Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: a comprehensive stability analysis

1 Vol.:(0123456789) Scien i ic Repo s | (2024) 14:6930 | h ps://doi.o g/10.1038/s41598-024-56944-z www.na u e.com/scien i ic epo s Nume ical simula ion o a ac ional s ochas ic delay di e en ial equa ions using spec al scheme: a comp ehensi e s abili y analysis Shuo Li 1, Sami Ullah Khan 2*, Muhammad Bilal Riaz 3,4, Salman A. AlQah ani 5 & A i M. Alam i 6 The ac ional s ochas ic delay di e en ial equa ion (FSDDE) is a powe ul ma hema ical ool o modeling complex sys ems ha exhibi bo h ac ional o de dynamics and s ochas ici y wi h ime delays. The pu pose o his s udy is o explo e he s abili y analysis o a sys em o FSDDEs. Ou s udy emphasizes he in e ac ion be ween ac ional calculus, s ochas ici y, and ime delays in unde s anding he s abili y o such sys ems. Analyzing he momen s o he sys em’s solu ions, we in es iga e s ochas ici y’s in luence on FSDDS. The a icle p o ides p ac ical insigh in o sol ing FSDDS e icien ly using a ious nume ical echniques. Addi ionally, his esea ch ocuses bo h on asymp o ic as well as Lyapuno s abili y o FSDDS. The local s abili y condi ions a e clea ly p esen ed and also he e ec s o a ac ional o de s wi h delay on he s abili y p ope ies a e examine. Th ough a comp ehensi e es o a s abili y c i e ia, p ac ical examples and nume ical simula ions we demons a e he complexi y and challenges conce n wi h he analyzing FSDDEs. Keywo ds F ac ional s ochas ic delay di e en ial equa ions, S abili y analysis, Spec al me hod, Legend e– Gauss–Loba o nodes In ecen esea ch, he ield o analysis o dynamical sys ems and ma hema ical modeling has wi nessed exp es- si e ad ancemen s, speci ically in he sys ems exhibi ing s udy o complica e he s ochas ic componen s and empo al beha io s. The sys ems has one such class o ha has accumula e he subs an ial a en ion is he ac- ional s ochas ic delay di e en ial sys ems (FSDDS). Such sys ems compass has b oad spec um o a eal wo ld phenomena, anging om enginee ing applica ions o biological p ocesses, whe e ime delays and andomness play c i ical oles in adap hei dynamics. The ac ional calculus, wi h i s oo s da ing back o he wo k o Eule and Leibniz, has ound a golden age in coinciden al enginee ing and science due o i s abili y o cap u e non-local in e ac ions and memo y e ec s which a e mo e accu a ely han classical calculus1–4. F ac ional de i a i es a e a exclusi e ma hema ical concep used o illus a e he sys ems wi h non-in ege dynamics o de . A ex ensi e in oduc ion o ac ional calculus can be pionee in Igo Podlubny’s7 and along wi h i s many applica ions. F ac ional calculus can be used o analyze complex sys ems and da a by he G igolini and Wes 5. The p ac ical applica ions and de ini ions co esponding ac ional calculus, a e clea ly explained by Magin6. S uden s and esea che s will also ind Igo Podlubny’s 1999 book7 which is use ul o unde s anding he ac ional di e en ial equa ions and hei solu ions. The modeling o FSDDS appea ed as a powe ul s uc u e and analyzing he complex sys ems wi h bo h andomness and memo y e ec s, as a esul o he con e gence o s ochas ic p ocesses o a ac ional calculus8–10. The e is a well-es ablished exe cise o assimila ing delays in o dynamic sys ems o na a i e o phenomena such as biological eac ion imes, ini e p opaga ion speeds, and communica ion delays in con ol sys ems11,12. A ich a ay o ma hema ical challenges and oppo uni ies a ises om he in e ac ion be ween ac ional calculus, OPEN 1School o Ma hema ics and Da a Sciences, Changji Uni e si y, Changji 831100, Xinjiang, People’s Republic o China. 2Depa men o Ma hema ics, Ci y Uni e si y o Science and In o ma ion Technology, Peshawa , KP 2500, Pakis an. 3IT4Inno a ions, VSB- Technical Uni e si y o Os a a, Os a a, Czech Republic. 