Elec onic
Resea ch A chi e
h p://www.aimsp ess.com/jou nal/e a
ERA, 32(2): 733–761.
DOI: 10.3934/e a.2024035
Recei ed: 26 Sep embe 2023
Re ised: 03 Decembe 2023
Accep ed: 15 Decembe 2023
Published: 10 Janua y 2024
Resea ch a icle
Exis ence, uniqueness and nume ical solu ion o s ochas ic ac ional
di e en ial equa ions wi h in ege and non-in ege o de s
Seda IGRET ARAZ1,2,*, Mehme Aki CETIN3and Abdon ATANGANA2,4,5
1Sii Uni e si y, Depa men o Ma hema ics Educa ion, Sii , Tu key
2Ins i u e o G oundwa e S udies, Facul y o Na u al and Ag icul u al Sciences, Uni e si y o he
F ee S a e, Sou h A ica
3ALTSO Voca ional School, Alanya Alaaddin Keykuba Uni e si y, An alya, Tu key
4Depa men o Medical Resea ch, China Medical Uni e si y Hospi al, China Medical Uni e si y,
Taichung, Taiwan
5IT4Inno a ions, VSB–Technical Uni e si y o Os a a, Os a a-Po uba 70800, Czech Republic
*Co espondence: Email: [email p o ec ed].
Abs ac : The pa ame ized app oach is ex ended in his s udy o ind solu ions o di e en ial equa ions
wi h ac al, ac ional, ac al- ac ional, and piecewise de i a i es wi h he inclusion o a s ochas ic
componen . The exis ence and uniqueness o he solu ion o he s ochas ic A angana-Baleanu ac-
ional di e en ial equa ion a e es ablished using Ca a heodo y’s exis ence heo em. Fo he solu ion
o di e en ial equa ions using piecewise di e en ial ope a o s, which ake in o accoun combining
de e minis ic and s ochas ic p ocesses u ilizing ce ain signi ican ma hema ical ools such as ac al and
ac al- ac ional de i a i es, he applicabili y o he pa ame ized echnique is being examined. We
discuss he c osso e beha io s o he model ob ained by including hese ope a o s and we p esen some
illus a i e examples o some p oblems wi h piecewise di e en ial ope a o s.
Keywo ds: Ca a heodo y’s condi ions; ac al- ac ional di e en ia ion; piecewise calculus;
pa ame ized me hod
1. In oduc ion
F ac ional analysis is a heo y ha s a ed wi h Leibniz asking i he e is a de i a i e o o de 1
/
2
o a unc ion. This heo y in e es ed many esea che s when di e en ypes o ac ional de i a i es
we e in oduced. One well-known de ini ion is he Riemann-Liou ille ac ional de i a i e whe e
he powe -law ke nel is inco po a ed. Capu o [1] in oduced a de i a i e wi h a modi ica ion on he
734
Riemann-Liou ille ac ional de i a i e [2] because i was use ul in heo y bu no app op ia e o
sol ing eal-li e p oblems. These ope a o s, which a e used o model powe law p ocesses, ha e
beha io ha is bo h nonlocal and singula . E en hough some p ocesses a e unique, ano he o m o
ma h is needed o desc ibe p ocesses ha beha e di e en ly. Capu o and Fab izio [3] ha e c ea ed
a ma hema ical concep called a de i a i e wi h ading memo y, which uses an exponen ial pa e n.
This de i a i e deals wi h p ocesses ha beha e p edic ably and wi hin a small a ea. Howe e , we
needed a de i a i e ha is p edic able bu ac s o e a la ge a ea. The A angana-Baleanu ac ional
de i a i e [4] is a ma hema ical ool ha mee s his equi emen , and i u ilizes he Mi ag-Le le
unc ion. The ac al de i a i e o Hausdo de i a i e [5] is a di e en kind o de i a i e used o
measu ing ac als in ac al geome y. F ac al de i a i es we e made o s udy how hings sp ead in a
s ange way when no mal ways o s udying do no conside he ac al shape o he hing ha hings a e
sp eading h ough. A ac al measu e
changes i s size in ela ion o
aised o he powe o
β
. This ype
o de i a i e is only used in a speci ic a ea, unlike he ac ional de i a i e, which is used in a simila
way. La e , A angana in oduced ac al- ac ional de i a i es [6] by combining he concep s o ac al
and ac ional de i a i es. Al hough he e is no doub ha ac ional di e en ial ope a o s a e use ul in
modeling ele an p ocesses [7–11], hese ope a o s canno be used o model c osso e p ocesses such
as om s ochas ic o powe -law o om ading memo y o s ochas ic [12–15]. Concluding ha a new
class o di e en ial ope a o s was needed o his, A angana and A az in oduced piecewise di e en ial
ope a o s [16], which can be c ea ed by including a ious di e en ial ope a o s o model such p ocesses.
These ope a o s, which can be used o desc ibe many p ocesses, om modeling he di e en a es (o
e en s opping) o an indi idual’s hea bea o e a pe iod o ime, o modeling he sp ead o a i us, i s
cumula i ely and hen daily, ha e become ocus o a en ion o esea che s.
In o de o be e unde s and and analyze he p ocesses discussed, i is necessa y o sol e he equa ions
ha ep esen hese p ocesses. Because i is di icul o sol e hese equa ions using analy ic me hods
when he ope a o s men ioned abo e and he nonlinea i y o he associa ed equa ions a e in ol ed, we
ha e o use nume ical me hods o ob ain solu ions o such equa ions. The pa ame ized me hod, which
deals wi h he app oxima ion o a unc ion wi h cons an s depending on a pa ame e , is one o he
