An Efficient Approach for Solving Differential Equations in the Frame of a New Fractional Derivative Operator

Ci a ion: A ia, N.; Akgül, A.; Seba,
D.; Nou , A.; la Sen, M.D.; Bay am, M.
An E icien App oach o Sol ing
Di e en ial Equa ions in he F ame o
a New F ac ional De i a i e
Ope a o . Symme y 2023,15, 144.
h ps://doi.o g/10.3390/sym15010144
Academic Edi o s: Dong ang Li and
Caloge o Ve o
Recei ed: 17 No embe 2022
Re ised: 10 Decembe 2022
Accep ed: 27 Decembe 2022
Published: 3 Janua y 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
symme y
S
S
A icle
An E icien App oach o Sol ing Di e en ial Equa ions in he
F ame o a New F ac ional De i a i e Ope a o
Nou hane A ia 1,* , Ali Akgül 2,3,4 , Djamila Seba 5, Abdelkade Nou 5, Manuel De la Sen 6
and Mus a a Bay am 7
1
Ecole Na ionale Supé ieu e des Sciences de la Me e de l’Aménagemen du Li o al, Campus Uni e si ai e de
Dely Ib ahim, Bois des Ca s, B.P. 19, Alge 16320, Alge ia
2
Depa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Bei u 1102 2801, Lebanon
3A and Science Facul y, Depa men o Ma hema ics, Sii Uni e si y, Sii 56100, Tu key
4Ma hema ics Resea ch Cen e , Depa men o Ma hema ics, Nea Eas Uni e si y, Nea Eas Boule a d,
Nicosia 99138, Tu key
5Dynamic o Engines and Vib oacous ic Labo a o y, Facul y o Enginee ’s Sciences, Uni e si y M’hamed
Bouga a o Boume des, Boume des 35000, Alge ia
6Depa men o Elec ici y and Elec onics, Ins i u e o Resea ch and De elopmen o P ocesses,
Facul y o Science and Technology, Uni e si y o he Basque Coun y, 48940 Leioa, Bizkaia, Spain
7Depa men o Compu e Enginee ing, Bi uni Uni e si y, Topkapı, Is anbul 34010, Tu key
*Co espondence: [email p o ec ed]
Abs ac :
Recen ly, a new ac ional de i a i e ope a o has been in oduced so ha i p esen s he
combina ion o he Riemann–Liou ille in eg al and Capu o de i a i e. This pape aims o enhance
he ep oducing ke nel Hilbe space me hod (RKHSM, o sho ) o sol ing ce ain ac ional
di e en ial equa ions in ol ing his new de i a i e. This is he i s ime ha he applica ion o he
RKHSM is employed o sol ing some di e en ial equa ions wi h he new ope a o . We illus a e
he con e gence analysis o he applicabili y and eliabili y o he sugges ed app oaches. The esul s
con i m ha he RKHSM inds he ue solu ion. Addi ionally, hese nume ical esul s indica e he
e ec i eness o he p oposed me hod.
Keywo ds:
ac ional di e en ial equa ions; p opo ional-Capu o hyb id ope a o ; cons an p opo ional-
Capu o ope a o ; ep oducing ke nel Hilbe space me hod
MSC: 46E22; 34A08
1. In oduc ion
The subjec o he de i a i e concep has become conside ably impo an and popula
due o i s many applica ions in he b oad disciplines o chemis y, biology, enginee ing,
applied physics, and many o he s. F ac ional calculus has an impo an ole in modeling
a ious ascina ing complex phenomena in he o m o o dina y o pa ial di e en ial
equa ions: o men ion a ew, ime- ac ional Sch ödinge equa ions [
1
,
2
], he ime- ac ional
Benjamin–Bona-Mahony equa ion [
2
], ime- ac ional Bu ge s equa ion [
3
], ime- ac ional
Ko eweg–de V ies equa ion [
4
], and ime- ac ional Ku amo o–Si ashinsky equa ion [
5
].
The symme ic and an i-symme ic solu ions o he ac ional Sch ödinge equa ion ha e
been s udied in [
6
]. In [
7
], he au ho s ex ended he Lie symme y analysis o he ime
ac ional gene alized KdV equa ions. Howe e , using he classical concep o de i a i e
does no ill he gap ha exis s in di e en ields. This la e need mode nized he classical
concep o he de i a i e o he ich concep o ac ional de i a i e [
8
,
9
]. Fo he mos
pa , he non-local beha io s o many p ocesses can be o mula ed as ac ional di e en ial
equa ions (FDEs, o sho ) [
10
]. He e, he exac solu ions o he FDEs a e oo complica ed
o be ound. F om his poin , some well-known nume ical echniques ha e been needed o
ind an app oxima e app oach o hese equa ions [11–13].
Symme y 2023,15, 144. h ps://doi.o g/10.3390/sym15010144 h ps://www.mdpi.com/jou nal/symme y
Symme y 2023,15, 144 2 o 25
Many esea che s wo ked on de eloping he ac ional calculus concep and in es i-
ga ing new ways o de ine ac ional de i a i es which ange om Riemann–Liou ille o
new hyb id p opo ional-Capu o ac ional de i a i es. The hyb id p opo ional-Capu o is
he no el sugges ed ac ional ope a o [
14
]. In hei pape , hey used an elemen a y FDE