4Depa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Byblos, Lebanon. 5Compu e Enginee ing Depa men , College o Compu e and In o ma ion Sciences, King Saud Uni e si y, Riyadh, Saudi A abia. 6So wa e Enginee ing Depa men , College o Compu e and In o ma ion Sciences, King Saud Uni e si y, Riyadh, Saudi A abia. *email: [email p o ec ed] 2 Vol:.(1234567890) Scien i ic Repo s | (2024) 14:6930 | h ps://doi.o g/10.1038/s41598-024-56944-z www.na u e.com/scien i ic epo s/ s ochas ici y, and ime delays13,14. As well as o heo e ical pu poses, unde s anding he beha io o FSDDS has p ac ical applica ions in ields such as biology, ecology, economics, inance, and enginee ing15. The p oposed ma hema ical backg ound o FSDDEs combines he s ochas ic p ocesses, ac ional calculus and ime delays. The ac ional calculus handles non-in ege -o de de i a i es, cap u ing he anomalous di usion and long memo y e ec s. Whe e he s ochas ic p ocesses in oduce he andomness in o dynamics, while ime delays in oduces he lags in a sys em’s esponse. Howe e , FSDEs in eg a e he abo e concep s: s ochas ic luc ua ions, memo y e ec s and ime delays o model he complex sys ems o inding he applica ions ac oss di e se ields. An unde s anding o dynamic sys ems elies hea ily on s abili y analysis16,17. S abili y analysis plays a c ucial ole in he con ex o ac ional s ochas ic delay di e en ial equa ions (FSDDEs), a po en ma hema ical ame- wo k o modeling complex sys ems displaying bo h ac ional o de dynamics and s ochas ici y18. F ac ional calculus, s ochas ici y, and ime delays a e h ee undamen al concep s ha come oge he uniquely in hese equa ions19–21. By ex ending he classical concep s o di e en ia ion and in eg a ion o non-in ege o de s, ac- ional calculus enables us o see memo y e ec s and abno mal di usion in sys ems19,22. In o de o po ay he in insic unce ain y p esen in eal-wo ld p ocesses, s ochas ici y in uses unp edic abili y in o he dynamics23,24. Fu he mo e, whe e eedback o communica ion be ween sys em componen s is no ins an aneous, ime delays a e equen ly seen in a a ie y o ields, including as biology, economics, and enginee ing25. Hence, he in es- iga ion o s abili y in FSDDEs becomes a mul i ace ed challenge, necessi a ing a comp ehensi e explo a ion o he in e play be ween hese elemen s26. In his a icle, we emba k on a de ailed jou ney in o he s abili y analysis o FSDDEs, shedding ligh on he in ica e ela ionships be ween ac ional calculus, s ochas ici y, and ime delays and elucida ing he me hods and echniques essen ial o comp ehending he s abili y p ope ies o hese complex sys ems. In his a icle, using he spec al me hods, which ha e gained p ominence as a e sa ile nume ical ech- nique o sol ing di e en ial equa ions in a wide ange o applica ions27–29. These me hods le e age he spec al decomposi ion o ope a o s o app oxima e solu ions in e ms o basis unc ions, o en leading o accu a e and e icien compu a ional s a egies. Spec al me hods ha e been widely applied o s anda d di e en ial equa ions and pa ial di e en ial equa ions (PDEs), bu hei adap a ion o FSDDEs ep esen s a compelling a enue o esea ch and applica ion30–32. F ac ional s ochas ic di e en ial equa ions (FSDDEs) a e a e sa ile class o ma hema ical models ha ex end adi ional s ochas ic di e en ial equa ions (SDEs) o encompass ac ional de i a i es, enabling he desc ip ion o complex sys ems wi h non-Ma ko ian, long-memo y e ec s. The e a e se e al uses o hese equa ions ac oss nume ous disciplines. FSDDEs help o be e p edic ma ke ola ili y and p ice changes in inance by cap u ing long- ange dependencies and imp o ing knowledge o asse p ice dynamics. They a e also a use ul mechanism in physics o ende ing excep ional di usion p ocesses and helping in diso de ed media o desc ibe he pa icle mo ion. Addi ionally, FSDDEs a e e y use ul in he ield o biological sciences because hey make i simple o