well-known nume ical me hods. While he pa ame e ized me hod is p esen ed in he li e a u e [8–10]
o classical di e en ial equa ions, A angana and A az [17] ex ended his me hod o sol e ac ional and
ac al- ac ional di e en ial equa ions. The pa ame ized me hod was compa ed wi h exis ing me hods
in he li e a u e in [17] and i was shown ha he me hod is mo e e ec i e han o he me hods, especially
when he pa ame e is close o 1.
Howe e , in [17], he applica ion o he ele an me hod o s ochas ic di e en ial equa ions wi h
ac ional, ac al- ac ional and piecewise de i a i es [16] is no aken in o accoun . The e o e, in
his s udy, we p esen he de i a ion o his me hod o s ochas ic di e en ial equa ions wi h ac ional,
ac al- ac ional and piecewise de i a i es. We employ he pa ame ized me hod o sol e di e en
ypes o equa ions ob ained by inco po a ing hese ma hema ical ools in o di e en ial equa ions.
Be o e p esen ing he associa ed me hod, i s he de ini ions o he abo e-men ioned ac ional, ac al
ac ional and piecewise de i a i es will be p esen ed. In he ollowing sec ion, wi h he help o
Ca a h
´
eodo y condi ions [15,16], he exis ence and uniqueness o he solu ion o A angana-Baleanu
s ochas ic di e en ial equa ions [18] will be in es iga ed. In he emaining sec ions, in addi ion o
he de i a ion o he pa ame ized me hod wi h hese de i a i es, some illus a i e examples will be
included.
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
735
2. P elimina ies
In his sec ion, he de ini ions o ac ional de i a i es wi h powe law beha io , ading memo y
and exhibi ing powe law beha io a e ading memo y, ac al- ac ional de i a i es and piecewise
de i a i es, which can be ep esen ed in di e en ways by including ac ional and ac al- ac ional
de i a i es, will be discussed.
The Capu o-Fab izio ac ional de i a i e [3] o he unc ion ( )∈H1(0,T)is de ined by
CF
0Dα
( )=1
1−αZ
0
0(τ)exp −α
1−α( −τ)dτ, (2.1)
whe e 0 <α<1 and H1(0,T)desc ibes he Hilbe space. The associa ed in eg al is gi en as
CF
0Jα
( )=(1−α) ( )+αZ
0
(τ)dτ. (2.2)
The Capu o ac ional de i a i e [1] o he unc ion ( )∈H1(0,T)is de ined by
C
0Dα
( )=1
Γ(1−α)Z
0
0(τ) ( −τ)−αdτ, (2.3)
whe e 0
< α ≤
1 and he Riemann-Liou ille ac ional de i a i e o he unc ion
( )∈C(0,T)
is
de ined by
RL
0Dα
( )=1
Γ(1−α)
d
d Z
0
(τ) ( −τ)−αdτ. (2.4)
The in eg al wi h powe -law ke nel [2] is gi en by
RL
0Jα
( )=1
Γ(α)Z
0
(τ) ( −τ)α−1dτ. (2.5)
The ollowing o mulas desc ibe he A angana-Baleanu ac ional de i a i e [4], which has he c osso e
beha io om s e ched exponen ial o powe -law,
ABC
0Dα
( )=1
1−αZ
0
0(τ)Eα−α
1−α( −τ)αdτ, (2.6)
and
ABR
0Dα
( )=1
1−α
d
d Z
0
(τ)Eα−α
1−α( −τ)αdτ. (2.7)
The abo e ope a o s a e called A angana-Baleanu ac ional de i a i e in he Capu o sense and A angana-
Baleanu ac ional de i a i e in he Riemann-Liou ille sense [4], espec i ely. The associa ed in eg al is
gi en by
AB
0Jα
( )=(1−α) ( )+α
Γ(α)Z
0
(τ) ( −τ)α−1dτ. (2.8)
The concep o ac al- ac ional di e en ia ion and in eg a ion has appea ed p e iously wi h he idea o
combining he ac al and ac ional de i a i es. The ac al- ac ional de i a i e [6] wi h powe -law
ke nel is de ined by
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
736
FFP
0Dα,β
( )=1
Γ(1−α)
d
d βZ
0
(τ) ( −τ)−αdτ, (2.9)
whe e he de ini ion o ac al de i a i e [5] is
d
d β ( )=lim
→ 1
( )− ( 1)
β− β
1
.(2.10)
The associa ed ac al- ac ional in eg al [6] wi h powe -law ke nel is gi en by
FFP
0Jα
( )=1
Γ(α)Z
0
βτβ−1 (τ) ( −τ)α−1dτ. (2.11)
The ac al- ac ional de i a i e wi h Mi ag-Le le ke nel [6] is de ined by
FFM
0Dα,β
( )=1
1−α
d
d βZ
0
(τ)Eα−α
1−α( −τ)αdτ(2.12)
and he associa ed ac al- ac ional in eg al is gi en by
FFM
0Jα
( )=(1−α)β β−1 ( )+α
Γ(α)Z
0
βτβ−1 (τ) ( −τ)α−1dτ. (2.13)
The ac al- ac ional de i a i e wi h exponen ial decay ke nel [6] is de ined by
FFE
0Dα,β
( )=1
1−α
d
d βZ
0
(τ)exp −α
1−α( −τ)dτ(2.14)
and he associa ed ac al- ac ional in eg al is gi en by
FFE
0Jα
( )=(1−α)β β−1 ( )+αZ
0
βτβ−1 (τ)dτ. (2.15)
We now p esen he de ini ions o he piecewise de i a i e and in eg al ope a o s, which made signi ican
con ibu ion o li e a u e [16].
The piecewise de i a i e wi h classical and ac ional de i a i e wi h powe -law ke nel such ha i
can be aken as [16]
PRL
0Dα
y( )=(y0( )i 0 ≤ ≤ 0
RL
0Dα
y( )i 0≤ ≤T(2.16)
whe e
PRL
0Dα
ep esen s he classical de i a i e wi hin 0
≤ ≤ 0
and he Riemann-Liou ille ac ional
de i a i e wi hin 0≤ ≤T.
The piecewise wi h Capu o de i a i e is gi en as [16]
PC
0Dα
y( )=(y0( )i 0 ≤ ≤ 0
C
0Dα
y( )i 0≤ ≤T(2.17)
whe e he unc ion
y( )
is con inuous bu no necessa ily di e en iable in
[ 0,T].
He e,
PRL
0Dα
ep esen s
he classical de i a i e on 0
≤ ≤ 0
and he Capu o ac ional de i a i e [1] on
0≤ ≤T
. The
associa ed piecewise in eg al o yis gi en as [16]
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
737
PPLI y( )=





