademul, A. R., Babatola, O.P. & Odekule, M. R. (). Ratoaled mplct Ruge-Kutta ceme fo te tegato of tff oda dffeetal equato. Bagale Joual of pue ad appled cece, ()6-3. [] Abdulme, C.E. & Uloko, A. J. (). A cla of a mplct tage-two Ratoal Ruge-Kutta Metod fo oluto of Oda Dffeetal Equato. Joual of Appled Matematc & Bofomatc, (3)7-3. [3] Lambet, J. D. (973). Computatoal metod oda dffeetal equato. ew Yok: Jo Wle & So Ccete. [4] Mlte G.. & Tetakov M. (4). Stocatc umec fo Matematcal Pc, Spge Velag, Bel. [5] Heade, D. B. & Spgle, R. (99). A-tablt of Ruge-Kutta metod fo tem wt addtve oe, BIT 3, [6] Yua, C. & Mao, X. 4. Stablt dtbuto of umecal oluto fo tocatc Dffeetal equato, Stoc. aal. appl.,33-5. [7] Sato, Y. & Mtu, T. (993). T-tablt of umecal ceme fo tocatc dffeetal equato, WSSIAA, [8] Sato, Y. & Mtu, T. (996). Stablt aal of umecal ceme fo tocatc dffeetal equato, SIAM J. ume. Aal. 33, [9] Plate, E. (999). A toducto to umecal Metod fo tocatc dffeetal equato, Acta, umeca 8, [] Soel, R. A. (8). Stocatc Ruge-Kutta Metod Wt Weak ad Stog Covegec. Iteatoal Joual of Cotempoa Matematcal Scece, 3(9), 4 48.