A Computational Scheme for the Numerical Results of Time-Fractional Degasperis–Procesi and Camassa–Holm Models

Ci a ion: Nadeem, M.; Ja a i, H.;
Akgül, A.; De la Sen, M. A
Compu a ional Scheme o he
Nume ical Resul s o Time-F ac ional
Degaspe is–P ocesi and
Camassa–Holm Models. Symme y
2022,14, 2532. h ps://doi.o g/
10.3390/sym14122532
Academic Edi o : Se kan A aci
Recei ed: 12 No embe 2022
Accep ed: 25 No embe 2022
Published: 30 No embe 2022
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symme y
S
S
A icle
A Compu a ional Scheme o he Nume ical Resul s o
Time-F ac ional Degaspe is–P ocesi and
Camassa–Holm Models
Muhammad Nadeem 1, Hossein Ja a i 2,3,4 , Ali Akgül 5,6 and Manuel De la Sen 7,*
1School o Ma hema ics and S a is ics, Qujing No mal Uni e si y, Qujing 655011, China
2Depa men o Ma hema ical Sciences, Uni e si y o Sou h A ica, UNISA, P e o ia 0003, Sou h A ica
3Depa men o Ma hema ics and In o ma ics, Aze baijan Uni e si y, Jeyhun Hajibeyli, 71,
AZ1007 Baku, Aze baijan
4Depa men o Medical Resea ch, China Medical Uni e si y Hospi al, China Medical Uni e si y,
Taichung 110122, Taiwan
5Depa men o Ma hema ics, A and Science Facul y, Sii Uni e si y, 56100 Sii , Tu key
6Depa men o Ma hema ics, Ma hema ics Resea ch Cen e , Nea Eas Uni e si y, Nea Eas Boule a d,
Me sin 10, 99138 Nicosia, Tu key
7Depa 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, Spain
*Co espondence: [email p o ec ed]
Abs ac :
This a icle p esen s an idea o a new app oach o he soli a y wa e solu ion o he
modi ied Degaspe is–P ocesi (mDP) and modi ied Camassa–Holm (mCH) models wi h a ime-
ac ional de i a i e. We combine Laplace ans o m (
L
T) and homo opy pe u ba ion me hod
(HPM) o o mula e he idea o he Laplace ans o m homo opy pe u ba ion me hod (
L
HPTM).
This s udy is conside ed unde he Capu o sense. This p oposed s a egy does no depend on any
assump ion and es ic ion o a iables, such as in he classical pe u ba ion me hod. Some nume ical
examples a e demons a ed and hei esul s a e compa ed g aphically in 2D and 3D dis ibu ion.
This app oach p esen s he i e a ions in he o m o a se ies solu ions. We also compu e he absolu e
e o o show he e ec i e pe o mance o his p oposed scheme.
Keywo ds:
Laplace ans o m; homo opy pe u ba ion me hod; mDP and mCH models; se ies solu ion
1. In oduc ion
Symme ies play an impo an ole in he s udy o nonlinea physical phenomena,
including he s udy o a di e en ial p oblem in a eal-wo ld p oblem. Recen ly, a ious
physical phenomena in ol ing ac ional di e en ial equa ions ha e become impo an
s udy o some applica ions o science and enginee ing. A a ie y o undamen al ac ional
de i a i e de ini ions we e p esen ed by A angana–Baleanu, Capu o–Fab izio, Liou ille–
Capu o, Riemann–Liou ille, and Hadama d, among o he s [
1
–
4
]. The Capu o ac ional
de i a i e compu es an o dina y de i a i e i s , ollowed by a ac ional in eg al and
hen p o ide he desi ed o de o a ac ional de i a i e. The Riemann–Liou ille ac ional
de i a i e is compu ed in e e se o de . The Capu o ac ional de i a i e only pe mi s he
p esence o adi ional ini ial and bounda y condi ions, whe eas he Riemann–Liou ille
ac ional de i a i e pe mi s ini ial condi ions in e ms o ac ional in eg als and hei
de i a i es [
5
]. Nume ous applica ions in science and enginee ing ha e been desc ibed
wi h he nonlinea models such as as ophysics, hyd ological, nuclea enginee ing, me e-
o ology, and as obiology [
6
,
7
]. The majo i y o he nonlinea models o ac ional o de
a e s ill challenging o esol e. As a esul , hese models a e c ucial o examining he
p ecise and nume ical solu ions. The complexi y o hese nonlinea ac ional issues can be
signi ican ly educed h ough he use o in eg al ans o m echniques. The e a e nume ous
Symme y 2022,14, 2532. h ps://doi.o g/10.3390/sym14122532 h ps://www.mdpi.com/jou nal/symme y
Symme y 2022,14, 2532 2 o 11
widely used and success ul s a egies o deal wi h hese nonlinea beha io when hey ha e
ac ional o de such as Laplace ans o m [
8
],
F
-Expansion scheme [
9
], (
´
G
/G)-expansion
app oach [
10
], Sumudu ans o m [
11
], T ial equa ion app oach [
12
], Va ia ional i e a ion
me hod (VIM) [
13
], Sub-equa ion [
14
], HPM [
15
], and Fini e di e ence scheme [
16
]. The e
a e se e al models o ocean wa e wa es which a e nonlinea dispe si e by na u e.
