Ci a ion: Ghazan a , S.; Ahmed, N.;
Iqbal, M.S.; Akgül, A.; Bay am, M.;
De la Sen, M. Imaging Ul asound
P opaga ion Using he Wes e el
Equa ion by he Gene alized
Kud yasho and Modi ied
Kud yasho Me hods. Appl. Sci.
2022,12, 11813. h ps://doi.o g/
10.3390/app122211813
Academic Edi o : Panagio is G.
As e is
Recei ed: 13 Oc obe 2022
Accep ed: 16 No embe 2022
Published: 21 No embe 2022
Publishe ’s No e: MDPI s ays neu al
wi h ega d o ju isdic ional claims in
published maps and ins i u ional a il-
ia ions.
Copy igh : © 2022 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/).
applied
sciences
A icle
Imaging Ul asound P opaga ion Using he Wes e el Equa ion
by he Gene alized Kud yasho and Modi ied
Kud yasho Me hods
Sid a Ghazan a 1, Nauman Ahmed 1, Muhammad Sajid Iqbal 2, Ali Akgül 3,4,* , Mus a a Bay am 5
and Manuel De la Sen 6
1Depa men o Ma hema ics and S a is ics, Uni e si y o Laho e, Laho e 54000, Pakis an
2Depa men o Humani ies & Basic Science, Mili a y College o Signals, NUST, Islamabad 44000, Pakis an
3Depa men o Ma hema ics, A and Science Facul y, Sii Uni e si y, 56100 Sii , Tu key
4Depa men o Ma hema ics, Ma hema ics Resea ch Cen e , Nea Eas Uni e si y, Nea Eas Boule a d,
99138 Nicosia, Tu key
5Depa men o Compu e Enginee ing, Bi uni Uni e si y, 34010 Is anbul, Tu key
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, Spain
*Co espondence: [email p o ec ed]
Abs ac :
This a icle deals wi h he s udy o ul asound p opaga ion, which p opaga es he mechani-
cal ib a ion o he molecules o o he pa icles o a ma e ial. I measu es he speed o sound in ai .
Fo his eason, he hi d-o de non-linea model o he Wes e el equa ion was chosen o be s udied,
as he solu ions o such p oblems ha e much impo ance o physical pu poses. In his a icle, we
discuss he exac soli a y wa e solu ions o he hi d-o de non-linea model o he Wes e el equa-
ion o an acous ic p essu e
p
ep esen ing he equa ion o ul asound wi h high in ensi y, as used
in acous ic omog aphy. Mo eo e , he non-linea coe icien
B/A
(being a pa o space-dependen
coe icien
K
), has also been in es iga ed in his li e a u e. This p oblem is sol ed using he Gene -
alized Kud yasho me hod along wi h a compa ison o he Modi ied Kud yasho me hod. All o
he solu ions ha e been discussed wi h bo h su ace and con ou plo s, which shows he beha io o
he solu ion. The images a e p epa ed in a well-es ablished way, showing he p oduc ion o issues
inside he human body.
Keywo ds:
ul asound imaging; soli a y wa es; modi ied Kud yasho me hod; gene alized Kud yasho
me hod
1. In oduc ion
Finding he solu ion o non-linea p oblems s ill aces many di icul ies in he ield
o ma hema ical physics. Non-linea pa ial di e en ial equa ions (NPDEs) [
1
] play a
signi ican ole in physical and ma hema ical models [
2
]. They de ine hei anges om
g a i a ion [
3
] o luid dynamics [
4
], desc ibing many di e en physical sys ems. They a e
mos ly ela ed o he ac ha hese ypes o equa ions ace he p oblem o inding hei
in eg abili y. The e a e almos no pe asi e echniques ha can be used o all p oblems,
and usually, e e y sepa a e model wo ks as an indi idual p oblem.
