Wave analysis in generalized fractional Tzitzéica-type nonlinear PDEs: Contributions to nonlinear sciences

Alexand ia Enginee ing Jou nal 92 (2024) 102–116 A ailable online 4 Ma ch 2024 1110-0168/© 2024 THE AUTHORS. Published by Else ie BV on behal o Facul y o Enginee ing, Alexand ia Uni e si y. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Con en s lis s a ailable a ScienceDi ec Alexand ia Enginee ing Jou nal jou nal homepage: www.else ie .com/loca e/aej O iginal A icle Wa e analysis in gene alized ac ional Tzi zéica- ype nonlinea PDEs: Con ibu ions o nonlinea sciences Naeem Ullah a, Hamood U Rehman b, Muhammad Im an Asjad a,∗, Muhammad Bilal Riaz c,d, Tasee Muhammad e aDepa men o Ma hema ics, Uni e si y o Managemen and Technology, Laho e, Pakis an bDepa men o Ma hema ics, Uni e si y o Oka a, Oka a, Pakis an cIT4Inno a ions, VSB–Technical Uni e si y o Os a a, Os a a, Czech Republic dDepa men o Compu e Science and Ma hema ics, Lebanese Ame ican Uni e si y, Byblos, Lebanon eDepa men o Ma hema ics, College o Science, King Khalid Uni e si y, Abha, Saudi A abia A R T I C L E I N F O A B S T R A C T Keywo ds: Ex ended di ec algeb aic me hod T a eling wa e s uc u es Non-linea Tzi zéica ype equa ions In his pape , he ex ended di ec algeb aic app oach wi h he gene al ac ional de i a i e is employed o a ain a ious no el wa e s uc u es o he non-linea ac ional Tzi zéica ype non-linea e olu ion equa ions in he o m o kink, singula , pe iodic singula , da k, b igh and da k-b igh combine soli ons. Fo he pu pose o illus a e he physical beha io o he acqui ed solu ions, some o he ex ac ed esul s a e ske ched in he pa e n o 3-D and 2-D plo s which show he efficiency and au hen ici y o he p oposed me hod. The unde conside a ion me hod can also u ilize o any o he non-linea model appea s in op ics and enginee ing. 1. In oduc ion Non-linea physical phenomena a e dominan in na u al li e and ha e exci ed he cu iosi y o esea che s o decades. In physics and enginee ing, de eloping he analy ical o nume ical solu ions o ac ional ma hema ical models o ce ain phenomena ha e become significan opics. Fo highly unde s anding he beha io o hese complica ed na u al phenomena ha e mos ly been es ablished o e hese models. Recen ly, ac ional ope a o s a e conside ed and ma hema ically dignified. The se e al p ope ies o ac ional ope a o s ha e de eloped a g ea in e es in ac ional calculus in nowadays, also an ex ensi e di e si y o applica ions in he field o fluid dynamics, plasma physics, op ical fibe , a omic science, enginee ing, ma hema ical biology, and se e al o he s [26–32,36,37]. Du ing he p e ious 20 yea s, ac ional calculus has become mo e popula and significan . Leibni z w o e a le e o he hospi al ega ding he defini ion o a non-in ege de i a i e, in his way a wide domain’s his o y opened. The o dina y diffe en ial equa ions (DEs) a e changed in o ac ional DEs which a e u ilized in a di e si y o ma hema ical modeling in diffe en a eas like as heology, epidemiology and compu e science. Non-locali y shows an ene ge ic ole in nume ous non-in ege de i a i e models. Recen ly, many schola s ha e e ealed ha non-linea ac ional diffe en ial equa ions (NFDEs) ha e a g ea impac in diffe en a enas, as well as physics, enginee ing, biology, wa e dynamics, con ol heo y and many mo e. F ac ional de i a i e is defined in diffe en ways like, Capu o, Riemann-Liou ille, Hadama d, Juma ie and Weyl ha has been used posi i ely in a ious fields, howe e all o hese defini ions ha e hei benefi s as well as d awbacks. The Riemann-Liou ille de