In his wo k, we conside a amily o impo an physically equa ion which is called a
modi ied β-equa ion in he ollowing o m [17]:
Dαϑθ−ϑςςθ + (β+1)ϑ2ϑς−βϑςϑςς −ϑϑςςς =0. (1)
Se ing β=3, we can ob ain mDP model such as
Dαϑθ−ϑςςθ +4ϑ2ϑς−3ϑςϑςς −ϑϑςςς =0, (2)
and β=2 in Equa ion (1), and we can ob ain mCH model such as
Dαϑθ−ϑςςθ +3ϑ2ϑς−2ϑςϑςς −ϑϑςςς =0, (3)
whe e
ϑ
symbolizes a ho izon al elemen o he luid eloci y,
ς
, and
θ
ep esen s he spa ial
and empo al elemen s. Liu and Ouyang [
18
] used some nume ical simula ions and de i ed
some new soli a y wa e solu ions o his model. The incomp essible Eule equa ion is
app oxima ed by he mDP and mCH models, which was ound o be ully in eg able wi h
a Lax pai and appea s in shallow wa e . [
19
]. Behe a and Meh a [
20
] de eloped wa ele
op imized ini e di e ence me hod o in es iga e he app oxima e solu ions o mDP and
mCH models. Dubey e al. [
21
] in oduced a q-homo opy analysis app oach combined
wi h a new app oach o ob ain he signi ican esul s ime- ac ional mDP and mCH models.
Yousi e al. [
22
] in oduced wo app oaches, namely, VIM and HPM o sol ing mDP and
mCH models, and ounded he esul s in good ag eemen . Kade and La i [
23
] used a Lie
symme y echnique o p esen ew unique b igh and da k soli on esul s o he mDP and
mCH models in he shape o Jacobi ellip ic unc ions and Weie s ass ellip ic unc ions.
Ano he e ec i e me hod o sol ing nonlinea challenges has been de eloped by
Ji-Huan He [
24
,
25
] wi h some ecen de elopmen s. La e , se e al scien is s demons a ed
he eliabili y and accu acy o his s a egy [
26
–
28
]. Gup a e al. [
29
] de i ed he analy ical
esul s o he amily o ime ac ional mCH model. Baleanu and Wu [30] p o ided some
undamen al esul s o ac ional di e ence equa ions by use o he
L
T and showed ha
L
T is e y use ul in s abili y analysis and explici solu ions o linea sys ems. Khu i and
Say y [
31
] p esen ed a me hod o pa icula a ie ies o di e en ial challenges. La e ,
Anjum and He [
32
] used his s a egy o add ess he nonlinea oscilla o issue. Nadeem
and Li [33] p oposed an idea ha has excellen esul s o he nonlinea ib a ion sys ems
and hen Zhang e al. [
34
] modi ied his scheme o ackle he p esence o nonlinea models;
howe e , all o hese ha e some es ic ions and p esump ions.