R 0
0y(τ)dτi 0 ≤ ≤ 0
1
Γ(α)R
0y(τ) ( −τ)α−1dτi 0≤ ≤T(2.18)
whe e
PPL
0Iα
ep esen s he classical in eg al on 0
≤ ≤ 0
and he in eg al wi h powe -law ke nel on
0≤ ≤T.
The piecewise de i a i e wi h classical de i a i e and exponen ial decay ke nel is gi en as [16]
PCF
0Dα
y( )=(y0( )i 0 ≤ ≤ 0
CF
0Dα
y( )i 0≤ ≤T(2.19)
whe e
PCF
0Dα
is he classical de i a i e on 0
≤ ≤ 0
and he Capu o-Fab izio ac ional de i a i e [3]
on
0≤ ≤T.
He e, i is assumed ha he unc ion
y( )
is di e en iable. A piecewise in eg al is gi en
as [16]
PCF I y( )=





R 0
0y(τ)dτi 0 ≤ ≤ 0
1−α
M(α)y( )+α
M(α)R
0y(τ)dτi 0≤ ≤T.(2.20)
The piecewise de i a i e wi h classical de i a i e and Mi ag-Le le ke nel is de ined by [16]
PAB
0Dα
y( )=(y0( )i 0 ≤ ≤ 0
ABC
0Dα
y( )i 0≤ ≤T(2.21)
whe e
PAB
0Dα
ep esen s he classical de i a i e on 0
≤ ≤ 0
and he A angana-Baleanu ac ional
de i a i e [4] on 0≤ ≤T.The associa ed piecewise in eg al is gi en as [16]
PABI y( )=





R 0
0y(τ)dτi 0 ≤ ≤ 0
(1−α)y( )+α
Γ(α)R
0y(τ) ( −τ)α−1dτi 0≤ ≤T.(2.22)
Lemma 1. (The gene aliza ion o he G onwall inequali y) Assume ha
b≥
0
, α >
0
,
and
x( )
is a
nonnega i e unc ion locally in eg able on 0
≤ <T,
and assume ha
y( )
is nonnega i e and locally
in eg able on 0≤ <T wi h
y( )≤x( )+bZ
0
y(τ) ( −τ)α−1dτ. (2.23)
Then,
y( )≤x( )+Z
0
∞
X
n=1
(bΓ(α))n
Γ(nα)y(τ) ( −τ)nα−1x(τ)dτ. (2.24)
De ini ion 1. (S i ling o mula) The S i ling o mula o he Gamma unc ion is o mula ed by
Γ(x)∼√2πe−xxx−1
2.(2.25)
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.

738
3. Ca a heodo y’s heo y o exis ence and uniqueness o a gene al Cauchy p oblem wi h
s ochas ic A angana-Baleanu ac ional de i a i e
In his sec ion, we p o e he exis ence and uniqueness o he solu ion o he A angana-Baleanu
s ochas ic di e en ial equa ion [18] by employing Ca a h
´
eodo y’s exis ence heo y [19,20], which is a
mo e gene al e sion o Peano’s exis ence heo em. I is wo h no ing ha he exis ence and uniqeness
o he solu ion o s ochas ic di e en ial equa ions wi h he Capu o ac ional de i a i e is p esen ed
in [21]. He e, we will examine he exis ence and uniqueness o he s ochas ic di e en ial equa ion wi h
A angana-Baleanu ac ional de i a i e. The di e en ial equa ion unde in es iga ion is ep esen ed in
he o m:
AB
0Dα
y( )= 1( ,y)d +σy( )dB ( ), ≥0 (3.1)
y( 0)=y0
unde he condi ions
E1) Fo all y,¯y∈H, he e is a cons an k>0 such ha
| 1( ,y)− 1( ,¯y)|2,| 2( ,y)− 2( ,¯y)|2≤k|y−¯y|2, ≥0.(3.2)
E2) Fo all y∈H, he e is a cons an ¯
k>0 such ha
| 1( ,y)|2,| 2( ,y)|2≤¯
k1+|y|2, ≥0 (3.3)
whe e
H
is a Banach space. No e ha condi ions E1 and E2 a e known as he Lipschi z condi ion and
he g ow h condi ion, espec i ely.
Theo em 1. Fo each
y0∈L2(Ω,H)
, Eq (26) has a unique mild solu ion
y∈C[0,T],L2(Ω,H)
=
S
such ha
sup
0≤ ≤T
E|y|2<∞.
P oo .
Fo he p oo , we will use he con ac ion mapping p inciple. Be o e p oceeding wi h he p oo ,
we de ine he no m
kηk2
γ=sup
0≤ ≤T
E|η( )|2(3.4)
whe e Edeno es he expec a ion.
Fo any
∈[0,T]
and
y∈S
, we de ine he mapping subjec o Ω =
C[0,T],L2(Ω,H)→
C[0,T],L2(Ω,H)
(Λy) ( )=y0+(1−α) 1( ,y)+(1−α)σy( )B0( )(3.5)
+α
Γ(α)Z
0
1(s,y) ( −s)α−1ds +ασ
Γ(α)Z
0
y(s) ( −s)α−1dB (s).
Thus, we w i e
E|(Λy) ( )−(Λ¯y) ( )|2=E
(1−α) ( 1( ,y)− 1( ,¯y))
+(1−α)σ(y( )−¯y( )) B0( )
+α
Γ(α)R
0( 1(s,y)− 1(s,¯y)) ( −s)α−1ds
+ασ
Γ(α)R
0(y(s)−¯y(s)) ( −s)α−1dB (s)