Pa ial di e en ial equa ions (PDEs) gi e solu ions in an ul ima e s a e om he pas
ew yea s while, o exempli y he solu ions o NPDEs [
5
], we can w i e hei solu ions wi h
some special cla i ica ion. To exempli y he mos impo an applica ions o NPDEs om he
his o ical poin o iew, we can highligh one o ou ocuses as he Wes e el equa ion [
6
],
which is a non-linea ma hema ical model, widely used o wa e p opaga ion, ha can
be speci ied by he possible physical measu emen s leading o o e -posed da a. O he
undamen al models, which can also be men ioned he e, a e he Eule and Na ie –S okes
Appl. Sci. 2022,12, 11813. h ps://doi.o g/10.3390/app122211813 h ps://www.mdpi.com/jou nal/applsci
Appl. Sci. 2022,12, 11813 2 o 9
equa ions in luid dynamics [
7
], non-linea Sch ödinge [
8
], Klein–Go don equa ion [
9
], he
Bol zmann equa ion in gas dynamics [10], and many mo e.
To w i e he solu ions o NPDEs explici ly [
11
], we can educe he gi en equa ion o
he equa ion o one dimension, o which, he p ocess o con e sion is applied on NPDE o
con e i in o an o dina y di e en ial equa ion (ODE).
In he pas ew yea s, many analy ical and nume ical echniques ha e been p ojec ed
o ob ain solu ions o NPDEs, o example, Be noulli unc ional me hodology [
12
], he
F-expansion echnique [
13
], he auxilia y equa ion echnique [
14
], he simples ex ended
equa ion echnique [
15
], he (G
0
/G)-expansion echnique [
16
], he sub-ODE echnique [
1
],
he gene alized Kud yasho echnique(GKM) [
17
], and many mo e. The collec i e heme
o all hese echniques is o con e he PDEs o ODEs using wa e ans o ma ions [
18
]. In
his ield, he s udy o soli ons [
19
] is playing an impo an ole in cons uc ing a ious
amilies o analy ic a eling wa e solu ions [
20
], which de ines he dynamics o soli ons
lea ing a ema kable posi ion in non-linea op ics. Acco ding o some heo e ical esea ch,
a ious modes o plasma (i.e., pe iodic, a ional, soli ons, shock-like, explosi e) [
21
] show
wa e p opaga ion in di e en na u es o non-linea wa es.
The eason behind choosing he gene alized o m o he Kud yasho me hod is ha
i app oaches he mos consis en solu ions o he NPDEs. I is also a e y use ul and
e icien app oach o inding he solu ions o non-linea e olu ion equa ions. The modi ied
Kud yasho me hod is a e y s ong solu ion scheme ha shows many ways owa ds
he exac solu ion o he NPDE p oblem in ma hema ical physics and biology. Due o
he e icien wo k o his me hod in he ield o ma hema ics, i has ecei ed signi ican
a en ion owa ds i . Simila ly, a highe -o de non-linea Sch ödinge equa ion (NDNLSE)
can be sol ed wi h he help o his powe ul me hod. I is a success ul applica ion ha can
be pe o med in se e al wo ks jus like in [22,23].
Ul asound imaging [
24
,
25
] is being used in a well-es ablished way o p oduce pic u es
o issues inside he body o human beings. They a e modeled in non-linea wa e equa ions
wi h high in ensi y.
In many medical and indus ial applica ions, high in ensi y- ocused ul asound (HIFU)
is one o he c ucial p ocedu es, which uses high-ene gy sound wa es di ec ly a an
a ea o abno mal issues o he body o igh en and li he skin. I also ea s emo s,
u e ine ib oids, and umo s in ce ain condi ions. I includes ul asound o welding,
he mo he apy, sonochemis y, and li ho ipsy.
Fo he pu pose o medical imaging, a spa ially a ying coe icien can be used, which
is called acous ic nonlinea i y pa ame e omog aphy [
26
]. High-in ensi y ul asound
p opaga ion [
27
] is being desc ibed wi h his pa ame e , which appea s in he o m o PDEs.
These ela ed imaging p oblems hus become a coe icien iden i ica ion o hem.
Conside ing he imaging ask in he o m o he Wes e el equa ion, consis s o an
iden i ied Kin he acous ic p essu e p o mula ion, ep esen ed as:
p −c2∆p−b∆p =K(p2) , in Ω×(0, T). (1)
I can also be o mula ed in e ms o he acous ic eloci y po en ial as:
ψ −c2∆ψ−b∆ψ =χ(ψ2) , in Ω×(0, T). (2)
wi h p=ςψ .