unde s and he complex memo y ela ed phenomena like he ansmission o disease and he dynamics o bio- logical popula ions. Also, he depend o S ochas ic dynamics o Michaelis-Men en kine ics on umo -immune in e cou se by33 and complexi y in he umo -immune model wi h a ime-delayed by34. Mo eo e , FSDDEs play a c i ical ole in he con ol sys ems analysis, modeling sys ems wi h inhe i ed e ec s o delayed and signal p ocessing in he applica ions o enginee ing. O e all, FSDDEs a e a i al ool o an icipa ing and unde s and- ing he complex sys ems complexi ies h ough a a ie y o special y due o hei e sa ili y in cap u ing he non-Ma ko ian beha io . F ac ional calculus empowe he modeling o anomalous di usion and long memo y e ec s along wi h non-in ege -o de de i a i es. Howe e , he s ochas ic p ocesses in oduce e lec ing inhe en unce ain ies, andomness, while ime delays ob ain he empo al lags be ween he esponses and he sys em s a es. These wo elemen s oge he in FSDDEs o e a comp ehensi e amewo k o modeling complex sys ems wi h memo y, andomness, and empo al dependencies ac oss a ious disciplines. We p esen an o e iew o he in e ac ion be ween spec al app oaches and FSDDEs, emphasizing hei impo ance and po en ial. I has been demons a ed ha spec al app oaches can be applied independen ly35,36 in o de o unde s and FSDDEs. In addi ion, we will discuss how spec al app oaches can be applied o scien i ic esea ch. The ad an ages and disad an ages o FSDDEs a e desc ibed, as well as how hey migh e ol e in he u u e. The pu pose o his a icle is o desc ibe he p ocess o modeling and unde s anding memo y- ich, in ica e sys ems. The combina ion o spec al app oaches and FSDDEs in his esea ch a ea is pa icula ly in iguing because, in addi ion o p o iding us wi h a highe le el o compu a ional e iciency when modeling and e alua - ing eal-wo ld phenomena, i also p o ides us wi h a be e le el o p ecision37. Real-wo ld applica ions o ac ional s ochas ic delay di e en ial equa ions (FSDDEs) o ep esen ing memo y- ela ed, nonlinea , and ochas ic sys ems ha e been inc edibly success ul. F ac ional de i a i es can be in eg a ed in o FSDDE o ep esen non-in ege -o de dynamics, including long- ange in e ela ionships and anomalous beha io . Fo he analysis o sys ems subjec o andom luc ua ions, s ochas ic ea u es can be included in FSDDEs in physics, inance, biology, and enginee ing. Using s ochas ici y and ac ional p obabili y wi h calculus, we a e able o s udy sys ems ha we e p e iously elusi e, imp o ing ou unde s anding o complex eal-wo ld e en s. Th oughou he a icle, he ollowing sec ions a e p esen ed: “Ma hema ical backg ound” sec ion p esen s a b and-new ma hema ical model o FSDDEs, while “S abili y analysis o FSDDEs” sec ion p o ides a b ie analysis o he s abili y analysis o he spec al me hod. A b ie discussion o he nume ical esul s o he s udy is p esen ed in “P ac ical examples” sec ion, obs acles and u u e di ec ions a e discussed in “Challenges and u u e di ec ions” sec ion, and a conclusion is p esen ed in “Conclusion” sec ion. 3 Vol.:(0123456789) Scien i ic Repo s | (2024) 14:6930 | h ps://doi.o g/10.1038/s41598-024-56944-z www.na u e.com/scien i ic epo s/ Ma hema ical backg ound F ac ional calculus In ac ional calculus di e en ia ion and in eg a ion a e ex ended o non-in ege o de s. F ac ional de i a i es a e commonly de ined by Riemann-Liou ille and Capu o me hods. I has been demons a ed ha ac ional calculus can be use ul o desc ibing complex beha io s since i is capable o ep esen ing sys ems ha ha e memo y and anomalous di usion. A non-in ege o ac ional o de o di e en ia ion and in eg a ion is in oduced as pa o ac ional calculus. As a esul o he inco po a ion o ac ional-o de de i a i es, ac ional calculus is essen ial o unde s anding he beha io o he sys em unde FSDDE condi ions. The e a e wo widely used de ini ions