o disco e ha hei new ope a o is deeply connec ed wi h he bi a ia e Mi ag–Le le
unc ion, which a ises na u ally om he modeling o ce ain sys ems o he eal wo ld.
In addi ion, hei new hyb id ac ional ope a o may be use ul as a ool o s udy he
anomalous beha io o dynamical sys ems in di e se eal da a, such as chao ic sys ems,
isco-elas ici y, elec o-chemis y, and physics.
In his esea ch, mo i a ed by he wo k o Baleanu e al. [
14
], we apply he RKHSM o
ac ional di e en ial equa ions wi h espec o he new hyb id ac ional ope a o .
The RKHSM is a widely used nume ical me hod o sol ing non-linea sys ems. This
me hod was p oposed in 1908 [
15
] and is an e ec i e nume ical me hod o complex non-
linea p oblems wi hou disc e iza ion. Many esea che s applied i o sol e se e al ypes
o equa ions [
16
–
21
]. I s p incipal ad an ages a e he ea u e ha i is easy o be applied,
especially because i is mesh ee, and i s capabili y o deal wi h di e se complex di e en ial
equa ions. The highligh s o he manusc ip can be summa ized as ollows: (i) an e icien
nume ical echnique is employed o sol ing some di e en ial equa ions wi h he new
ope a o ; (ii) he e ec o he new ac ional de i a i e is shown in he ob ained ou comes;
(iii) he supe io pe o mance o he used me hod is con i med ia compa ing he nume ical
solu ions wi h he ue ones.
In Sec ion 2, we will discuss some basic ools o apply he RKHSM. A e doing he
p epa a ions we need, we will desc ibe how o apply he RKHSM in Sec ion 3. In Sec ion 4,
some applica ions a e p esen ed. Finally, he conclusion is gi en.
2. Ma hema ical Concep s
2.1. The New F ac ional De i a i e Ope a o
De ini ion 1. The Capu o de i a i e o o de o }(τ)is desc ibed by [9]
C
0Dγ
τ}(τ) =RL
0I1−γ
τ}0(τ) = 1
Γ(1−γ)Zτ
0
}0(η)(τ−η)−γdη, (1)
whe e RL
0Iγ
τis he Riemann–Liou ille in eg al which is gi en by he de ini ion below.
De ini ion 2.
Le
}
be an in eg able unc ion. The Riemann–Liou ille in eg al o o de
γ>
0o
}
is gi en by [9]
RL
0Iγ
τ}(τ) = 1
Γ(γ)Zτ
0
}(η)(τ−η)γ−1dη. (2)
The mo e essen ial p ope ies ela ed o he abo e ope a o s can be ob ained om [
22
].
De ini ion 3.
Le 0
≤γ≤
1and
K0
,
K1∈C([0, 1]×R,R+)
.The p opo ional de i a i e
ope a o o o de γo a di e en iable unc ion }is gi en by [23]
PDγ}(τ) = K1(γ,τ)}(τ) + K0(γ,τ)}0(τ), (3)
whe e he unc ions K0and K1sa is y he ollowing condi ions:
lim
γ→0+
K0(γ,τ) = 0; lim
γ→1−
K0(γ,τ) = 1; K0(γ,τ)6=0, 0 <γ≤1; ∀τ∈R, (4)
lim
γ→0+
K1(γ,τ) = 1; lim
γ→1−
K1(γ,τ) = 0; K1(γ,τ)6=0, 0 ≤γ<1; ∀τ∈R. (5)
Rema k 1.
Fo he special cases o
γ
,we can ob ain om
(3)
–
(5)
wo di e en cases. The i s
one, i
γ=
0, hen
(3)
educes o he unc ion i sel , i.e.,
PD0}(τ) = }(τ)
.In he case
γ=
1,
(3)
educes o he s anda d di e en ia ion ope a o , i.e., PD1}(τ) = d
dτ}(τ) = }0(τ).
Symme y 2023,15, 144 3 o 25
Rema k 2.
The p opo ional (con o mable) de i a i e goes back o Khalil e al. [
24
]. In [
25
],
some p ope ies o he con o mable de i a i e we e in es iga ed. In he same yea , a modi ied
p opo ional de i a i e was explo ed in [
26
] wi h mo e p ope ies. De ini ion 3 ep esen s a p ecise
de ini ion o he p opo ional de i a i e. Fo mo e de ails ela ed o he p opo ional ope a o see, o
ins ance, [27].
I is in e es ing o no e ha he p opo ional de i a i e ope a o (3) o o de γcan be
exp essed as a special case whe e
K0
and
K1
depend only on
γ
, hey a e cons an unc ions
wi h espec o τ. So as a consequence, his pa icula case can be de ined as ollows.
De ini ion 4.
Le 0
≤γ≤
1and
K0
,
K1∈C([0, 1],R+)
.The cons an p opo ional (CP, o sho )
de i a i e ope a o o o de γo a di e en iable unc ion }is gi en by [14]
CPDγ}(τ) = K1(γ)}(τ) + K0(γ)}0(τ), (6)
whe e he unc ions K0and K1sa is y he ollowing condi ions:
lim
γ→0+
K0(γ) = 0; lim
γ→1−
K0(γ) = 1; K0(γ)6=0, 0 <γ≤1; (7)
lim
γ→0+
K1(γ) = 1; lim
γ→1−
K1(γ) = 0; K1(γ)6=0, 0 ≤γ<1. (8)
Recen ly, Baleanu e al. [
14
] in oduced he concep o a new hyb id ac ional ope a o
which can be de ined in wo possible ways. One is o combine bo h p opo ional and Capu o
de ini ions. The o he is o combine bo h cons an p opo ional and Capu o de ini ions.
The concep o hese ope a o s is o malized as ollows.
De ini ion 5.
Le he unc ion
}
be di e en iable and le
}
wi h i s de i a i e
}0
be locally
L1
unc ions on R+[14].
1. The p opo ional-Capu o (PC, o sho ) hyb id ope a o o o de γ, o }is gi en by
PC
0Dγ
τ}(τ) =RL
0I1−γ
τhPDγ}(τ)i
=