He e,
p
is he acous ic p essu e,
b
is he di usi i y o sound, and
c
is he known cons an ,
which ep esen s he speed o sound. In he abo e equa ions, he
K
and
χ
ha e he ollowing
in e dependence
K=βa
λ
,
βa=
1
+B
2A
, whe e
B
A
signi ies he pa ame e o nonlinea i y,
ς
is
wo king as he mass densi y, and λ=ςc2ac ing as bulk modulus, whe e χ=ςK.
The spa ial domain
Ω⊂R∈ {
1, 2, 3
}
is supposed o be smoo h and bounded on
which he gi en PDEs a e assumed o hold.
Appl. Sci. 2022,12, 11813 3 o 9
2. P oblem S a emen
A Wes e el equa ion in p essu e o mula ion wi h acous ic p essu e
p
, di usi i y o
sound
b
, a known cons an
c
, and
K=βa
λ
, whe e
βa=
1
+B
2A
is he nonlinea pa ame e ,
can be w i en in he ollowing o m:
2Kpp +p2
+c2pxx +bpxx −p =0. (3)
Ou goal is basically o sol e his PDE analy ically. We will ind he exac solu ions o
his equa ion, wi hou assuming he ini ial and bounda y condi ions.
3. Basic Idea
To exempli y he concep o one o he p oposed echniques, a nonlinea PDE can be
aken as:
S(u,u ,ux,uxx,u ,uxx , . . .) = 0, (4)
which shows ha
S
con ains
u
and i s pa ial de i a i es. This PDE can be con e ed o he
ollowing ODE as:
T(u,u0,u00,u000, . . .) = 0, (5)
wi h he help o he ollowing a eling wa e ans o ma ion:
u(x, ) = u(η),η=αx+e , (6)
whe e
α
is he non-ze o a bi a y cons an and
e
is he speed o he a eling wa e. To
demons a e his me hod in de ail, we can desc ibe i as:
4. The Gene alized Kud yasho Me hod
Suppose he ini ial solu ion o Equa ion (5) is as ollows:
u(η) = ∑P
i=0aiRi(η)
∑Q
j=0bjRj(η), (7)
wi h
ai
, whe e
(i=
0, 1, 2,
. . .
,
P)
;
bj
, whe e
(j=
0, 1, 2,
. . .
,
Q)
,
(aP6=
0,
bQ6=
0
)
a e ound
o be unknown coe icien s; and R=R(η)is he solu ion o
dR
dη=R2(η)−R(η), (8)
which can be embodied as
R(η) = 1
1+C1eη,C1is he cons an o in eg a ion. (9)
By using he homogeneous balance p inciple, we ob ain he alues o
P
and
Q
in
Equa ion (7)
o a ain he polynomial
R
by subs i u ing he Equa ions (7) and (8) in o
Equa ion (5)
. Now, equa ing all he coe icien s o polynomials o ze o, we ob ain he sys em
o algeb aic equa ions. To ind he alues o unknown coe icien s
ai(i=
0, 1, 2,
. . .
,
P)
,
bj(j=
0, 1, 2,
. . .
,
Q)
,
(aP6=
0,
bQ6=
0
)
, we sol e he sys em o algeb aic equa ions. Las ly,
we de elop he soli a y wa e solu ion o he sugges ed equa ion.
Appl. Sci. 2022,12, 11813 4 o 9
5. The Modi ied Kud yasho Me hod
Conside ing he same non-linea PDE as men ioned abo e in Sec ion 3and ollowing
he same abo e men ioned s eps, we may ha e he ini ial solu ion o he Equa ion (5) can
be exp essed as he ini e se ies as ollows:
u(η) =
P
∑
i=0
aiRi(η), (10)
wi h
ai
, whe e
(i=
0, 1, 2,
. . .
,
P)
;
(aP6=
0
)
is ound o be an unknown coe icien and
R=R(η)is he solu ion o
dR
dη=ln(a)R2(η)−R(η), (11)
which can be embodied as
R(η) = 1
1+C1aη,C1is he cons an o in eg a ion. (12)
No e ha ais any andom cons an numbe .