o ac ional de i a- i es: Riemann-Liou ille and Capu o. Bo h o hese will be in es iga ed using ma hema ics. Riemann–Liou ille ac ional de i a i e Below is a desc ip ion o how he Riemann-Liou ille ac ional de i a i e o a unc ion ( ) o o de α>0 is de ined.: Ŵ(.) ep esen s he gamma unc ion. This o mula ion cap u es he his o ical de elopmen o ac ional calculus as well as he memo y e ec s in FSDDEs. Wha is he ac ional de i a i e o o de ? A decaying weigh ing unc ion is applied o an a e age o all p e ious alues o ( ). Capu o ac ional de i a i e Fo a unc ion ( ) o o de α>0 , he Capu o ac ional de i a i e is de ined as ollows: In his case, n ep esen s he smalles in ege la ge han α . Capu o de i a i es a e commonly used in p ac ical applica ions due o hei gua an ee o a well-de ined ini ial condi ion o he FSDDE and hei a oidance o non-local e ms such as Riemann-Liou ille de i a i es. F ac ional o de di e en ial equa ions The ac ional dynamics o he sys em a e ep esen ed by ac ional o de de i a i es in FSDDEs. The ollowing is a ma hema ical ep esen a ion o a gene al FSDDE: whe e • dα d αx( ) ep esen s a ac ion o de de i a i e o he s a e a iable x( ). • F, Wi h he sys em is cap u ed nonlinea ly dynamics. • The delayed s a e is ep esen ed by x( −τ) a ( −τ) . • The s ochas ic p ocess o noise is ep esen ed by ξ( ) a ec ing he sys em. To asce ain he s abili y o equilib ium poin s, asymp o ic beha io , and long- e m beha io o he FSDDE sys em, ea u es o solu ions o hese equa ions a e equen ly s udied in s abili y analysis. A applicable ool o comp ehensi e dynamics o FSDDEs is a ac ional calculus, which empowe he mod- eling o complex sys ems wi h anomalous di usion, memo y e ec and non-local in e ac ions. Howe e , he pa icula scene o he desi ed ma hema ical cha ac e is ics o model will de e mined whe he o adop a Capu o ac ional de i a i e o Riemann–Liou ille. S ochas ic p ocesses The sys em dynamics a e b ough in o he disa angemen by s ochas ici y. He e he di e en s ochas ic p ocesses a e heS a ono ich in eg als, B ownian mo ion and I a e equen ly used in FSDDEs. A s ic u e o including andomness o unce ain y in he ma hema ical modeling o ac ual sys ems is d i en by he s ochas ic calculus. Time delays Sys ems wi h communica ion be ween componen s and non-ins an aneous eedback ime delays. S abili y p ob- lems and complex dynamics may esul om hem. Howe e , ac ional o de de i a i es and delayed e ms bo h mus be aken in o accoun while s udying FSDDEs wi h he empo al delays. S abili y analysis o FSDDEs Ma hema ical componen s o s abili y analysis o he FSDDEs a e co e ed in his sec ion. In o de o in e - oga e he s abili y o he p oposed equa ions, we mus demons a e he undamen al me hods and ideas ha will be applied. (1) d α d α ( )= 1 Ŵ( 1 −α) d d  0 ( −τ)−α (τ)dτ , (2) d α d α ( )= 1 Ŵ(n − α)  0 ( −τ)n−α−1d n dτ n (τ)dτ . (3) dα d αx( )=F[x( ),x( −τ),ξ( )] , 4 Vol:.(1234567890) Scien i ic Repo s | (2024) 14:6930 | h ps://doi.o g/10.1038/s41598-024-56944-z www.na u e.com/scien i ic epo s/ Local s abili y analysis Analyzing he local s abili y con ain FSDDE beha io close o he equilib ium poin s. To achie e he s abili y o ixed poin s, ce ainly linea iza ion me hods like cha ac e is ic equa ions and Laplace ans o ma ion a e used. In p esen s udy we examine how ac ional o de de i a i es and s ochas ici y a ec he local s abili y. The ini ial s ep in app ehend he beha io o FSDDEs is local s abili y analysis a ound he equilib ium poin s. Conside FSDDE gi en in Eq.3 o we linea ize a ound an equilib ium poin by conside ing small pe u ba ions δdx( ) a ound x∗ : whe e • ∂F ∂x | x=x∗ ep esen s Jacobian ma ix o he F wi h espec o x e alua ed a x∗ . • ∂F ∂x(−τ) |x=x ∗ deno es Jacobian ma ix o F wi h espec o he x(−τ) e alua ed a x∗ . • ∂F ∂ξ |x=x ∗ ep esen s he sensi i i y o F o he s ochas ic p ocess ξ( ) e alua ed a x∗ . To