1
Γ(1−γ)Rτ
0(K1(γ,η)}(η) + K0(γ,η)}0(η))(τ−η)−γdη, 0 <γ<1,
Rτ
0}(η)dη,γ=0,
}0(τ),γ=1.
(9)
2.
The cons an p opo ional-Capu o (CPC, o sho ) hyb id ope a o o o de
γ
, o
}
is gi en by
CPC0Dγ
τ}(τ) =RL
0I1−γ
τhCPDγ}(τ)i
=




1
Γ(1−γ)Rτ
0(K1(γ)}(η) + K0(γ)}0(η))(τ−η)−γdη, 0 <γ<1,
Rτ
0}(η)dη,γ=0,
}0(τ),γ=1.
(10)
P oposi ion 1. The hyb id CPC and PC ac ional ope a o s a e non-local and singula [14].
Theo em 1. The Laplace ans o m o he hyb id CPC ope a o CPC
0Dγ
τis ep esen ed by [14]
LhCPC
0Dγ
τ}(τ)i=K1(γ)
s+K0(γ)sγˆ
}(s)−K0(γ)sγ−1}(0), (11)
whe e he unc ion
}(τ)
is di e en iable.
}
, wi h i s de i a i e
}0
, a e locally
L1
unc ions on
R+
,
and ˆ
}(s)exis s.
Symme y 2023,15, 144 4 o 25
Among he main p ope ies o he PC and CPC ac ional ope a o s, we men ion he
ollowing ones, namely, he in e sion ela ions:
•PC
0Iγ
τPC
0Dγ
τ}(τ) = }(τ)−exp−Rτ
0
K1(γ,s)
K0(γ,s)ds}(0),
•CPC
0Iγ
τCPC
0Dγ
τ}(τ) = }(τ)−exp−K1(γ)
K0(γ)τ}(0),
•PC
0Dγ
τPC
0Iγ
τ}(τ) = CPC
0Dγ
τCPC
0Iγ
τ}(τ) = }(τ)−τ−γ
Γ(1−γ)lim
τ→0
RL
0Iγ
τ}(τ),
whe e
PC
0Iγ
τ
and
CPC
0Iγ
τ
deno e he in e se o he ope a o s
PC
0Dγ
τ
and
CPC
0Dγ
τ
, espec i ely.
They we e cons uc ed in [14] as ollows.
P oposi ion 2.
The in e se ope a o s o he CPC and PC ac ional de i a i es a e de ined, espec-
i ely, as [14]
PC
0Iγ
τ}(τ) = Zτ
0exp−Zτ
η
K1(γ,s)
K0(γ,s)dsRL
0D1−γ
η}(η)
K0(γ,η)dη, (12)
CPC
0Iγ
τ}(τ) = 1
K0(γ)Zτ
0exp−K1(γ)
K0(γ)(τ−η)RL
0D1−γ
η}(η)dη. (13)
These sa is y he in e sion ela ions which a e men ioned jus abo e. The
RL
0Dγ
τ
deno es he
Riemann–Liou ille de i a i e which is de ined as ollows:
RL
0Dγ
τ}(τ) = 1
Γ(1−γ)
d
dτZτ
0
}(η)(τ−η)−γdη. (14)
Fo mo e de ails on CPC and PC hyb id ac ional ope a o s see [14].
Rema k 3. O special in e es , Baleanu e al. [14] ealized ha he speci ic case
K0(γ,τ) = γτ1−γ,K1(γ,τ) = (1−γ)τγ. (15)
will no be use ul in applica ions because o he lack o dimensional ag eemen in
(3)
while his is
impo an o physical consis ency.
Rema k 4. We shall pay special a en ion in his pape o wo speci ic cases when
1. Fo any σ∈(0, +∞),we ake
K0(γ) = γσ1−γ,K1(γ) = (1−γ)σγ(16)
2. Fo any σ∈(0, +∞),we ake
K0(γ) = γσ1+γ,K1(γ) = (1−γ)σγ(17)
2.2. The Rep oducing Ke nel Theo y
The ollowing a e some undamen al de ini ions and heo ems equi ed om he
ep oducing ke nel heo y.
De ini ion 6.
We say ha a unc ion
K:S×S→C
is a ep oducing ke nel o he space
H
p o ided [28]:
1. K(·, )∈H,∀ ∈S.
2. h ,K(·, )i= ( ),∀ ∈H and ∀ ∈S.
whe e H is a Hilbe space o e S and S 6=∅.
Rema k 5. No e ha he asse ion (2)is called he ep oducing p ope y (RP).
Rema k 6. An RKHS “H” is a Hilbe space endowed wi h an RK “K”.
Symme y 2023,15, 144 5 o 25
De ini ion 7
([
28
])
.
The unc ion space
W2
2[
0,
T]
consis s o all unc ions
o which
and
0
a e
absolu ely con inuous unc ions on [0, T], 00 ∈L2[0, T],and (0) = 0.
De ini ion 8 ([28]).I ,g∈W2
2[0, T], hen he inne p oduc and no m a e
h ,giW2
2=
1
∑
ı=0
(ı)(0)g(ı)(0) + ZT
0 (m)( )g(m)( )d ,
and
k kW2
2=qh , iW2
2.
Theo em 2. We ob ain he RK unc ion Sη( )o W2
2[0, T]as:
Sη( ) =  η+1/2 η 2−1/6 3, ≤η,
−1/6 η3+1/2 η2+ η, >η.(18)
P oo . We mus p o e
 ,SηW2
2= (η).
We ha e
 ,SηW2
2= (0)Sη(0) + 0(0)S0
η(0) + 0(0)S0
η(0) + ZT
0 00( )S00
η( )d .
Applying in eg a ion by pa s, we ob ain:
 ,SηW2
2= (0)Sη(0) + 0(0)S0
η(0) + 0(T)S00
η(T)− 0(0)S00
η(0)−ZT
0 0( )S(3)
η( )d .
Since ( )∈W2
2[0, T], we ha e
(0) = 0. (19)
Then
 ,SηW2
2= 0(0)S0
η(0) + 0(T)S00
η(T)− 0(0)S00
η(0)−ZT
0 0( )S(3)
η( )d .
We need o compu e S0
η(0),S00
η(0), and S00
η(T):
S0
η(0) = η,
S00
η(0) = η,
S00
η(T) = 0.
By using he abo e equa ions, we ob ain:
 ,SηW2
2=−ZT
0 0( )S(3)
η( )d . (20)
We ha e
S(3)
η( ) = −1 , ≤η,
0 , >η.