Wi h he help o he homogeneous balance p inciple, we ob ain he alue o
P
in
Equa ion (10)
o a ain he polynomial
R
by subs i u ing Equa ions (10) and (11) in o
Equa ion (5). Now we will equa e all he coe icien s o polynomials o ze o o ob ain
he sys em o algeb aic equa ions. Now, o ind he alues o unknown coe icien s
ai(i=0, 1, 2, . . . , P),(aP6=0)
, we sol e he sys em o algeb aic equa ions. Las ly, we de-
elop he soli a y wa e solu ion o he sugges ed equa ion.
6. Applica ions o he Gene alized Kud yasho Me hod on he Wes e el Equa ion
Using he wa e ans o ma ion
p(x
,
) = p(η)
,
η=αx+e
, we can educe
Equa ion (3)
o he ODE as ollows:
e2h2Kpp00 + (p0)2−p00i+α2ebp000 +α2c2p00 =0. (13)
To ind he solu ion o he ackled model, we balance
(p000)
and
(p0)2
by using he
homogeneous balance p inciple o ind he alue o
0N0
and ound i o be
N=M+
1.
Since
M
is a ee pa ame e , we can se i as
M=
0, which allows us o se he alue o
N
as
N=1. Thus, he solu ion o Equa ion (13) akes he o m:
P(η) = a0+a1R
b0
. (14)
Subs i u ing Equa ion (14) in o (13) along wi h Equa ion (8), and equa ing each coe -
icien o he equa ion o ze o, we ob ain he sys em o equa ions as ollows. The code o
Maple was used o ind he ollowing esul s:
R3: 6 α2beb0+6Ke2a1=0,
R2:−12 α2beb0+2α2c2b0+4Ke2a0−2e2b0−10 Ke2a1=0,
R: 7 α2beb0−3α2c2b0−6Ke2a0+3e2b0+4Ke2a1=0,
1 : −α2beb0+α2c2b0+2Ke2a0−e2b0=0,
(15)
By sol ing he abo e sys em o equa ions, we ge he ollowing esul :
Appl. Sci. 2022,12, 11813 5 o 9
a0=b0α2be−α2c2+e2
2Ke2,
a1=−α2bb0
Ke. (16)
whe e
e
is he wa e speed. Subs i u ing hese alues in Equa ion (14) using Equa ion (9),
we ob ain he inal solu ion as:
p(x, ) = −α2be+α2beC1eη−α2c2−α2c2C1eη+e2+e2C1eη
2e2K(1+C1eη). (17)
whe e η=αx+e and C1is an a bi a y cons an .
7. Compa ison wi h he Modi ied Kud yasho Me hod on he Wes e el Equa ion
Applying he modi ied o m o he Kud yasho me hod on he Wes e el equa ion o
ha e a compa ison be ween bo h me hod esul s. We may ha e he ollowing cases a e
applying he modi ied Kud yasho me hod:
Case I: When
a0=α2be−α2c2+e2
2Ke2,
a1=−α2b
Ke. (18)
whe e
e
is he wa e speed. Subs i u ing hese alues in Equa ion (14) using Equa ion (9),
we ob ain he inal solu ion as:
p(x, ) = −α2c2−α2c2C1aη+e2+e2C1aη−α2be+α2beC1aη
2e2K(1+C1aη). (19)
whe e η=αx+e and C1is an a bi a y cons an .
Case II: When
a0=α2be−α2c2+e2
2Ke2,
a1=−α2b(3+ln(a))
2Ke(ln(a)+1). (20)
Subs i u ing hese alues in Equa ion (14) using Equa ion (9), we ob ain he inal
solu ion as:
p(x, ) = −α2c2+e2+α2be
2Ke2−α2b(3+ln(a))
2Ke(ln(a)+1)(1+C1aη). (21)
whe e η=αx+e and C1is an a bi a y cons an .
Case III: When
a0=−−3α2be−e2ln(a)−2e2+2α2c2+α2c2ln(a)
2Ke2(2+ln(a)) ,
a1=−α2b
Ke.