de e mine he local s abili y o he equilib ium poin x∗ , we analyze he eigen alues o he esul ing linea ized equa ion. I all eigen alues ha e nega i e eal pa s, he equilib ium is locally s able. O he wise, i any eigen alue has a non-nega i e eal pa , he equilib ium is uns able. Theo em: local s abili y analysis o FSDDEs using he spec al me hod S a emen Conside he FSDDE o he o m: whe e x( ) is he s a e ec o , A, B, and C a e ma ices, α is a ac ional o de in he ange (0,1), τ is he delay pa ame e , and dW( ) d ep esen s a s ochas ic p ocess. Assume ha he FSDDE is linea . P oo 1. Linea iza ion Begin by linea izing he FSDDE a ound he equilib ium poin x∗=0 o ob ain whe e δx( ) ep esen s small pe u ba ions om he equilib ium. 2. Laplace ans o m Apply he Laplace ans o m o he linea ized equa ion o ge whe e s is he Laplace a iable, and �X(s) , W(s) a e he Laplace ans o ms o δx( ) and dW( ) d espec i ely. 3. Eigen alue analysis Use he Laplace- ans o med equa ion o de i e an algeb aic equa ion in s by isola ing �X(s) on one side. 4. Cha ac e is ic equa ion The algeb aic equa ion can be w i en in he o m o a cha ac e is ic equa ion: Whe e I is he iden i y ma ix and he solu ions o his equa ion a e he eigen alues o he Laplace- ans- o med ma ix sαI−A+Be−sτ . 5. Spec al me hod We need o ind he eigen alues o he ma ix sαI−A+Be−sτ as s a ies. The s abili y analysis o his sys em elies on analyzing he eal pa s o hese eigen alues. 6. S abili y analysis Fo s abili y c i e ia, all he eigen alues o s in he en i ons o ze o mus ha e nega i e eal po ions o he e o be local asymp o ic s abili y. To sa is ied his condi ion is o ally depending on he accu a e alues o A, B, and C, as well as he delay pa ame e τ and also he ac ional o de α . Howe e , he calcula ion o eigen alues o a ma ix wi h ac ional de i a i es and analyzing he cha ac e is ic equa ion is di icul and challenging ask. I may be necessa y o c ea e local s abili y equi emen s and in es iga e eigen alues o he FSDDE using specialized app oaches and nume ical me hods. To come a a conclusion on he local s abili y o he equilib ium poin , he calcula ions and ac ual e idence may equi e sma nume i- cal and ma hema ical echniques because hey especially depend on he p ecise alues o he ma ices and pa ame e s used in he p oposed FSDDE. (4) dα d α  δx( )  = ∂F ∂x |x=x∗δx( )+ ∂F ∂x(−τ) |x=x∗δx( −τ)+ ∂F ∂ξ |x=x∗δξ( ) , dα d αx( )=−Ax( )+Bx( −τ)+C dW( ) d , dα d αδx( )=−Aδx( )+Bδx( −τ)+C dW( ) d , sα�X(s)−δx( 0 )=−A�X(s)+Be−sτ�X(s)+CW(s), � X(s)= − δx(0) + CW(s) s α +A−Be −sτ . sαI−A+Be−sτ=0. 5 Vol.:(0123456789) Scien i ic Repo s | (2024) 14:6930 | h ps://doi.o g/10.1038/s41598-024-56944-z www.na u e.com/scien i ic epo s/ Lyapuno –K aso skii unc ionals Fo global s abili y in FSDDEs esea ching, he Lyapuno –K aso skii unc ionals a e an aluable esou ce. We desc ibe he c ea ion o applicable unc ionals and how o use hem o deli e he global s abili y esul s while accoun ing o s ochas ici y along wi h ac ional calculus. In p esen subsec ion we discuss he ma hema ical ools known as Lyapuno –K aso skii unc ionals used o examine he s abili y o FSDDEs. These unc ionals a e essen ial in o de o gua an ee he asymp o ic s abili y o equilib ium poin s in p oposed FSDDEs. In he con ex o FSDDEs le ’s gi e a b ie ma hema ical de ini ion o Lyapuno –K aso skii unc ionals: He e we assume Eq.3 o de ine he Lyapuno –K aso skii unc ional de ine by V[x( )] o his FSDDEs is de ined as a eal- alued unc ion sa is ies he ollowing condi ions: Posi i i y V[x( )] is a posi i e de ini e o each x( ) excep a equilib ium poin : V[x( )]>0 o all x( ) = 0 . Ze o a equilib ium The p oposed V[x∗]=0 a he equilib ium poin . Non-inc easing de i a i e The unc ional V[x( )] a ime de i a i e along he ajec o ies o FSDDEs is non- posi i e: d d V[x( )]≤ 0 o all x( ). Lyapuno –K aso skii unc ional se es as a expec an Lyapuno