Symme y 2023,15, 144 6 o 25
Thus
 ,SηW2
2=−Zη
0 0( )S(3)
η( )d −ZT
η 0( )S(3)
η( )d
=Zη
0 0( )d
= (η)− (0),
and, since ( )∈W2
2[0, T], we deduce
 ,SηW2
2= (η).
De ini ion 9
([
28
])
.
The unc ion space
W1
2[
0,
T]
consis s o all unc ions
o which
is absolu ely
con inuous unc ion on [0, T]and 0∈L2[0, T].
De ini ion 10 ([28]).I ,g∈W1
2[0, T], hen he inne p oduc and no m a e
h ,giW1
2= (0)g(0) + ZT
0 0( )g0( )d ,
and
k kW1
2=qh , iW1
2.
Theo em 3. We ob ain he RK unc ion Rη( )o W1
2[0, T]as:
Rη( ) = 1+ , ≤η,
η+1 , >η.(21)
P oo . We mus p o e
 ,RηW1
2= (η).
We ha e
 ,RηW1
2= (0)Rη(0) + ZT
0 0(x)R0
η(x)dx.
We need o compu e Rη(0):
Rη(0) = 1,
By using he abo e equa ion, we ob ain:
 ,RηW1
2= (0) + ZT
0 0( )R0
η( )d .
We ha e
R0
η( ) = 1 , ≤η,
0 , >η.
Thus
 ,RηW1
2= (0) + Zη
0 0( )R0
η( )d +ZT
η 0( )R0
η( )d
= (0) + Zη
0 0( )d
= (0) + (η)− (0),
Symme y 2023,15, 144 7 o 25
so, we deduce
 ,RηW1
2= (η).
3. The RKHS App oach
Conside ing he ollowing ac ional ini ial alue p oblem:
CPC
0Dγ
( ) = φ( , ( )), ∈[0, T],
(0) = µ.(22)
whe e CPC
0Dγ
( )is gi en in (10).
One skill ul way o in es iga e he conside ed p oblem by using RKHSM is o homog-
enize he ini ial condi ion (0) = µ. To do so, he ans o ma ion has he o m:
g( ) = ( )−µ.
Then, (22) becomes
CPC
0Dγ
g( ) = Λ( ,g( )), ∈[0, T],
g(0) = 0. (23)
whe e Λ( ,g( )) = φ( ,g( ) + µ)−µ 1−γK1(γ)/Γ(2−γ).
The i s s ep is o de ine a linea ope a o O:W2
2[0, T]→W1
2[0, T]de ined by
Og( ) = CPC
0Dγ
g( ). (24)
Theo em 4. The ope a o O:W2
2[0, T]→W1
2[0, T]is bounded and linea .
P oo . Fo checking he linea i y, le g( ),m( )∈W2
2[0, T]. Then,
O(g+m)( ) = CPC
0Dγ
(g+m)( ),
=1
Γ(1−γ)Z
0K1(γ)(g+m)(τ) + K0(γ)(g+m)0(τ)( −τ)−γdτ,
=CPC
0Dγ
g( ) + CPC
0Dγ
m( ),
=Og( ) + Om( ).
Addi ionally, le g( )∈W2
2[0, T]and ξ∈R. Then
O(ξg)( ) = CPC
0Dγ
(ξg)( ),
=1
Γ(1−γ)Z
0K1(γ)(ξg)(τ) + K0(γ)(ξg)0(τ)( −τ)−γdτ,
=ξCPC
0Dγ
g( ),
=ξOg( ).
We can now p o e ha Ois bounded. This shows ha
kOgkW1
2≤ΞkgkW2
2, wi h Ξ>0.
F om De ini ion 10, we ha e
kOg( )k2
W1
2=hOg( ),Og( )iW1
2=[Og(0)]2+ZT
0Og0( )2d .
Symme y 2023,15, 144 8 o 25
By i ue o he RP, we ha e
g( ) = hg(),S ()iW2
2.
In addi ion,
Og( ) = hg(),OS ()iW2
2,
Og0( ) = hg(),∂ (OS ())iW2
2.
Using he Schwa z inequali y and he con inui y o S () o ob ain
|Og( )|=hg(),OS ()iW2
2≤kgkW2
2kOS ()kW2
2≤Ξ1kgkW2
2. (25)
In he same way,
Og0( )≤Ξ2kgkW2
2.
Hence
kOg( )k2
W2
2≤Ξ2
1kgk2
W2
2+ZT
0
Ξ2
2kgk2
W2
2d ,
=Ξ2
1+TΞ2
2kgk2
W2
2. (26)
F om
(26)
, we conclude ha
Hence
kOg( )k2
W2
2≤Ξ2
1kgk2
W2
2+ZT
0
Ξ2
2kgk2
W2
2d ,
=Ξ2
1+TΞ2
2kgk2
W2
2.(26)
F om (26), we conclude ha kOg(τ)kW1
2≤ΞkgkW2
2,whe e Ξ=Ξ2
1+TΞ2
2.
By applying (24), we can ew i e (23) as ollow