(22)
Subs i u ing hese alues in Equa ion (14) using Equa ion (9) we ob ain he inal
solu ion as:
Appl. Sci. 2022,12, 11813 6 o 9
p(x, ) = −−3α2be−e2ln(a)−2e2+2α2c2+α2c2ln(a)
2Ke2(2+ln(a)) −α2b
Ke(1+C1aη). (23)
whe e η=αx+e and C1is an a bi a y cons an .
The g aphical beha io o soli ons o he abo e-men ioned Wes e el equa ion has
been shown in he igu es gi en below. To unde s and he physical p ope ies o he
a ained ou comes, some o he esul an s a e ep esen ed by selec ing di e en alues o
pa ame e s. Fo example, Figu es 1–4a e ep esen ing he beha io o soli ons in he o m
o su ace and con ou plo s whe e he pa ame e s a e men ioned below he igu es.
-5
0
x
5
10
5
0
10
20
30
-10
0
-20
-5
p
x
-4 -2 0 2 4 6 8
-4
-3
-2
-1
0
1
2
Figu e 1.
The abo e g aphs show he g aphical illus a ion o soli ons in he o m o he su ace
plo s (on he le ) and con ou plo s (on he igh side) o acous ic p essu e
p
whe e he alues o
pa ame e s a e men ioned below in A. This g aph ep esen s he beha io o soli ons o Equa ion (17)
which is a lump wa e wi h a backg ound o a lump wa e wi h a kink backg ound.
4
2
0
-2
-410
5
x
0
-5
8
2
4
5
6
7
3
p1
x
-4 -2 0 2 4 6 8
-4
-3
-2
-1
0
1
2
Figu e 2.
The abo e g aphs show he g aphical illus a ion o soli ons in he o m o he su ace plo s
(on he le ) and con ou plo s (on he igh side) o acous ic p essu e
p
whe e he alues o pa ame e s
a e men ioned below in B. This g aph ep esen s he beha io o soli ons o Case I Equa ion (19)
which is a mixed lump ain wa e wi h a kink backg ound, he dynamical ea he wa e (as eloci y
and ampli ude) has e ained he same alue along he x-axis.
Appl. Sci. 2022,12, 11813 7 o 9
-4
-2
0
2
410
5
x
0
-5
4.2
4.4
4.6
4.8
5.4
6.2
6
5.8
5.6
5
5.2
p2
x
-4 -2 0 2 4 6 8
-4
-3
-2
-1
0
1
2
Figu e 3.
The abo e g aphs show he g aphical illus a ion o soli ons in he o m o he su ace plo s
(on he le ) and con ou plo s (on he igh side) o acous ic p essu e
p
whe e he alues o pa ame e s
a e men ioned below in C. This g aph ep esen s he beha io o soli ons o Case II Equa ion (21)
showing 3D and hei con ou plo s o b ea he s dis ibu ion unde soli a y wa e backg ound.
5
0
-5
10
8
6
x
4
2
0
-2
-4
4
3
5
2
6
p3
x
-4 -2 0 2 4 6 8
-4
-3
-2
-1
0
1
2
Figu e 4.
The abo e g aphs show he g aphical illus a ion o soli ons in he o m o he su ace plo s
(on he le ) and con ou plo s (on he igh side) o acous ic p essu e
p
whe e he alues o pa ame e s
a e men ioned below in D. This g aph ep esen s he beha io o soli ons o Case III Equa ion (23)
which shows 3D and hei con ou plo s o b ea he s dis ibu ion unde soli a y wa e backg ound.
A. e=5, α=c=b=1, C1=−1, b0=K=0.1, [x, ] = (−4:0.1:9, −4:0.5:2.5).
B. e=5, α=c=b=1, a=−5, C1=−1, b0=K=0.1, [x, ] = (−4:0.1:9, −4:0.5:2.5).
C.
e=
5,
α=c=b=
1,
a=−
2.5,
C1=−
1,
b0=K=
0.1,
[x
,
] =
(
−
4:0.1:9,
−4:0.5:2.5).
D.
e=
5,
α=c=b=
1,
a=−
0.5,
C1=−
1,
b0=K=
0.1,
[x
,
] =
(
−
4:0.1:9,
−4:0.5:2.5).