2
.(3.6)
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
739
Taking 2
α−
1
>
0, by he Cauchy-Schwa z inequali y, I o’s isome y o mula and he Lipschi z
condi ion [22], we ha e
E|(Λy) ( )−(Λ¯y) ( )|2≤4(1−α)2kσ1+|B0|2E|y−¯y|2(3.7)
+(T+1)4α2k
Γ2(α)Z
0
E|y−¯y|2( −s)2α−2ds
≤4(1−α)2σk1+|B0|2ky−¯ykγ
+(T+1)4α2k
Γ2(α)
2α−1
(2α−1)ky−¯ykγ
≤4σ(1−α)2k 1+sup
∈[0,T]|B0|2!ky−¯ykγ
+(T+1)4α2k
Γ2(α)
2α−1
(2α−1)ky−¯ykγ
≤4σ(1−α)2k1+kB0k∞ky−¯ykγ
+(T+1)4σα2k
Γ2(α)
2α−1
(2α−1)ky−¯ykγ
≤˜
kky−¯ykγ,
whe e
˜
k=4σ(1−α)2k1+kB0k∞+(T+1)4σα2k
Γ2(α)
2α−1
(2α−1).(3.8)
Using he gene alized G onwall inequali y [23], we w i e
EΛ2y( )−Λ2¯y( )≤4σ(1−α)2k1+kB0k∞E|Λy−Λ¯y|2(3.9)
+4σα2k(T+1)
Γ2(α)Z
0
( −s)2α−2E|Λy−Λ¯y|2ds
≤4σ(1−α)2k1+kB0k∞"4(1−α)2k(1+kB0k∞)
+(T+1)4α2k
Γ2(α)
2α−1
(2α−1)#(3.10)
+4σα2k(T+1)
Γ2(α)Z
0
( −s)2α−2"4(1−α)2k(1+kB0k∞)
+(T+1)4α2k
Γ2(α)
s2α−1
(2α−1)#ds(3.11)
≤

4σ(1−α)2k(1+kB0k∞)2
+(T+1)4σα2k
Γ2(α)
T2α−1
(2α−1)4(1−α)2(1+kB0k∞)
+4σ(1−α)2k(1+kB0k∞)4α2k(T+1)
Γ2(α)T2α−1
(2α−1)
+4σα2k(T+1)
Γ2(α)2Γ2(2α−1)
Γ(4α−2)
T4α−2
(2α−1)


ky−¯ykγ.
By he induc ion o mula o n, we can hen w i e
E|(Λny) ( )−(Λn¯y) ( )|≤

4σ(1−α)2(1+kB0k∞)n
+4σ(1−α)2(1+kB0k∞) (T+1)4α2k
Γ2(α)
T2α−1
(2α−1)n−1
+4σ(1−α)2(1+kB0k∞)n−14σα2k(T+1)
Γ2(α)Tn(2α−1)
Γ(n(2α−1))
+4σα2k(T+1)
Γ2(α)nTn(2α−1)
(2α−1)
Γn(2α−1)
Γ(n(2α−1))

ky−¯ykγ(3.12)
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
740
≤Lky−¯ykγ
whe e
L=

4σ(1−α)2(1+kB0k∞)n
+4σ(1−α)2(1+kB0k∞) (T+1)4α2k
Γ2(α)
T2α−1
(2α−1)n−1
+4σ(1−α)2(1+kB0k∞)n−14σα2k(T+1)
Γ2(α)Tn(2α−1)
Γ(n(2α−1))
+4σα2k(T+1)
Γ2(α)nTn(2α−1)
(2α−1)
Γn(2α−1)
Γ(n(2α−1))

.(3.13)
To p o e he heo em holds, we will show ha
L<
1 o su icien la ge
n
. Le us conside he ollowing
se ies o posi i e e ms
∞
n=1

4σ(1−α)2(1+kB0k∞)n
+4σ(1−α)2(1+kB0k∞) (T+1)4α2k
Γ2(α)
T2α−1
(2α−1)n−1
+4σ(1−α)2(1+kB0k∞)n−14σα2k(T+1)
Γ2(α)Tn(2α−1)
Γ(n(2α−1))
+4σα2k(T+1)
Γ2(α)nTn(2α−1)
(2α−1)
Γn(2α−1)
Γ(n(2α−1))

.(3.14)
Using he d’Alembe disc iminan me hod
lim
n→∞ 4α2k(T+1)
Γ2(α)n+1T(n+1)(2α−1)
(2α−1)
Γn+1(2α−1)
Γ((n+1)(2α−1))
4σα2k(T+1)
Γ2(α)nTn(2α−1)
(2α−1)
Γn(2α−1)
Γ(n(2α−1))
<1 (3.15)
which is equi alen o
lim
n→∞ 4σα2k(T+1)
Γ2(α)T(2α−1)Γ(2α−1)Γ(n(2α−1))
Γ((n+1) (2α−1)) <1.(3.16)
Using he S i ling o mula [21], we ha e he ollowing o las e m
lim
n→∞ 4σα2k(T+1)
Γ2(α)Γ(2α−1)T(2α−1)e(2α−1)
√n+1
√nn
n+1n(2α−1)1
((n+1)(2α−1))(2α−1)=0 (3.17)
and knowing ha α < 1,we can ha e
lim
n→∞

4σ(1−α)2(1+kB0k∞)n
+4σ(1−α)2(1+kB0k∞) (T+1)4α2k
Γ2(α)
T2α−1
(2α−1)n−1
+4σ(1−α)2(1+kB0k∞)n−14σα2k(T+1)
Γ2(α)Tn(2α−1)
Γ(n(2α−1))
+4σα2k(T+1)
Γ2(α)nTn(2α−1)
(2α−1)
Γn(2α−1)
Γ(n(2α−1))