Og( ) = Λ( , g( )), ∈[0, T],
g(0) = 0.(27)
whe e Λ( , g( )) = φ( , g( ) + µ)−(µ 1−γK1(γ)) /Γ(2 −γ).
P io o cons uc he nume ical solu ion o (27), we should i s cons uc he o hogonal
unc ion sys em o W2
2[0, T].Fo his, le us in oduce he use ul unc ions:
κı( ) = R ı( )and ψı( ) = O∗κı( ),
whe e
• R ı( ) ep esen s he RKF associa ed wi h W1
2[0, T].
•O∗is he o mal adjoin o O.
• { ı}∞
ı=1 is a dense coun able se in [0, T].
The G am-Schmid ’s p ocess gi es he ollowing o hono mal sys em {¯
ψı}∞
ı=1:
¯
ψı( ) =
ı
X
k=1
$ıkψk( ), $ıı >0, ı = 1,2,.... (28)
He e {ψı}∞
ı=1 deno es a unc ion sys em in W2
2[0, T]whe e i s exp ession can be de e mined by
he ollowing way:
$ı =






1
kψ1k o ı== 1,
1
eı o ı=6= 1,
−1
eıPı−1
k=Cık$k o ı > ,
(29)
whe e eı=qkψık2−Pı−1
k=1 C2
ık, Cık =ψı,¯
ψkW2
2
.
12
.
By applying (24), we can ew i e (23) as ollows
Og( ) = Λ( ,g( )), ∈[0, T],
g(0) = 0. (27)
whe e Λ( ,g( )) = φ( ,g( ) + µ)−µ 1−γK1(γ)/Γ(2−γ).
P io o cons uc ing he nume ical solu ion o
(27)
, we should i s cons uc he
o hogonal unc ion sys em o W2
2[0, T]. Fo his, le us in oduce he use ul unc ions:
κı( ) = R ı( )and ψı( ) = O∗κı( ),
whe e
•R ı( ) ep esen s he RKF associa ed wi h W1
2[0, T].
•O∗is he o mal adjoin o O.
•{ ı}∞
ı=1is a dense coun able se in [0, T].
The G am–Schmid p ocess gi es he ollowing o hono mal sys em {¯
ψı}∞
ı=1:
¯
ψı( ) =
ı
∑
k=1
ıkψk( ), ıı >0, ı=1, 2, . . . . (28)
He e,
{ψı}∞
ı=1
deno es a unc ioning sys em in
W2
2[
0,
T]
whe e i s exp ession can be de e -
mined by he ollowing way:
ı=