unc ion, and in FSDDEs i s p ope ies a e used o p o e he global asymp o ic s abili y o equilib ium poin x∗ . Howe e , i he abo e unc ional can be ound, and i s ime de i a i e sa is ies all he men ioned p ope ies, hen i signi y he equilib ium poin x∗ globally asymp o ically s able. In he s abili y analysis o FSDDEs he Lyapuno –K aso skii unc ionals cons uc ion and analyzing hei p ope ies is a c i ical s ep, as i p o ides a ma hema ical amewo k o impose long- e m beha io o complex sys ems p oduce he s ochas ici y, ac ional o de dynamics and ime delays. Spec al me hod Sol ing FSDDEs using a spec al me hod o nume ical simula ions can be a powe ul app oach. To expand he solu ion in o o hogonal unc ions, i u ilizes echniques such as Fou ie se ies and Chebyshe polynomials. A simpli ied FSDDE will be sol ed using he spec al me hod. When dealing wi h SDDEs ha a e complex, you may need o adap he me hod and u ilize mo e ad anced spec al me hods. Le ’s conside he ollowing FSDDE: whe e  d α d αx( )  ep esen s he ac ional de i a i e o x( ) wi h o de α . k is a cons an . σ is he ampli ude o he s ochas ic p ocess W( ). W( ) is a s anda d Wiene p ocess (B ownian mo ion). β is a cons an . x( −τ) ep e- sen s he delayed s a e a ime −τ . We will sol e his equa ion using he spec al me hod wi h a unca ed Fou ie se ies expansion o x( ). S ep 1: Disc e iza ion To apply he spec al me hod, we i s disc e ize he ime domain. Le n=n , whe e  is he ime s ep, and n is he ime index. We choose a la ge enough N such ha N=T , whe e T is he inal simula ion ime. S ep 2: Spec al expansion We exp ess x( ) as a unca ed Fou ie se ies: whe e M is he numbe o Fou ie modes (a pa ame e you choose). Xj a e he complex Fou ie coe icien s. ωj = T 2πj a e he angula equencies. S ep 3: Disc e ize he FSDDE He e a e he spec al ep esen a ions o he de i a i es in he FSDDE: Once he Fou ie coe icien s a e aken in o accoun , he FSDDE becomes an algeb aic equa ion: S ep 4: Sol e he sys em A each ime s ep n , we ha e a sys em o algeb aic equa ions as a unc ion o Fou ie coe icien s Xj . Calcula - ing Xj a each ime s ep can be done wi h nume ical me hods, including he Eule me hod and an adap i e sol e . S ep 5: In e se Fou ie ans o m A e compu ing he Fou ie coe icien s Xj a each ime s ep, you can use he in e se Fou ie ans o m o ob ain x( ) in he ime domain: (5) dαx( ) d α=−kx( )+σ dW( ) d +βx( − τ), x ( )≈ M  j= 1 Xjeiωj , (6) dα d αx( )≈ M  j =1 (iωj)αXjeiωj . (7) M  j =1 (iωj)αXjeiωj n=−k M  j =1 Xjeiωj n+σdW( n) d +β M  j =1 Xjeiωj( n−τ) . 6 Vol:.(1234567890) Scien i ic Repo s | (2024) 14:6930 | h ps://doi.o g/10.1038/s41598-024-56944-z www.na u e.com/scien i ic epo s/ To simula e he beha io o x( ) o e he desi ed ime in e al, epea his p ocess o all ime s eps. By app oxima ing he solu ion o he FSDDE nume ically wi h a ini e numbe o Fou ie modes, his spec- al me hod p o ides a nume ical solu ion. A ade-o be ween accu acy and compu a ional complexi y can be achie ed by adjus ing he numbe o modes (M). A nume ical me hod mus also be s able and ha e con e gence p ope ies o p ac ical applica ions. Validi y and accu acy o he p oposed s abili y me hod In addi ion o ac ional s ochas ic di e en ial equa ions (FSDDEs), he Spec al Colloca ion Me hod (SCM) has p o en o be an e ec i e nume ical echnique. Se e al ma hema ical conside a ions can be used o assess he alidi y and accu acy o he Spec al Colloca ion Me hod o FSDDEs: Spec al accu acy To app oxima e he solu ion, he Spec al Colloca ion Me hod uses basis unc ions, o en o hogonal polynomials o igonome ic unc ions. Based on hese basis unc ions, spec al accu acy is achie ed wi h excellen con e gence p ope ies. In compa ison o many o he nume ical me hods, his me hod ypically con e ges exponen ially as , p o id- ing a highly accu a e ep esen a ion o he solu ion wi h ewe deg ees o eedom. Consis ency wi