=0.(3.18)
This gua an ees ha
L<
1 holds. This p o es ha Λ
y( )
is a con ac ion mapping, which comple es he
p oo .
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
741
4. Pa ame ized me hod o a gene al Cauchy p oblem wi h s ochas ic ac ional de i a i es
In his sec ion, we de elop he pa ame ized app oach o nume ically sol ing di e en ial equa ions
wi h ac ional de i a i es ha inco po a e s ochas ic componen s. Be o e p esen ing he ex ension o
he me hod o he solu ions o di e en di e en ial equa ions, we shall ecall he o mula ion o he
pa ame ized app oach [17,24–26]. The app oach is o mula ed by he ollowing:
ϕ1( ,y)≈" 1−1
2ξ!ϕ1 k,yk+1
2ξϕ1 k+1,˜yk+1#.(4.1)
4.1. Pa ame ized me hod o a gene al Cauchy p oblem wi h s ochas ic componen
To de i e he associa ed me hod, in his subsec ion, we conside a gene al Cauchy p oblem wi h
s ochas ic componen gi en by
dy ( )=ϕ1( ,y)d +σy( )dB ( ).(4.2)
We con e he abo e in o an in eg al equa ion, by applying on bo h sides he classical in eg al
y( )=y(0)+Z
0
ϕ1(τ, y)dτ+Z
0
σy(τ)dB (τ).(4.3)
A = k+1,we w i e
y( k+1)=y(0)+Z k+1
0
ϕ1(τ, y)dτ+Z k+1
0
σy(τ)dB (τ)(4.4)
and a = k
y( k)=y(0)+Z k
0
ϕ1(τ, y)dτ+Z k
0
σy(τ)dB (τ).(4.5)
Subs ac ing hese wo equali ies gi es
y( k+1)=y( k)+Z k+1
k
ϕ1(τ, y)dτ+Z k+1
k
σy(τ)dB (τ).(4.6)
The unc ion
ϕ1(τ, y)
can be app oxima ed by using he pa ame ized app oach [17,24
–
26] p esen ed
ea lie . A e simpli ica ion, we ha e he p edic o -co ec o o mula [27]
yk+1=yk+h" 1−1
2ξ!ϕ1 k,yk+1
2ξϕ1 k+1,˜yk+1#(4.7)
+σy(ck) (B( k+1)−B( k)) ,
whe e ck∈[ k, k+1]and he p edic o e m
˜yk+1=yk+hϕ1 k,yk.(4.8)
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
748
5.3. Pa ame ized me hod o a gene al Cauchy p oblem wi h s ochas ic ac al- ac ional de i a i e
wi h powe -law ke nel
In his sec ion, we ob ain he nume ical solu ion o a gene al Cauchy p oblem wi h s ochas ic
ac al- ac ional de i a i e [18] wi h powe -law ke nel by using he pa ame ized me hod [17]. The
associa ed p oblem unde conside a ion is ep esen ed by
(FFP
0Dα
y( )=ϕ1( ,y)+σy( )dB ( ),i >0,
y(0)=y0,i =0.(5.11)
Applying he ac al- ac ional de i a i e [6] wi h powe -law ke nel, we ha e
y( )=β
Γ(α)Z
0
τβ−1ϕ1(τ, y) ( −τ)α−1dτ+β
Γ(α)Z
0
σy(τ)τβ−1( −τ)α−1dB (τ).(5.12)
A = k+1,we ha e
y( k+1)=β
Γ(α)
k
X
n=0Z n+1
n
τβ−1ϕ1(τ, y) ( k+1−τ)α−1dτ(5.13)
+β
Γ(α)
k
X
n=0Z n+1
n
τβ−1σy(τ) ( k+1−τ)α−1B0(τ)dτ.
Replacing he unc ion ϕ1(τ, y)by i s pa ame ized app oxima ion, we ha e
yk+1=β
Γ(α)
k
X
n=0" 1−1
2ξ!ϕ1( n,yn)+1
2ξϕ1 n+1,˜yn+1#(5.14)
×Z n+1
n
τβ−1( k+1−τ)α−1dτ
+β
hΓ(α)
k
X
n=0
σy(cn) (B( n+1)−B( n))
×Z n+1
n
τβ−1( k+1−τ)α−1dτ.
The in eg al on he igh hand side o he abo e equa ion is calcula ed by using he change o a iables
τ= k+1uand dτ= k+1du as ollows:
Z n+1
n
τβ−1( k+1−τ)α−1dτ= α+β−1
k+1Z n+1
n
uβ−1(1−u)α−1du
= α+β−1
k+1 B n+1
k+1
, β, α!−B n
k+1
, β, α!!,
whe e he unc ion
B(·,·,·)
is he incomple e Be a unc ion. By calcula ion o hese in eg als, he
ollowing nume ical scheme is ob ained:
yk+1=β
Γ(α)
k
X
n=0" 1−1
2ξ!ϕ1( n,yn)+1
2ξϕ1 n+1,˜yn+1#(5.15)
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.