In his cu en s udy, we cons uc an idea o a new scheme ha enables us o ob ain
he app oxima e solu ion o mDP and mCH models wi h ac ional o de in he Capu o
sense. This scheme
L
T coupled wi h HPM is easy o implemen , s aigh o wa d, and
e ec i e o nonlinea p oblems in science and enginee ing. This a icle is o ganized as
ollows: In Sec ion 2, we de ine a ew undamen al cha ac e is ics o calculus heo y. We
p esen he o mula ion o
L
HPTM o ob ain he solu ion o mDP and mCH models in
Sec ion 3. In Sec ion 4, we demons a e he easibili y and pe o mance o
L
HPTM by
conside ing some nume ical examples and compa ed wi h he exac solu ion. Finally, we
p esen he esul s and discussion and e eal he conclusion in Sec ions 5and 6.
2. P elimina y View
This sec ion explains a ew ac ional p ope ies o calculus heo y ha plays an
impo an ole in he cons uc ion o his p oposed scheme.
Symme y 2022,14, 2532 3 o 11
De ini ion 1.
The Capu o ac ional de i a i e ope a o o o de
α
unc ion
ϑ(ς)
is desc ibed
as [34]:
Dαϑ(ς) = Jk−αDkϑ(ς) = 1
Γ(k−α)Zθ
0(θ−η)k−α−1 k(θ)d ,
o k −1<α≤k,k∈N,θ>0, ϑ∈Ck
−1
De ini ion 2. The LT o unc ion ϑ(θ)is desc ibed as [3,6]:
L[Dmα
ςϑ(ς,θ)] = snαF(s)−
m−1
∑
k=0
smα−k−1ϑ(k)
ς(0, θ),m−1<α≤m
De ini ion 3. Le ϑ(θ) = θα, so LT is [34]:
L[θα] = Z∞
0e−s θαd =Γ(α+1)
s(α+1)
whe e s is he independen a iable o he ans o med unc ion θ.
De ini ion 4. The Capu o ac ional de i a i e ope a o o unc ion (ς,θ) o o de α>0,
Dγϑ(ς,θ) = 






1
Γ(k−α)Rθ
0(θ−η)k−γ−1∂kϑ(ς,θ)
∂ηkdη,k−1<γ<k,
∂kϑ(ς,θ)
∂θk,γ=k∈N
3. Fundamen al Concep o LHPTM
This segmen p esen s he cons uc ion o
L
HPTM o he app oxima e solu ion o
he ime ac ional mDP model. We s a his p ocedu e by assuming a nonlinea ac ional
model such as [35]
Dα
θϑ(ς,θ) = τ1[ϑ(ς,θ)] + τ2[ϑ(ς,θ)] + g(ς,θ),ς∈R,n−1<α≤n(4)
He e, we conside
Dα
θ=∂α
∂θα
in he Capu o sense,
τ1
is linea and
τ2
is a nonlinea
ope a o , and g(ς,θ)is conside ed as a sou ce e m.
Using LT o Equa ion (4), we ob ain
LhDα
τϑ(ς,θ)i=Lhτ1ϑ(ς,θ) + τ2ϑ(ς,θ) + g(ς,θ)i.
Applying LT, we gain
sαL[ϑ(ς,θ)] −sα−1hϑ(ς, 0)i=Lhτ1ϑ(ς,θ) + τ2ϑ(ς,θ) + g(ς,θ)i.
Ope a ing in e se LT, we ob ain
ϑ(ς,θ) = W(ς,θ) + L−1"1
sαLnτ1ϑ(ς,θ) + τ2ϑ(ς,θ)o#, (5)
whe e W(ς,θ) = L−1h1
sϑ(ς, 0) + 1
sαLg(ς,θ)i.
Now, applying he HPM [24] on Equa ion (5):
ϑ(ς,θ) =
∞
∑
n=0
pnϑn(ς,θ), (6)
Symme y 2022,14, 2532 4 o 11
whe e “p” is homo opy pa ame e and also we may calcula e τ2as
τ2ϑ(ς,θ) =
∞
∑
n=0
pnHn(ϑ). (7)
We can ob ain he polynomials using he ollowing p ocedu e:
Hn(ϑ0+ϑ1+· · · +ϑn) = 1
n!