1
kψ1k o ı==1,
1
eı o ı=6=1,
−1
eı∑ı−1
k=Cık k o ı>,
(29)
whe e eı=qkψık2−∑ı−1
k=1C2
ık,Cık =hψı,¯
ψkiW2
2.
Symme y 2023,15, 144 9 o 25
Rema k 7. No ha he o mula o ψı( )is gi en by
ψı( ) = O∗κı( ) = hO∗κı(η),S (η)iW2
2=hκı(η),OS (η)iW1
2=hR ı(η),OS (η)iW1
2=OηS (η)|η= ı.
The ope a o Oηmeans ha Ois applied o η a iables.
Theo em 5.
Assume
{ ı}∞
ı=1
is dense on
[
0,
T]
, hen
{ψı}∞
ı=1
is he comple e sys em o he space
W2
2[0, T].
P oo . We see easily ha ψı( )∈W2
2[0, T]. The e o e, o each ixed g( )∈W2
2[0, T],
hg( ),ψı( )iW2
2=0, ı=1, 2, . . . ,
Since
hg( ),ψı( )iW2
2=hg( ),O∗κı( )iW2
2=hOg( ),κı( )iW1
2=Og( ı) = 0.
Due o he densi y o { ı}∞
ı=1in [0, T], one can ge
Og( ) = 0.
Addi ionally, he exis ence o O−1implies.
g( ) = 0.
Lemma 1. Assume g( )∈W2
2[0, T], hen


g(ı)( )

C≤zkg( )kW2
2,ı=0, 1.
zis non-nega i e and kg( )kC=max
∈[0,T]|g( )|.
P oo . ∀ ∈[0, T]we ha e
g(ı)( ) = Dg(),∂(ı)
S ()EW2
2
,ı=0, 1,
and 

∂(ı)
S 

W2
2
≤zı,ı=0, 1.
As a esul ,
g(ı)( )=Dg(),∂(ı)
S ()EW2
2
≤

∂(ı)
S 

W2
2
kgkW2
2≤zıkgkW2
2,ı=0, 1. (30)
He e z=max
ı=0,1{zı}.
Theo em 6.
Assume
{ ı}∞
ı=1
is dense in
[
0,
T]
and he e exis s a unique solu ion
g( )
o
(27)
in
W2
2[0, T], hen he solu ion’s ep esen a ion o (27)is gi en by
g( ) =
∞
∑
ı=1
ı
∑
k=1
ıkΛ( k,g( k)) ¯
ψı( ), (31)
Symme y 2023,15, 144 16 o 25
Figu e 3. Compa ison o nume ical solu ions o he RKHSM by he ES o he FS o Example 1.