749
× α+β−1
k+1 B n+1
k+1
, β, α!−B n
k+1
, β, α!!
+β
hΓ(α)
k
X
n=0
σy(cn) (B( n+1)−B( n))
× α+β−1
k+1 B n+1
k+1
, β, α!−B n
k+1
, β, α!!.
We know ha he e m ˜yn+1is p edic ed by he ollowing:
˜yn+1=y0+β
Γ(α)
k
X
n=0
ϕ1( n,yn) α+β−1
k+1 B n+1
k+1
, β, α!−B n
k+1
, β, α!!.(5.16)
5.4. Pa ame ized me hod o a gene al Cauchy p oblem wi h s ochas ic ac al- ac ional de i a i e
wi h Mi ag-Le le ke nel
To examine he solu ion o a gene al Cauchy p oblem wi h s ochas ic ac al- ac ional de i a i e
wi h Mi ag-Le le ke nel [18], we conside he ollowing p oblem:
(FFM
0Dα
y( )=ϕ1( ,y)+σy( )dB ( ),i >0,
y(0)=y0,i =0(5.17)
A e aking he associa ed in eg al, he abo e can be a anged as ollows:
y( )=(1−α)ϕ1( ,y)+(1−α)σy( )dB ( )(5.18)
+αβ
Γ(α)Z
0
τβ−1ϕ1(τ, y) ( −τ)α−1dτ
+αβ
Γ(α)Z
0
σy(τ)τβ−1( −τ)α−1B0(τ)dτ.
A = k+1,we ha e
y( )=(1−α)ϕ1 k+1,yk+1+(1−α)σy( k+1)dB ( k+1)(5.19)
+αβ
Γ(α)Z k+1
0
τβ−1ϕ1(τ, y) ( k+1−τ)α−1dτ
+αβ
Γ(α)Z k+1
0
σy(τ)τβ−1( k+1−τ)α−1dB (τ).
Using he ϕ1(τ, y)app oxima ions, we ha e
yk+1=(1−α)ϕ1 k+1,yk+1+(1−α)σy(ck+1) (B( k+1)−B( k)) (5.20)
+αβ
Γ(α)
k
X
n=0" 1−1
2ξ!ϕ1( n,yn)+1
2ξϕ1 n+1,˜yn+1#
×Z n+1
n
τβ−1( k+1−τ)α−1dτ
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
750
+αβ
hΓ(α)
k
X
n=0
σy(cn) (B( n+1)−B( n))
×Z n+1
n
τβ−1( k+1−τ)α−1dτ.
Using he calcula ions o hese in eg als and a anging he abo e, we ha e
yk+1=(1−α)ϕ1 k+1,yk+1+(1−α)σy(ck+1) B( k+1)−B( k)
h!(5.21)
+αβ
Γ(α)
k
X
n=0" 1−1
2ξ!ϕ1( n,yn)+1
2ξϕ1 n+1,˜yn+1#
× α+β−1
k+1 B n+1
k+1
, β, α!−B n
k+1
, β, α!!
+αβ
hΓ(α)
k
X
n=0
σy(cn) (B( n+1)−B( n))
× α+β−1
k+1 B n+1
k+1
, β, α!−B n
k+1
, β, α!!,
whe e he p edic o o mula is s a ed as:
˜yk+1=y0+(1−α)β β−1
k+1ϕ1 k+1,yk+1+αβ
Γ(α)
k
X
n=0
ϕ1( n,yn)(5.22)
× α+β−1
k+1 B n+1
k+1
, β, α!−B n
k+1
, β, α!!.
6. Pa ame ized me hod o a gene al Cauchy p oblem wi h piecewise de i a i e
In his sec ion, we de i e he pa ame ized me hod [17] o some e sions o nonlinea di e en ial
equa ions wi h piecewise di e en ia ion. We shall s a wi h he e sion o nonlinea di e en ial
equa ions wi h piecewise de i a i e [16], in which classical p ocesses can be used in he i s ime
in e al, p ocesses wi h powe -law a e ading memo y in he second ime in e al, and s ochas ic
p ocesses can be used in he hi d ime in e al. The associa ed model is ep esen ed by he ollowing:



























dy
d =ϕ( ,y), i 0 ≤ ≤ 1
y(0)=y0,
ABC
1Dα
y=ϕ( ,y),i 1≤ ≤ 2
y( 1)=y1,
dy ( )=ϕ( ,y)d +σydB ( ), i 2≤ ≤T
y( 2)=y2
.(6.1)
The unc ion
ϕ( ,y)
can be app oxima ed by employing he pa ame ized o mula ion [17], hus in e-
g a ing wi hin [ n, n+1], we ha e he ollowing co ec o o mula wi h p edic o e m:
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
751
yk+1=








y0+hk1
j1=0h1−1
2ξϕ1 j1,yj1+1
2ξϕ1 j1+1,˜yj1+1i,
i 0 ≤ ≤ 1
,(6.2)





































y1+(1−α)ϕ1 k2+1,˜yk2+1
+(1−α)σyck2+1B k2+1−B k2
+hα
Γ(α)
k2
j2=k1+1h1−1
2ξϕ1 j2,yj2+1
2ξϕ1 j2+1,˜yj2+1i
×(k2−j2+1)α−(k2−j2)α
+hα−1
Γ(α)
k2
j2=k1+1σycj2B j2+1−B j2
×(k2−j2+1)α−(k2−j2)α,
i 1≤ ≤ 2,











y2+hk
j3=k2+1h1−1
2ξϕ1 j3,yj3+1
2ξϕ1 j3+1,˜yj3+1i
+σy(ck) (B( k+1)−B( k)) ,
i 2≤ ≤T.
The p edic o componen s o each in e al a e calcula ed as



















n˜yk1+1=y0+hk1
j1=0ϕ1 j1,yj1,i 0 ≤ ≤ 1,







˜yk2+1=y1+(1−α)ϕ1 k2,yk2+hα
Γ(α)
k2
j2=k1+1ϕ1 j2,yj2
×(k2−j2+1)α−(k2−j2)α,i 1≤ ≤ 2,
n˜yk3+1=y2+hk
j3=k2+1ϕ1 j3,yj3,i 2≤ ≤T.
(6.3)
Now, we p oceed wi h an ano he e sion o nonlinea di e en ial equa ions wi h piecewise de i a i es
[16]. In he i s ime in e al, ading memo y p ocesses can be u ilized, while s ochas ic p ocesses
can be used in he second ime in e al. Fo he hi d ime in e al, p ocesses ha deal wi h powe -law
beha io s ha ing ac al p ope ies can be employed. The model ha explains he p ocess p esen ed
he e is shown as ollows:



























CF
0Dα
y=ϕ( ,y), i 0 ≤ ≤ 1
y(0)=y0,
dy ( )=ϕ( ,y)d +σydB ( ),i 1≤ ≤ 2
y( 1)=y1,
FFP
2Dα
y=ϕ( ,y), i 2≤ ≤T
y( 2)=y2.
(6.4)
Using he a o emen ioned concep o nume ical scheme, he nume ical scheme o he Cauchy p oblem
in he amewo k o piecewise de i a i e [16] is achie ed as
yk+1=





















y0+(1−α)ϕ1 k1+1,yk1+1
+αhk1
j1=0h1−1
2ξϕ1 j1,yj1+1
2ξϕ1 j1+1,˜yj1+1i,
i 0 ≤ ≤ 1
(6.5)











y1+hk2
j2=k1+1h1−1
2ξϕ1 j2,yj2+1
2ξϕ1 j2+1,˜yj2+1i
+σyck2B k2+1−B k2,
i 1≤ ≤ 2
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
752



