8. Conclusions
In his pape , he gene alized Kud yasho and modi ied Kud yasho me hods we e
applied o a Wes e el equa ion showing ul asound imaging, which p oduces di e en
pic u es o human body issues. I includes he de ails o he Wes e el equa ion, which
p opaga es he imaging o highly in ense ul asound wa es. We ha e ound he exac
solu ions and discussed di e en cases ha ep esen a eling wa es bo h ma hema ically
and g aphically. All possible solu ions ha e been accessed wi h di e en ypes o soli ons
o ob ain he a eling wa e solu ion wi h di e en pa ame ic alues. The abo e wo k
clea ly shows he e icien applica ions o NPDEs.
No e: In he sequel o inding be e solu ions, i we ake his p oblem in he ime
ac ional pa ial di e en ial equa ion, hen he ac ional pa ame e can be adjus ed ac-
co ding o he p oblems on he physical side. The e o e, we ecommend o he u u e
and o ou sel es o conside he ac ional e sion o his p oblem and ind whe he he
Appl. Sci. 2022,12, 11813 8 o 9
solu ions a e compa able and how hey a e be e con e gen w. . he compa ison o he
in ege o de o he ac ional o de PDEs.
Au ho Con ibu ions:
S.G., Concep ualiza ion; N.A., Da a cu a ion; M.S.I., Fo mal analysis; A.A.,
In es iga ion; M.B., Me hodology and M.D.l.S., Supe ision. All au ho s ha e ead and ag eed o he
published e sion o he manusc ip .
Funding:
The au ho s a e g a e ul o he Basque Go e nmen o i s suppo h ough G an s IT1555-22
and KK-2022/00090; 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 .
Re e ences
1.
Alha bi, A.; Alma a i, M. Ricca i–Be noulli sub-ODE app oach on he pa ial di e en ial equa ions and applica ions. In . J. Ma h.
Compu . Sci. 2020,15, 367–388.
2. Logan, J.D. An In oduc ion o Nonlinea Pa ial Di e en ial Equa ions; John Wiley & Sons: Hoboken, NJ, USA, 2008; Volume 89.
3. De Sabba a, V.; Gaspe ini, M. In oduc ion o G a i a ion; Wo ld Scien i ic Publishing Company: Singapo e, 1986.
4. Ba chelo , C.K.; Ba chelo , G. An In oduc ion o Fluid Dynamics; Camb idge Uni e si y P ess: Camb idge, UK, 2000.
5. Rosinge , E.E. Gene alized Solu ions o Nonlinea Pa ial Di e en ial Equa ions; Else ie : Ams e dam, The Ne he lands, 1987.
6.
Ka amalis, A.; Wein, W.; Na ab, N. Fas ul asound image simula ion using he wes e el equa ion. In P oceedings o he
In e na ional Con e ence on Medical Image Compu ing and Compu e -Assis ed In e en ion, Beijing, China, 20–24 Sep embe
2010; Sp inge : Be lin/Heidelbe g, Ge many, 2010; pp. 243–250.
7.
Hughes, T.J.; F anca, L.P.; Malle , M. A new ini e elemen o mula ion o compu a ional luid dynamics: I. Symme ic o ms o
he comp essible Eule and Na ie -S okes equa ions and he second law o he modynamics. Compu . Me hods Appl. Mech. Eng.
1986,54, 223–234. [C ossRe ]
8.
Del ou , M.; Fo in, M.; Pay , G. Fini e-di e ence solu ions o a non-linea Sch ödinge equa ion. J. Compu . Phys.
1981
,
44, 277–288. [C ossRe ]
9. De weile , S. Klein-Go don equa ion and o a ing black holes. Phys. Re . D 1980,22, 2323. [C ossRe ]
10.
Rei z, R.D. One-dimensional comp essible gas dynamics calcula ions using he Bol zmann equa ion. J. Compu . Phys.
1981
,
42, 108–123. [C ossRe ]
11.
Galak iono , V.A.; S i shche skii, S.R. Exac Solu ions and In a ian Subspaces o Nonlinea Pa ial Di e en ial Equa ions in Mechanics
and Physics; Chapman and Hall/CRC: Boca Ra on, FL, USA, 2006.
12.
Koka , M.M. Cope : A me hodology o lea ning in a ian unc ional desc ip ions. In Machine Lea ning; Sp inge :
Be lin/Heidelbe g, Ge many, 1986; pp. 151–154.