Symme y 2023,15, 144 17 o 25
(a) Exac and he RKHSM’s solu ions.
(b) Absolu e e o o he RKHSM.
Figu e 4. Compa ison o nume ical solu ions o he RKHSM by he ES o he SS o Example 1.
Symme y 2023,15, 144 18 o 25
(a) Exac and he RKHSM’s solu ions.
(b) Absolu e e o o he RKHSM.
Figu e 5. Compa ison o nume ical solu ions o he RKHSM by he ES o he SS o Example 1.
Symme y 2023,15, 144 19 o 25
Figu e 6. Compa ison o nume ical solu ions o he RKHSM by he ES o he SS o Example 1.
Example 2. Conside ing he ollowing simple p oblem:
CPC
0Dγ
( ) = , 0 < ≤1,
(0) = 0. (35)
The exac solu ion o he abo e p oblem akes he ollowing o m
( ) = exp− K1(γ)
K0(γ) γ−Γ(1+γ)+Γ1+γ,− K1(γ)
K0(γ)− K1(γ)
K0(γ)−γ
Γ(1+γ)K1(γ).
1. Fi s si ua ion (FS, o sho ): K0(γ) = γσ1−γ,K1(γ) = (1−γ)σγ,σ=1
2.
2. Second si ua ion (SS, o sho ): K0(γ) = γσ1+γ,K1(γ) = (1−γ)σγ,σ=1
2.
This RKHSM is es ed wi h he g id poin s
ı=ı
n
,
ı=
1, 2,
. . .
,
n
.The ES, AS, AE, and RE
o
(35)
a e shown in Tables 5–8when
γ∈ {
0.25, 0.5, 0.75, 0.9
}
and
∈[
0, 1
]
o bo h he FS and
SS. Addi ionally, he compa isons be ween he ES and RKHSM’s solu ion o bo h cases a e depic ed
in Figu e 7 o he FS and Figu e 8 o he SS, when
γ=
0.25, 0.5, 0.75, 0.9. F om he esul s, we
can see ha he nume ical solu ions a e ound o be in excellen ag eemen wi h he exac solu ions.
Symme y 2023,15, 144 20 o 25
Table 5. Fi s and second si ua ions: esul s o Example 2when γ=0.25.
Fi s Si ua ion Second Si ua ion
ES AS AE RE ES AS AE RE
0.1 0.278390226 0.278390225 8.0 ×10−10 2.873664108 ×10−90.366578617 0.366578616 1.7 ×10−94.637477257 ×10−9
0.2 0.559438322 0.559438324 2.5 ×10−94.468767876 ×10−90.694736216 0.694736218 1.3 ×10−91.871213807 ×10−9
0.3 0.794667664 0.794667667 3.2 ×10−94.026840585 ×10−90.941462921 0.941462918 2.9 ×10−93.080312495 ×10−9
0.4 0.986000956 0.986000987 3.1 ×10−83.184581091 ×10−81.125538939 1.125538968 2.9 ×10−82.576543467 ×10−8
0.5 1.141251776 1.141251787 1.1 ×10−89.638539217 ×10−91.265695498 1.265695503 5.0 ×10−93.950397238 ×10−9
0.6 1.268231856 1.268231894 3.8 ×10−82.996297548 ×10−81.375697794 1.375697830 3.6 ×10−82.616853800 ×10−8
0.7 1.373458393 1.373458428 3.5 ×10−82.548311633 ×10−81.464920275 1.464920304 2.9 ×10−81.979629915 ×10−8
0.8 1.462030835 1.462030879 4.4 ×10−83.009512450 ×10−81.539590658 1.539590698 4.0 ×10−82.598093187 ×10−8
0.9 1.537826398 1.537826474 7.6 ×10−84.942040278 ×10−81.603828534 1.603828600 6.6 ×10−84.115153123 ×10−8
1 1.603753900 1.603753849 5.1 ×10−83.180039032 ×10−80.001239376 1.660377912 3.0 ×10−81.806817547 ×10−8
Table 6. Fi s and second si ua ions: esul s o Example 2when γ=0.5.
Fi s Si ua ion Second Si ua ion
ES AS AE RE ES AS AE RE
0.1 0.064667410 0.064667411 9.9 ×10−10 1.530910242 ×10−80.124390490 0.124390493 2.9 ×10−92.331367938 ×10−8
0.2 0.175914718 0.175914719 1.5 ×10−98.526859025 ×10−90.326098542 0.326098549 6.5 ×10−91.993262514 ×10−8
0.3 0.311030443 0.311030447 4.4 ×10−91.414652521 ×10−80.556730844 0.556730856 1.2 ×10−82.227288130 ×10−8
0.4 0.461172981 0.461172979 1.7 ×10−93.686252383 ×10−90.798597356 0.798597370 1.4 ×10−81.803161493 ×10−8
0.5 0.621108506 0.621108511 4.7 ×10−97.567115817 ×10−91.042442942 1.265695503 1.9 ×10−81.822641756 ×10−8
0.6 0.787336310 0.787336312 2.1 ×10−92.667221077 ×10−91.283029637 1.375697830 3.6 ×10−82.805858881 ×10−8
0.7 0.957370780 0.957370788 8.2 ×10−98.565124582 ×10−91.517362373 1.464920304 4.2 ×10−82.767961161 ×10−8
0.8 1.129387212 1.129387215 3.0 ×10−92.656307746 ×10−91.743794486 1.539590698 4.6 ×10−82.637925603 ×10−8
0.9 1.302020083 1.302020093 1.0 ×10−87.680373084 ×10−91.961520086 1.603828600 8.3 ×10−84.231412368 ×10−8
1 1.474236919 1.474236919 0 0 2.170265034 2.170264989 4.5 ×10−82.073479473 ×10−8
Symme y 2023,15, 144 21 o 25
Table 7. Fi s and second si ua ions: esul s o Example 2when γ=0.75.
Fi s Si ua ion Second Si ua ion
ES AS AE RE ES AS AE RE
0.1 0.01738193388 0.01738193250 1.38 ×10−97.939277698 ×10−80.048404880 0.048404879 1.1 ×10−92.355134427 ×10−8
0.2 0.05796964429 0.05796964023 4.06 ×10−97.003665539 ×10−80.158979414 0.158979408 5.5 ×10−93.459567419 ×10−8
0.3 0.1168624847 0.1168624762 8.5 ×10−97.273506140 ×10−80.315694535 0.315694524 1.1 ×10−83.579409442 ×10−8
0.4 0.1917111802 0.1917111584 2.2 ×10−81.137127213 ×10−70.510261243 0.510261205 3.8 ×10−87.427567849 ×10−8
0.5 0.2809210058 0.2809209865 1.9 ×10−86.870258757 ×10−80.736856811 0.736856784 2.7 ×10−83.596356798 ×10−8
0.6 0.3832748550 0.3832748188 3.6 ×10−89.444919104 ×10−80.990974626 0.990974570 5.6 ×10−85.600547032 ×10−8
0.7 0.4977838273 0.4977837878 4.0 ×10−87.935171421 ×10−81.268951107 1.268951049 5.8 ×10−84.570704078 ×10−8
0.8 0.6236125669 0.6236125196 4.7 ×10−87.584837527 ×10−81.567719773 1.567719703 7.0 ×10−84.465083697 ×10−8
0.9 0.7600365805 0.7600365012 7.9 ×10−81.043370833 ×10−71.884664677 1.884664559 1.2 ×10−76.261060731 ×10−8
1 0.9064155258 0.9064155820 5.6 ×10−86.200246840 ×10−82.217524640 2.217524727 8.7 ×10−83.923293497 ×10−8
Table 8. Fi s and second si ua ions: esul s o Example 2when γ=0.9.
Fi s Si ua ion Second Si ua ion
ES AS AE RE ES AS AE RE
0.1 0.0081861790 0.0081861771 1.9 ×10−92.313655737 ×10−70.028351089 0.028351084 5.2 ×10−91.844726329 ×10−7
0.2 0.0304849118 0.0304849070 4.8 ×10−91.567988792 ×10−70.105006807 0.105006793 1.4 ×10−81.314200522 ×10−7
0.3 0.0657211749 0.0657211649 1.0 ×10−81.526144354 ×10−70.225161367 0.225161338 2.9 ×10−81.279082660 ×10−7
0.4 0.1132757197 0.1132756975 2.2 ×10−81.959819815 ×10−70.386004901 0.386004835 6.7 ×10−81.725366693 ×10−7
0.5 0.1727089417 0.1727089198 2.2 ×10−81.268029309 ×10−70.585395620 0.585395557 6.4 ×10−81.088152999 ×10−7
0.6 0.2436739618 0.2436739235 3.8 ×10−81.571772368 ×10−70.821549439 0.821549325 1.1 ×10−71.380318635 ×10−7
0.7 0.3258809599 0.3258809169 4.3 ×10−81.319500225 ×10−71.092910710 1.092910583 1.3 ×10−71.162034545 ×10−7
0.8 0.4190787537 0.4190787040 5.0 ×10−81.185934614 ×10−71.398085171 1.398085022 1.5 ×10−71.065743369 ×10−7
0.9 0.5230442467 0.5230441624 8.4 ×10−81.611718330 ×10−71.735800499 1.735800248 2.5 ×10−71.446018711 ×10−7
1 0.6375755634 0.6375756254 6.2 ×10−89.724337562 ×10−82.104880901 2.104881081 1.8 ×10−78.551552723 ×10−8