h FSDDE o mula ion This me hod is alid only i i is capable o accu a ely cap u ing FSDDE’s inhe en ea u es. The Spec al Colloca ion Me hod should accommoda e ac ional de i a i es and s ochas ic componen s ound in FSDDEs I is nume ically app oxima ed. I is impo an o ensu e ha he basis unc ions used in he me hod can ep esen ac ional-o de de i a i es accu a ely, and ha s ochas ic p ocesses a e app op ia ely handled. S abili y and con e gence FSDDEs, which a e o en based on bo h ac ional ope a o s and s ochas ic componen s, place a g ea deal o emphasis on s abili y and con e gence. The Spec al Colloca ion Me hod mus accommoda e s ochas ic p ocesses bo h spa ially and empo ally. L-s abili y analysis, o example, can be used o assess he s abili y o a me hod analy ically and nume ically. E o analysis In o de o quan i a i ely assess he spec al collloca ion me hod’s accu acy, i is impe a i e o pe o m a ho ough e o analysis. Ob aining his objec i e equi es an examina ion o bo h he con e gence a es and e o es ima es on a spa ial and empo al basis. Knowing how e o s beha e and how hey ela e o esolu ion and o he a iables is c ucial o de e mining he eliabili y o a me hod. Applicabili y and compu a ional e iciency In addi ion o being applied and compu a ionally e icien , his me hod is also alid and has a good deg ee o applicabili y. The e a e many di e en ypes o FSDDEs, and i is impo an o he FSDDE sys em o be able o handle hem all. Time and memo y equi emen s mus be con- side ed when calcula ing a me hod’s compu a ional cos . P ac ical examples A a ie y o examples, including epidemiology, inance, and neu oscience, a e used o illus a e he concep s discussed. These illus a ions illus a e a wide ange o p ac ical applica ions o FSDDEs. P ac ical si ua ions emphasize s abili y analysis. By using s ochas ic delay di e en ial equa ions wi h a non- linea ac ional ope a o and a ochas ic e m, a wide ange o eal-wo ld domains can be explo ed. These equa ions a e c ucial o in e p e ing he in luence o his o ical s a es on cu en dynamics in Spanish complex sys ems ha demons a e memo y and nonlinea i y. Finance, in pa icula , inds hei use essen ial o gaining an unde s anding o he implica ions o delayed ma ke esponses and unp edic able ma ke luc ua ions, which can be use ul in managing isks and making decisions in he inancial sec o o ecas ing inancial ou comes. Fu he mo e, hese equa ions acili a e he de elopmen o e ec i e s a egies o con ol and p e en disease by enabling epidemiologis s o simula e disease p opaga ion o e a long pe iod o ime. I is used in ecology o unde s and popula ion beha io as a consequence o p io condi ions and unce ain y in he en i onmen , in o de o imp o e he managemen and conse a ion o ecosys ems. Combining ac ional de i a i es and s o- chas ic componen s, SDDEs enable he analysis o in ica e ependencies in Spain’s a ied sec o s and acili a e in o med decision-making. No e ha o ollowing examples we use he Ma lab so wa e used o he nume ical compu a ion. Example 1 Complex FSDDE wi h nonlinea ac ional ope a o and s ochas ic e m Conside he FSDDE: In his example: dα d αx( ) ep esen s he ac ional de i a i e o x( ) wi h o de α . The e m −0.2x( ) ep esen s a damping e ec . 0.5 dW( ) d ep esen s a s ochas ic e m d i en by B ownian mo ion. 0.1x( −1) in oduces a ime delay o 1 ime uni . 0.05 dα d αx( −1) 2 ep esen s a nonlinea e m. x ( n)= M  j =1 Xjeiωj n . (8) d α d αx( )=−0.2x( )+0.5 dW( ) d +0.1x( −1)+0.05  d α d αx( −1) 2 . 