13.
Ka aman, B. The use o imp o ed-F expansion me hod o he ime- ac ional Benjamin–Ono equa ion. Re . Real Acad. Cienc.
Exac as Físicas Na . Se . A Ma . 2021,115, 1–7. [C ossRe ]
14.
L , X.; Lai, S.; Wu, Y. An auxilia y equa ion echnique and exac solu ions o a nonlinea Klein–Go don equa ion. Chaos Soli ons
F ac als 2009,41, 82–90. [C ossRe ]
15.
Zayed, E.M.; Shohib, R.M. Op ical soli ons and o he solu ions o Biswas–A shed equa ion using he ex ended simples equa ion
me hod. Op ik 2019,185, 626–635. [C ossRe ]
16.
Beki , A.; Gune , O.; Bh awy, A.H.; Biswas, A. Sol ing Nonlinea F ac ional Di e en ial Equa ions Using Exp-Func ion and
(G’/G)-Expansion Me hods. 2015. A ailable online: h ps:// jp.nipne. o/2015_60_3-4/RomJPhys.60.p360.pd (accessed on 12
Oc obe 2022).
17.
Gabe , A.; Aljohani, A.; Ebaid, A.; Machado, J.T. The gene alized Kud yasho me hod o nonlinea space– ime ac ional pa ial
di e en ial equa ions o Bu ge s ype. Nonlinea Dyn. 2019,95, 361–368. [C ossRe ]
18.
Ta a a i, S.; Ele he iades, G.V. Fou -dimensional wa e ans o ma ions by space- ime me asu aces. a Xi
2020
, a Xi :2011.08423.
19. D azin, P.G.; D azin, P.G.; Johnson, R. Soli ons: An In oduc ion; Camb idge Uni e si y P ess: Camb idge, UK, 1989; Volume 2.
20.
Yoku¸s, A.; Du u , H.; No al, T.A.; Abu-Zinadah, H.; Tuz, M.; Ahmad, H. S udy on he applica ions o wo analy ical me hods o
he cons uc ion o a eling wa e solu ions o he modi ied equal wid h equa ion. Open Phys. 2020,18, 1003–1010. [C ossRe ]
21.
Bashi , M.F.; Mu aza, G. E ec o empe a u e aniso opy on a ious modes and ins abili ies o a magne ized non- ela i is ic
bi-Maxwellian plasma. B az. J. Phys. 2012,42, 487–504. [C ossRe ]
22.
Kha e , A.; Seadawy, A.; Helal, M. Gene al soli on solu ions o an n-dimensional nonlinea Sch ödinge equa ion. Nuo o C. B
2000,115, 1303–1311.
23.
Kichenassamy, S.; Li man, W. Blow-up su aces o nonlinea wa e equa ions, I. Commun. Pa ial. Di e . Equ.
1993
,18, 431–452.
[C ossRe ]
24. Wells, P.N. Ul asound imaging. Phys. Med. Biol. 2006,51, R83. [C ossRe ] [PubMed]
25.
Ross, M.T.; An ico, M.; McMahon, K.L.; Ren, J.; Powell, S.K.; Pandey, A.K.; Allenby, M.C.; Fon ana osa, D.; Wood u , M.A.
Ul asound Imaging O e s P omising Al e na i e o C ea e 3-D Models o Pe sonalised Au icula Implan s. Ul asound Med.
Biol. 2022,48, 450–459. [C ossRe ] [PubMed]
Appl. Sci. 2022,12, 11813 9 o 9
26.
Zhang, D.; Gong, X.F. Expe imen al in es iga ion o he acous ic nonlinea i y pa ame e omog aphy o excised pa hological
biological issues. Ul asound Med. Biol. 1999,25, 593–599. [C ossRe ]
27.
Zhang, D.; Chen, X.; Gong, X.F. Acous ic nonlinea i y pa ame e omog aphy o biological issues ia pa ame ic a ay om
a ci cula pis on sou ce: Theo e ical analysis and compu e simula ions. J. Acous . Soc. Am.
2001
,109, 1219–1225. [C ossRe ]
[PubMed]