β
Γ(α)
k
j3=k2+1h1−1
2ξϕ1 j3,yj3+1
2ξϕ1 j3+1,˜yj3+1i
× α+β−1
k+1B j3+1
k+1, β, α−B j3
k+1, β, α
+β
hΓ(α)
k
j3=k2+1σycj3B j3+1−B j3
× α+β−1
k+1B j3+1
k+1, β, α−B j3
k+1, β, α,
i 2≤ ≤T.
The p edic o componen s o each in e al a e de e mined as























n˜yk1+1=y0+hk1
j1=0ϕ1 j1,yj1,i 0 ≤ ≤ 1,
n˜yk2+1=y1+hk2
j2=k1+1ϕ1 j2,yj2,i 1≤ ≤ 2,









˜yk1+1=(1−α)β β−1
k1ϕ1 k1,yk1+αβ
Γ(α)
k
j3=k2+1ϕ1 j3,yj3
× α+β−1
k1+1B j3+1
k1+1, β, α−B j3
k1+1, β, α,i 2≤ ≤T.
(6.6)
7. Illus a i e examples
In his sec ion, we will in es iga e he applicabili y o he pa ame ized me hod o di e en ial
equa ions wi h piecewise de i a i es wi h he help o some illus a i e examples. This will be pe o med
wi h he combina ion o de e minis ic and s ochas ic p ocesses whe e he concep s o classical, s ochas ic,
ac ional, and ac al- ac ional a e added. We will s a wi h a simple piecewise Cauchy p oblem in
which he i s pa is wi h classical de e minis ic, he second pa is wi h A angana-Baleanu de i a i e
and las pa is wi h he classical s ochas ic. Ano he simple scena io will be p esen ed wi h classical
de e minis ic, Capu o ac ional de i a i e and he classical s ochas ic. Finally, we will conside an
anxie y model [28] employing he di e en e sions o he piecewise de i a i e.
Example 1. We conside a gene al Cauchy p oblem wi h piecewise de i a i e



























dy
d =− , i 0 ≤ ≤ 1
y(0)=0,
ABC
1Dα
y=− ,i 1≤ ≤ 2
y( 1)=y1,
dy ( )=− d +σydB ( ), i 2≤ ≤T
y( 2)=y2.
(7.1)
The nume ical solu ion o abo e p oblem is ep esen ed by
yk+1=








y0+hk1
j1=0h−1−1
2ξ j1−1
2ξ j1+1i,
i 0 ≤ ≤ 1
,(7.2)































y1−(1−α) k2+1+(1−α)σyck2+1B k2+1−B k2
+hα
Γ(α)
k2
j2=k1+1h−1−1
2ξ j2−1
2ξ j2+1i
×(k2−j2+1)α−(k2−j2)α
+hα−1
Γ(α)
k2
j2=k1+1σycj2B j2+1−B j2
×(k2−j2+1)α−(k2−j2)α,
i 1≤ ≤ 2,
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
753











y2+hk
j3=k2+1h−1−1
2ξ j3−1
2ξ j3+1i
+σy(ck) (B( k+1)−B( k)) ,
i 2≤ ≤T.
The p edic o e ms a e as ollows:



















n˜yk1+1=y0+hk1
j1=0− j1,i 0 ≤ ≤ 1,







˜yk2+1=y1−(1−α) k2+1−hα
Γ(α)
k2
j2=k1+1 j2
×(k2−j2+1)α−(k2−j2)α,i 1≤ ≤ 2,
n˜yk3+1=y2−hk
j3=k2+1 j3,i 2≤ ≤ .
(7.3)
No ing ha he s ochas ic cons an σis aken as 0.1, he ollowing ini ial condi ions a e as ollows:
y(0)=0,(7.4)





















y( 1)=−45 i α=0.9
y( 1)=−47.9 i α=0.8
y( 1)=−48.49 i α=0.6
y( 1)=−48.8 i α=0.4
y( 1)=−48.2 i α=0.2
,





















y( 2)=−145 i α=0.9
y( 2)=−123.6 i α=0.8
y( 2)=−85.2 i α=0.6
y( 2)=−76 i α=0.4
y( 2)=−60.2 i α=0.2
.
In Figu e 1, he nume ical simula ion o he conside ed p oblem wi h piecewise de i a i e is pe o med
by conside ing di e en alues o ac ional o de s.
Figu e 1. The g aphical isualiza ion o he piecewise Cauchy p oblem.
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.

754
Example 2. We conside a gene al Cauchy p oblem wi h piecewise de i a i e



























dy
d =sin , i 0 ≤ ≤ 1
y(0)=1,
ABC
1Dα
y=sin ,i 1≤ ≤ 2
y( 1)=y1,
dy ( )=sin d +σydB ( ), i 2≤ ≤T
y( 2)=y2.
(7.5)
The nume ical solu ion o abo e p oblem is ep esen ed by
yk+1=








y0+hk1
j1=0h1−1
2ξsin  j1+1
2ξsin  j1+1i,
i 0 ≤ ≤ 1
,(7.6)































y1+(1−α)sin  k2+1+(1−α)σyck2+1B k2+1−B k2
+hα
Γ(α)
k2
j2=k1+1h1−1
2ξsin  j2+1
2ξsin  j2+1i
×(k2−j2+1)α−(k2−j2)α
+hα−1
Γ(α)
k2
j2=k1+1σycj2B j2+1−B j2
×(k2−j2+1)α−(k2−j2)α,
i 1≤ ≤ 2,











y2+hk
j3=k2+1h1−1
2ξsin  j3+1
2ξsin  j3+1i
+σy(ck) (B( k+1)−B( k)) ,
i 2≤ ≤T.
The p edic o componen s o each in e al a e calcula ed as



