7 Vol.:(0123456789) Scien i ic Repo s | (2024) 14:6930 | h ps://doi.o g/10.1038/s41598-024-56944-z www.na u e.com/scien i ic epo s/ Explana ion The equa ion models a sys em wi h damping ( −0.2x( ) ), s ochas ic luc ua ions ( 0.5 dW( ) d ), a ime delay ( 0.1x( −1) ), and a nonlinea in e ac ion e m ( 0.05 dα d αx( −1) 2 ). The ac ional de i a i e ( dα d α ) adds memo y e ec s o he sys em, making i complex and challenging o analyze. This equa ion appea s o desc ibe a complex sys em wi h memo y, damping, s ochas ic luc ua ions, and non-local in e ac ions. Analyzing such equa ions o en equi es specialized echniques om ac ional calculus and s ochas ic calculus. Depending on he speci ic alues o he pa ame e s and ini ial condi ions, he beha io o x( ) can exhibi a ious in e es ing and po en ially unp edic able pa e ns. In Fig.1, we ake he Schema ic diag am o he gi en FSDDEs model gi en in (Eq.8). In Fig.2, we ake he s ochas ic luc ua ion e m equal o ze o and ind he nume ical solu ion o he model Eq. (8). Using di e en al- ues o he ac ional pa ame e α=0.5, 0.6, 0.7, 0.8, 0.9, 1 ; he Fig.2 is d awn. In p esen igu e we clea ly see ha , as we inc ease he alue o ac ional pa ame e α he solu ion g adually app oaches o he con e gen solu ion. In Fig.3, we ake he nume ical solu ion o he model Eq. (8) along wi h s ochas ic luc ua ion. Using di - e en alues o he ac ional pa ame e α=0.6, 0.8, 1 ; he Fig.3 is d awn. In Fig.3 we clea ly see ha , as we inc ease he alue o ac ional pa ame e α he solu ion g adually con e ges o ze o. Simila ly, in Fig.4, we d aw he g apes o compa e he de e minis ic and s ochas ic solu ions o di e en pa ame e alues α=0.5, 0.9 . We clea ly see ha bo h solu ion ha ing a good ag eemen o each o he . Howe e , in Fig.5, we compa e he nume i- cal solu ion o he model Eq. (8) by Lgend e spec al and Chebyshe spec al me hods. We clea ly see ha bo h he solu ions a e in he e good ag eemen s. Example 2 FSDDE wi h nonlinea e ms and ime-dependen s ochas ic o cing Conside he FSDDE: Figu e1. Schema ic diag am o he gi en FSDDEs model gi en in (Eq.8), inco po a ing he “Spec al Me hod”. Figu e2. Solu ion ajec o y o FSDDE model gi en in (Eq.8) o di e en ac ional pa ame e alues α=0.5, 0.6, 0.7, 0.8, 0.9, 1; using s ochas ic e m equal o ze o. 8 Vol:.(1234567890) Scien i ic Repo s | (2024) 14:6930 | h ps://doi.o g/10.1038/s41598-024-56944-z www.na u e.com/scien i ic epo s/ In his example: dα d αx( ) ep esen s he ac ional de i a i e o x( ) wi h o de α . The e m −0.5x( ) ep esen s a damping e ec . 0.2 sin ( ) dW( ) d in oduces ime-dependen s ochas ic o cing. 0.2x( −1) ep esen s a ime delay o 1 ime uni . 0.02x( )3 is a cubic nonlinea e m. (9) dα d αx( )=−0.5x( )+0.2 sin( ) dW( ) d +0.2x( −1)+0.02x( )3 . Figu e3. Solu ion ajec o y o FSDDE model gi en in (Eq.8) o di e en ac ional pa ame e alues α=0.6, 0.8, 1 . Figu e4. Compa ison de e minis ic and s ochas ic solu ion o FSDDE model gi en in (Eq.8) o di e en ac ional pa ame e alues α=0.5, 0.9 . Figu e5. Compa ison o de e minis ic solu ion o FSDDE model gi en in (Eq.8) be ween Legend e Spec al and Chebyshe Spec al me hods o ac ional pa ame e alues α=0.9 . 9 Vol.:(0123456789) Scien i ic Repo s | (2024) 14:6930 | h ps://doi.o g/10.1038/s41598-024-56944-z www.na u e.com/scien i ic epo s/ Explana ion This equa ion models a sys em wi h bo h de e minis ic ( − 0.5 x( )+ 0.2 x( − 1 )+ 0.02 x( )3 ) and s ochas ic ( 0.2 sin ( ) dW( ) d ) componen s. - The ac ional de i a i e ( dα d α ) in oduces memo y e ec s and cap u es complex dynamics, including he cubic nonlinea i y. This equa ion combines elemen s o de e minis ic and s ochas ic dynamics, memo y e ec s, and nonlinea i y, making i a complex sys em o analyze. F ac ional de i a i es ( dα d α ) ex end he concep o o dina y de i a i es ( d d ) o non-in ege o de s, adding ano he laye o complexi y o he model. Figu e6. Solu ion ajec o y o FSDDE model gi en in (Eq.9) o di e en ac ional pa ame e alues α=0.2, 0.4, 0.6, 0.8, 1; using s ochas ic e m equal o ze o. Figu e7. Solu ion ajec o y o FSDDE model gi en in (Eq.9) o di e en ac ional pa ame e alues α=0.2, 0.6, 1 . Figu e8. Compa ison de e minis ic and s ochas ic solu ion o FSDDE model gi en in (Eq.9) o di e en ac ional pa ame e alues α=0.5, 0.9 .