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
.