∂n
∂pn τ2∞
∑
i=0
piϑi!p=0
.n=0, 1, 2, · · ·
Now, u ilize Equa ions (6) and (7) in Equa ion (5) o ob ain
∞
∑
n=0
pnϑn(ς,θ) = W(ς,θ) + p"L−1(1
sαL τ1
∞
∑
n=0
pnϑn(ς,θ) +
∞
∑
n=0
pnHn(ϑ)!)#. (8)
Co ela ing he alues o p, we ob ain
p0:ϑ0(ς,θ) = W(ς,θ)
p1:ϑ1(ς,θ) = −L−1"1
sαLτ1ϑ0(ς,θ) + H0#,
p2:ϑ2(ς,θ) = −L−1"1
sαLτ1ϑ1(ς,θ) + H1#,
p3:ϑ3(ς,θ) = −L−1"1
sαLτ1ϑ2(ς,θ) + H2#,
.
.
.
By p oceeding wi h hese i e a ions, we a e able o iden i y se ies solu ion in he ollow-
ing o m:
ϑ(ς,θ) = ϑ0(ς,θ) + p1ϑ1(ς,θ) + p2ϑ2(ς,θ) + p3ϑ3(ς,θ) + · · · .
Le ing p=1, he abo e se ies p o ides he app oxima e solu ion o Equa ion (4) as
ϑ(ς,θ) = ϑ0+ϑ1+ϑ2+· · · =lim
N→∞
N
∑
n=0
ϑn(ς,θ).
This se ies usually con e ges qui e as .
4. Nume ical P oblem
This sec ion inco po a es he concep o
L
HPTM o p o iding he soli a y wa e solu-
ion o mDP and mCH models wi h a ime- ac ional o de . This app oach p oduces high
accu acy a e a ce ain numbe o i e a ions. We demons a e he g aphical ep esen a ions
in 2D and 3D o m o he physical beha io o mDP and mCH models.
4.1. Example 1
Conside he ime ac ional mDP model such as
∂αϑ
∂θα−∂
∂θ ∂2ϑ
∂ς2+4ϑ2∂ϑ
∂ς −3∂ϑ
∂ς
∂2ϑ
∂ς2−ϑ∂3ϑ
∂ς3=0, (9)
Symme y 2022,14, 2532 5 o 11
wi h ini ial condi ion
ϑ(ς, 0) = −15
8sech2ς
2. (10)
U ilizing he LT on Equa ion (9), we ob ain:
Lh∂αϑ
∂θαi=Lh∂
∂θ ∂2ϑ
∂ς2−4ϑ2∂ϑ
∂ς +3∂ϑ
∂ς
∂2ϑ
∂x2+ϑ∂3ϑ
∂x3i,
sαL[ϑ(ς,θ)] −sα−1hϑ(ς, 0)i=Lh∂
∂θ ∂2ϑ
∂ς2−4ϑ2∂ϑ
∂ς +3∂ϑ
∂ς
∂2ϑ
∂ς2+ϑ∂3ϑ
∂ς3i,
L[ϑ] = ϑ(ς, 0)
s+1
sαLh∂
∂θ ∂2ϑ
∂ς2−4ϑ2∂ϑ
∂ς +3∂ϑ
∂ς
∂2ϑ
∂ς2+ϑ∂3ϑ
∂ς3i.
Wi h he aid o he in e se LT p ope y,
ϑ(ς,θ) = ϑ(ς, 0) + L−1"1
sαL∂
∂θ ∂2ϑ
∂ς2−4ϑ2∂ϑ
∂ς +3∂ϑ
∂ς
∂2ϑ
∂ς2+ϑ∂3ϑ
∂ς3#. (11)
Now, using he s a egy o HPM as de ined in Equa ion (6) o he abo e equa ion,
we ob ain
∞
∑
n=0
pnϑn=ϑ(ς, 0) + L−1"1
sαL∞
∑
n=0
pn∂
∂θ ∂2ϑn
∂ς2−4
∞
∑
n=0
pnϑ2
n
∞
∑
n=0
pn∂ϑn
∂ς +3
∞
∑
n=0
pn∂ϑn
∂ς
∞
∑
n=0
pn∂2ϑn
∂ς2+
∞
∑
n=0
pnϑn
∞
∑
n=0
pn∂3ϑn
∂ς3#, (12)
which is called he i e a i e o mula. Compa ing he componen s o
p
, we ob ain he
ollowing i e a ions:
p0:ϑ0=ϑ(ς, 0),
=−15
8sech21
2ς
p1:ϑ1=L−1"1
sαL∂
∂θ ∂2ϑ0
∂ς2−4ϑ2
0
∂ϑ0
∂ς +3∂ϑ0
∂x
∂2ϑ0
∂x2+ϑ0
∂3ϑ0
∂ς3#
=−450 csch5(ς)sinh6ς
2θα
Γ(1+α),
.