n˜yk1+1=y0+hk1
j1=0sin  j1,i 0 ≤ ≤ 1,







˜yk2+1=y1+(1−α)sin  k2+1+hα
Γ(α)
k2
j2=k1+1sin  j2
×(k2−j2+1)α−(k2−j2)α,i 1≤ ≤ 2,
n˜yk3+1=y2+hk
j3=k2+1sin  j3,i 2≤ ≤T.
(7.7)
No ing ha he s ochas ic cons an σis aken as 0.1, he ini ial condi ions a e as ollows:
y(0)=1,(7.8)





















y( 1)=1.6 i α=0.9
y( 1)=2 i α=0.8
y( 1)=2.5 i α=0.6
y( 1)=3.2 i α=0.4
y( 1)=3.4 i α=0.2
,





















y( 2)=2 i α=0.9
y( 2)=2.48 i α=0.8
y( 2)=3.36 i α=0.6
y( 2)=4.7 i α=0.4
y( 2)=4.27 i α=0.2
.
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
755
In Figu e 2, he nume ical simula ion o he conside ed p oblem wi h piecewise de i a i e is
pe o med by conside ing di e en alues o ac ional o de s.
Figu e 2. The g aphical isualiza ion o he piecewise Cauchy p oblem.
Example 3. (Ma hema ical modeling o anxie y o ma hema ics) Ins uc ional and social psychologi-
cal en i onmen a e some o he a i ude a ibu e o s uden s and possible ac o s a ec ing he s uden s’
disliking o liking o ma hema ics and ma hema ics anxie y is closely ela ed o a b oad spec um o
cogni i e, psychological, and beha io al p oblems [28,29]. We nex conside a ma hema ical model as-
socia ed wi h he anxie y o ma hema ics [28]. The ma hema ical model unde in es iga ion is p esen ed
by he ollowing:
dS
d =(1−ε)χ+ωR+ρ(1−η)P−θ
N(A+ϕQ)+κS(7.9)
dP
d =εχ −(κ+(1−η)ρ)P
dE
d =θ
N(A+ϕQ)S−(κ+υ)E
dA
d =(1−%)υE−(κ+δ+ς)A
dQ
d =δA−κQ
dR
d =ςA+%υE−(κ+ω)R
and he ini ial condi ions a e aken as
S(0)≥0,P(0)≥0,E(0)≥0,A(0)≥0,Q(0)≥0,R(0)≥0.(7.10)
He e,
S
: anxie y owa ds ma hema ics suscep ible s uden s;
P
: anxie y owa ds ma hema ics p o ec ed
s uden s;
E
: anxie y owa ds ma hema ics exposed s uden s;
A
: s uden s who ha e anxie y owa ds
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
756
ma hema ics;
Q
: s uden s who ha e pe manen anxie y owa ds ma hema ics;
R
: s uden s eco e ed
om anxie y owa ds ma hema ics.
Replacing he classical de i a i e by he piecewise di e en ial ope a o s and simpli ying he model
wi h piecewise de i a i e, we ge he ollowing modi ied model o anxie y:



























dU
d =ψ( ,U), i 0 ≤ ≤ 1
U(0)=U0,
C
1Dα
U=ψ( ,U),i 1≤ ≤ 2
U( i)=U1,
dU ( )=ψ( ,U)d +σiUdBi( ), i 2≤ ≤T
U( 2)=U2,
(7.11)
whe e
U=

S
P
E
A
Q
R

, ψ ( ,U)=

(1−ε)χ+ωR+ρ(1−η)P−θ
N(A+ϕQ)+κS
εχ −(κ+(1−η)ρ)P
θ
N(A+ϕQ)S−(κ+υ)E
(1−%)υE−(κ+δ+ς)A
δA−κQ
ςA+%υE−(κ+ω)R

.(7.12)
Using he sugges ed me hod o each in e al, he nume ical solu ion can be ob ained as
Uk+1=








U0+hk1
j1=0h1−1
2ξψ j1,Uj1+1
2ξψ j1+1,e
Uj1+1i,
i 0 ≤ ≤ 1
,(7.13)

























U1+hα
Γ(α+1)
k2
j2=k1+1h1−1
2ξψ j2,Uj2+1
2ξψ j2+1,e
Uj2+1i
×(k2−j2+1)α−(k2−j2)α
+hα−1
Γ(α+1)
k2
j2=k1+1σiUcj2Bi j2+1−Bi j2
×(k2−j2+1)α−(k2−j2)α,
i 1≤ ≤ 2,











U2+hk
j3=k2+1h1−1
2ξψ j3,Uj3+1
2ξψ j3+1,e
Uj3+1i
+σy(ck) (B( k+1)−B( k)) ,
i 2≤ ≤T.
The p edic o componen s o each in e al a e calcula ed as























ne
Uk1+1=U0+hk1
j1=0ψ j1,Uj1,i 0 ≤ ≤ 1,










e
Uk2+1=U1+hα
Γ(α+1)
k2
j2=k1+1ψ j2,Uj2"(k2−j2+1)α
−(k2−j2)α#,
i 1≤ ≤ 2,
ne
Uk3+1=U2+hk
j3=k2+1ψ j3,Uj3,i 2≤ ≤T.
(7.14)
In Figu e 3, we simula e he nume ical solu ion o he anxie y model wi h piecewise de i a i e o
α=0.9.
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.
757
Figu e 3. The g aphical isualiza ion o he anxie y model wi h piecewise se ing.
We p esen ano he case o ou model since we know ha he model can be modi ied wi h di e en
de i a i es in each in e als. Fo ano he case o ou model, i can be w i en as ollows:



























CF
0Dα
U=ψ( ,U), i 0 ≤ ≤ 1
U(0)=U0,
dU ( )=ψ( ,U)d +σUdB ( ),i 1≤ ≤ 2
U( 1)=U1,
FFP
2Dα,β
U=ψ( ,U), i 2≤ ≤T
U( 2)=U2.
(7.15)
Fo such a model, we ob ain
Uk+1=





















U0+(1−α)ψ k1+1,Uk1+1
+αhk1
j1=0h1−1
2ξψ j1,Uj1+1
2ξψ j1+1,e
Uj1+1i,
i 0 ≤ ≤ 1
(7.16)
Elec onic Resea ch A chi e Volume 32, Issue 2, 733–761.