Symme y 2023,15, 144 22 o 25
(a) Exac solu ions o di e en alues o γ.
(b) App oxima e solu ions o di e en alues o γ.
Figu e 7. Compa ison o nume ical solu ions o he RKHSM by he ES o he FS o Example 2.
Symme y 2023,15, 144 23 o 25
(a) Exac solu ions o di e en alues o γ.
(b) App oxima e solu ions o di e en alues o γ.
Figu e 8. Compa ison o nume ical solu ions o he RKHSM by he ES o he SS o Example 2.
5. Conclusions
In his pape , an e icien me hod has been applied success ully o sol ing FDEs. The
app oxima e solu ion
gn( )
and i s de i a i e bo h con e ge uni o mly. The accu acy and
applicabili y o he p oposed me hod a e alida ed by compu ing he nume ical solu ions
a many g id poin s. The boundedness o he linea ope a o is demons a ed. The esul s
show ha he p oposed me hod is a powe ul ool o app oxima e many o he non-linea
p oblems which a e desc ibed by he new hyb id ac ional de i a i e ope a o . This
esea ch opens he way o he use o he RKHSM o s udy he men ioned p oblem o
a ious new ac ional de i a i es. As pa o ou pu pose, we plan o apply he RKHSM
Symme y 2023,15, 144 24 o 25
o mul idimensional ac ional pa ial di e en ial equa ions ha a e desc ibed wi h CPC
de i a i e, which will be new in he li e a u e.
Au ho Con ibu ions:
Concep ualiza ion, D.S.; me hodology, N.A.; so wa e, A.A.; alida ion,
A.A. and D.S.; o mal analysis, N.A.; in es iga ion, A.A.; esou ces, M.D.l.S.; da a cu a ion, A.A.;
w i ing—o iginal d a p epa a ion, N.A.; w i ing— e iew and edi ing, N.A.; isualiza ion, A.A.;
supe ision, M.B.; p ojec adminis a ion, A.A., M.D.l.S., M.B., D.S. and A.N.; unding acquisi ion,
M.D.l.S. and M.B. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Da a A ailabili y S a emen : Da a a e included wi hin his esea ch.
Acknowledgmen s:
The au ho s a e g a e ul o he Basque Go e nmen o i s suppo h ough G an
IT1555-22 and o MCIN/AEI 269.10.13039/501100011033 o G an PID2021-1235430B-C21/C22.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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