.
.
Consequen ly, all he esul s a e shown as
ϑ(ς,θ) = ϑ0+ϑ1+ϑ2· · · ,
ϑ(ς,θ) = −15
8sech2ς
2−450 csch5(ς)sinh6ς
2θα
Γ(1+α)+· · · .(13)
Finally, we ob ain he ollowing esul a α=1
ϑ(ς,θ) = −15
8hsech21
2ς−5
2θi. (14)
4.2. Example 2
Conside he ollowing ime ac ional mCH model,

Symme y 2022,14, 2532 6 o 11
∂αϑ
∂θα−∂
∂θ ∂2ϑ
∂ς2+3ϑ2∂ϑ
∂ς −2∂ϑ
∂ς
∂2ϑ
∂ς2−ϑ∂3ϑ
∂ς3=0, (15)
wi h ini ial condi ion
ϑ(ς, 0) = −2 sech2ς
2. (16)
U ilizing he LT on Equa ion (15), we ob ain
Lh∂αϑ
∂θαi=Lh∂
∂θ ∂2ϑ
∂ς2−3ϑ2∂ϑ
∂ς +2∂ϑ
∂ς
∂2ϑ
∂ς2+ϑ∂3ϑ
∂ς3i,
sαL[ϑ(ς,θ)] −sα−1hϑ(ς, 0)i=Lh∂
∂θ ∂2ϑ
∂ς2−3ϑ2∂ϑ
∂ς +2∂ϑ
∂ς
∂2ϑ
∂ς2+ϑ∂3ϑ
∂ς3i,
L[ϑ] = ϑ(ς, 0)
s+1
sαLh∂
∂θ ∂2ϑ
∂ς2−3ϑ2∂ϑ
∂ς +2∂ϑ
∂ς
∂2ϑ
∂ς2+ϑ∂3ϑ
∂ς3i.
Wi h he aid o he in e se LT p ope y,
ϑ=ϑ(ς, 0) + L−1"1
sαL∂
∂θ ∂2ϑ
∂ς2−3ϑ2∂ϑ
∂ς +2∂ϑ
∂ς
∂2ϑ
∂ς2+ϑ∂3ϑ
∂ς3#. (17)
Now, using he s a egy o HPM as de ined in Equa ion (6) o he abo e equa ion,
we ob ain
∞
∑
n=0
pnϑn=ϑ(ς, 0) + L−1"1
sαL∞
∑
n=0
pn∂
∂θ ∂2ϑn
∂ς2−3
∞
∑
n=0
pnϑ2
n
∞
∑
n=0
pn∂ϑn
∂ς +2
∞
∑
n=0
pn∂ϑn
∂ς
∞
∑
n=0
pn∂2ϑn
∂ς2+
∞
∑
n=0
pnϑn
∞
∑
n=0
pn∂3ϑn
∂ς3#, (18)
which is called he i e a i e o mula. Compa ing he componen s o
p
, we ob ain he
ollowing i e a ions:
p0:ϑ0=ϑ(ς, 0),
=−2 sech2ς
2
p1:ϑ1=L−1"1
sαL∂
∂θ ∂2ϑ0
∂ς2−3ϑ2
0
∂ϑ0
∂ς +2∂ϑ0
∂ς
∂2ϑ0
∂ς2+ϑ0
∂3ϑ0
∂ς3#
=−384 csch5(ς)sinh6ς
2θα
Γ(1+α),
.
.
.
