Imaging Ultrasound Propagation Using the Westervelt Equation by the Generalized Kudryashov and Modified Kudryashov Methods

  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: 2Kpp +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=b0α2be−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]