Consequen ly, all o he esul s a e shown as
ϑ(ς,θ) = ϑ0+ϑ1+ϑ2· · · ,
ϑ(ς,θ) = −2 sech2ς
2−384 csch5(ς)sinh6ς
2θα
Γ(1+α)+· · · .(19)
Finally, we ob ain he ollowing esul a α=1
ϑ(ς,θ) = −2 sech2ς−θ
2. (20)
Symme y 2022,14, 2532 7 o 11
5. Resul s and Discussion
In his pa , we p o ide he esul s and discussion o ime ac ional mDP and mCH
models o demons a e he eliabili y o
L
HPTM h ough he g aphical ep esen a ions.
Figu e 1has been di ided in o wo pa s: (a) 3D su ace solu ion o Equa ion (13) a
α=
1; (b) 3D su ace solu ion o Equa ion (14), whe e
−
10
≤ς≤
10 and
θ=
0.05. Figu e 2
ep esen s he physical beha io o mDP model in 2D plo dis ibu ion a di e en ac ional
o de . We di ide i in o ou pa s: (a) compa ison be ween he app oxima e alues a
α=
0.25 and he exac alues (b) compa ison be ween he app oxima e alues a
α=
0.50
and he exac alues (c) compa ison be ween he app oxima e alues a
α=
0.75 and he
exac alues (d) compa ison be ween he app oxima e alues a
α=
1 and he exac alues.
We p esen his compa ison a −7.5 ≤ς≤7.5 and θ=0.01.
(a) (b)
Figu e 1.
Compa ison o app oxima e and exac solu ions a
α=
1 (
a
) he wo e ms’ app oxima e
solu ion o Equa ion (13); (b) he exac solu ion o Equa ion (14).
-6-4-2 2 4 6 θ
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
ϛ
α=0.25
Exac
ϑ(ϛ,θ)
(a)
-6-4-2 2 4 6 θ
-2.0
-1.5
-1.0
-0.5
ϛ
α=0.50
Exac
ϑ(ϛ,θ)
(b)
-6-4-2 2 4 6 θ
-2.0
-1.5
-1.0
-0.5
ϛ
α=0.75
Exac
ϑ(ϛ,θ)
(c)
-6-4-2 2 4 6 θ
-1.5
-1.0
-0.5
ϛ
α=1
Exac
ϑ(ϛ,θ)
(d)
Figu e 2.
Plo solu ion be ween app oxima e and exac solu ion a di e en ac ional o de . (
a
) plo
solu ion o Equa ions (13) and (14); (
b
) plo solu ion o Equa ions (13) and (14); (
c
) plo solu ion o
Equa ions (13) and (14); (d) plo solu ion o Equa ions (13) and (14).
Figu e 3has been di ided in o wo pa s: (a) 3D su ace solu ion o Equa ion (19) a
α=
1; (b) 3D su ace solu ion o Equa ion (20) whe e
−
1
≤ς≤
1 and
θ=
0.01. Figu e 4
ep esen s he physical beha io o he mCH model in 2D plo dis ibu ion a di e en
Symme y 2022,14, 2532 8 o 11
ac ional o de . We di ide i in o ou pa s: (a) compa ison be ween he app oxima e
alues a
α=
0.25 and he exac alues; (b) compa ison be ween he app oxima e alues a
α=
0.50 and he exac alues; (c) compa ison be ween he app oxima e alues a
α=
0.75
and he exac alues; (d) compa ison be ween he app oxima e alues a
α=
1 and he
exac alues. We p esen his compa ison a −5≤ς≤5 and θ=0.01.
(a) (b)
Figu e 3.
Compa ison o app oxima e and exac solu ions a
α=
1. (
a
) he wo e ms app oxima e
solu ion o Equa ion (19); (b) he exac solu ion o Equa ion (20).
-4-2 2 4 θ
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
ϛ
α=0.25
Exac
ϑ(ϛ,θ)
(a)
-4-2 2 4 θ
-2.0
-1.5
-1.0
-0.5
ϛ
α=0.50
Exac
ϑ(ϛ,θ)
(b)
-4-2 2 4 θ
-2.0
-1.5
-1.0
-0.5
ϛ
α=0.75
Exac
ϑ(ϛ,θ)
(c)
-4-2 2 4 θ
-2.0
-1.5
-1.0
-0.5
ϛ
α=1
Exac
ϑ(ϛ,θ)
(d)
Figu e 4.
Plo solu ion be ween app oxima e and exac solu ions a di e en ac ional o de . (
a
) plo
solu ion o Equa ions (19) and (20); (
b
) plo solu ion o Equa ions (19) and (20); (
c
) plo solu ion o
Equa ions (19) and (20); (d) plo solu ion o Equa ions (19) and (20).
This g aphical ep esen a ion shows ha
L
HPTM is e y easy o implemen and
achie es a high alidi y o esul s nea he exac solu ion. We also calcula e he absolu e
e o in Tables 1and 2among he app oxima e and he exac solu ions o di e en in ege s
o
ς
wi h
θ=
0.01 a
α=
0.50, 1, espec i ely. The absolu e e o ep esen s ha
L
HPTM
p o ides high easibili y wi h an inc ease o
ς
a
α=
1. Hence, we s a e ha he solu ions
wi h LHPTM a e in ou s anding coope a ion.
Symme y 2022,14, 2532 9 o 11
Table 1. Compa ison be ween mDP and he exac solu ions a θ=0.01.
ςϑapp ox a
α=0.50 ϑapp ox a α=1Exac Solu ion
(ϑexac )E o =|ϑexac −ϑapp ox |
1−1.92812 −1.51478 −1.49154 0.02324
2−1.0006 −0.806342 −0.802536 0.003806
3−0.385726 −0.342981 −0.34657 0.003589
4−0.140106 −0.133147 −0.1357 0.002553
5−0.0509675 −0.0499585 −0.0511053 0.0011468
6−0.0186525 −0.0185124 −0.0189647 0.0004523
7−0.00684765 −0.00682852 −0.00699915 0.00017063
8−0.00251713 −0.00251454 −0.00257789 0.00006335
9−0.000925732 −0.000825379 −0.000948764 0.000023385
10 −0.000340521 −0.000340473 −0.000349087 0.0000008614
Table 2. Compa ison be ween mCH and he exac solu ions a θ=0.01.
ςϑapp ox a
α=0.50 ϑapp ox a α=1Exac Solu ion
(ϑexac )E o =|ϑexac −ϑapp ox |
1−2.0096 −1.58015 −1.60719 3.18734
2−1.04519 −0.846361 −0.856068 0.009707
3−0.406574 −0.364698 −0.36496 0.00262
4−0.1486654 −0.14267 −0.141879 0.000791
5−0.0542504 −0.0537117 −0.0532682 0.0004435
6−0.0198801 −0.0199294 −0.0197437 0.0001857
7−0.00730199 −0.00735482 −0.00728336 0.00007146
8−0.00268465 −0.00270884 −0.00268212 0.00002672
9−0.000987407 −0.000996952 −0.000983064 0.000009888
10 −0.000363217 −0.000366816 −0.00036317 0.000003646
6. Conclusions
In his s udy, we p esen an idea o
L
HPTM o ob ain he soli a y wa e solu ion o
he mDP and mCH models wi h ac ional o de . The majo ad an age o his scheme is
ha i p o ides he signi ican esul s in he calcula ion o successi e i e a ions. We do
no equi e any assump ion o e en a small pe u ba ion o he cons uc ion o his new
scheme. I can easily be seen ha all he e ms a e ound in he o m o se ies solu ions.
On he o he hand, we use Ma hema ica so wa e 11.0.1 o e alua e he i e a ions and he
g aphical ep esen a ions in 2D and 3D plo dis ibu ion. These esul s demons a e he
easibili y and accu acy o
L
HPTM, and hus we can decla e ha ou solu ion p ocedu e is
signi ican ly s aigh o wa d. We in end o expand his app oach wi h he neu al ne wo k
me hod o ob aining he app oxima e solu ion o ac ional di e en ial p oblems o ou
u u e wo k in science and enginee ing phenomena.
Au ho Con ibu ions:
In es iga ion, Me hodology, So wa e, and W i ing—o iginal d a , M.N.;
W i ing— e iew and edi ing, and supe ision H.J.; Valida ion, Visualiza ion, A.A.; Concep ualiza ion,
Fo mal analysis, and Funding acquisi ion, M.D.l.S. All au ho s ha e ead and ag eed o submi
he manusc ip .
Funding: This esea ch unded by Basque Go e nmen h ough G an IT1155-22.
Ins i u ional Re iew Boa d S a emen : No applicable.