i a i e has decep i e mo e se e e p oblem is ha i is incapable o offe he de i a i e o a cons an equal o ze o. Mo eo e , i a unc ion is a cons an a he o igin, i s ac ional de i a i e has a uniqueness a he o igin, like as exponen ial and Mi ag-Le e unc ions. Because o hese d awbacks, he ange o applicabili y o Riemann-Liou ille ac ional de i a i es is es ic ed. While he Capu o de i a i e is powe less o deal p oblems wi h a singula ke nel. Due o hese limi s he schola s ha e s imula ed o disco e mo e app op ia e defini ions ha ha e a ac ional-o de and a e mo e comp ehensi e. A no el well-manne ed simple non-in ege de i a i e like he con o mable de i a i e, was defined by Khalil e al. [25] depending on he de i a i e’s undamen al limi o mula ion. Due o dis inc ion be ween Capu o o mula ions and Riemann-Liou ille, he con o mable de i a i e ulfils many * Co esponding au ho . E-mail add ess: [email p o ec ed] (M.I. Asjad). h ps://doi.o g/10.1016/j.aej.2024.02.045 Recei ed 3 Decembe 2023; Recei ed in e ised o m 6 Feb ua y 2024; Accep ed 22 Feb ua y 2024 Alexand ia Enginee ing Jou nal 92 (2024) 102–116 103 N. Ullah, H.U. Rehman, M.I. Asjad e al. c ucial ea u es. The au ho , es ablished in Abdelhakim [3] ha , o ce ain unc ions, he Capu o defini ion canno yield longe esul s han he con o mable defini ion in Khalil e al. [25]. In his s udy, ou aim is o u ilize a no el gene alized defini ion o he ac ional-o de de i a i e ha has benefi s o e o he ea lie defini ions o achie e di ec solu ions o NFDEs. Because o wide applica ions in many esea ch a eas, NFDEs ha e become mo e cha ming and a e ising g adually. Conside ing he essen ial pa o analy ical solu ions o NFDEs in non-linea esea ch, i would be aluable o explo e possible no el soli on solu ions o he non-linea ac ional model. One o he mos a is ic de elopmen s in heo e ic physics and non-linea science has been he es ablishmen o me hods o de e mining exac solu ions o NFDEs. Va ious success ul app oaches ha e been es ablished due o he apid ad ancemen in non-linea sciences o ins ance he soli a y wa e Ansa z me hod [8,9], he sine-Go don expansion echnique [16], Fan sub-equa ion app oach [15], sine-cosine me hod [43], F-expansion scheme [45], modified Kud yasho scheme [10], modified simple equa ion scheme [38], [21–23]; [33]; [46]; [39]; [13]; [11,12]; [7]; [5,6]; [2]; [24]; [34]; [14] and so on. One o he mos significan analy ical app oach o finding he exac solu ions o nonlinea PDEs is he ex ended di ec algeb aic me hod (EDAM). This app oach has been used success ully o cons uc many significan nonlinea models [1,8,9,19,35]. In ma hema ical modeling a ious ac ional models a e implemen ed by powe ul nume ical schemes o he analysis o many diseases dynamics, including HIV/AIDS, COVID-19, mala ia, ube culosis, also i is beneficial o con olling and moni o ing he diseases [17,18]. The main ad an age o he EDAM o e o he me hods is ha i offe s mo e gene alized solu ions, which p oduce some known solu ions by choosing app op ia e pa ame e s. The Tzi zéica-Dodd-Bullough-Mikhailo (TDBM), Tzi zéica-Dodd-Bullough (TDB), and Liou ille non-linea equa ions ha a ise in op ics. Tzi zéi- ca’s s udy [42]in 1910 es ablished he TDBM equa ion. Tzi zéica ype equa ion is he imp o ed o m o TDBM and engaged in a ious esea ches o he p e ious decades, comp ising in wo ks ([44]and [41]). The ac ional o m o hese equa ions is gi en like as: 𝑡𝐷𝐺𝐹𝐷 2𝛼𝑢−𝑥𝐷𝐺𝐹𝐷 2𝛽𝑢−𝑒𝑢+𝑒−2𝑢=0,(1) 𝑡𝐷𝐺𝐹𝐷 𝛼 𝑥𝐷𝐺𝐹𝐷 𝛽𝑢+𝑒𝑢+𝑒−2𝑢=0,(2) 𝑡𝐷𝐺𝐹𝐷 𝛼 𝑥𝐷𝐺𝐹𝐷 𝛽𝑢−𝑒𝑢−𝑒−2𝑢=0,(3) whe e 𝑡 >0and 0 <𝛼, 𝛽≤1. Se e al scien ific demons a ions a e c ea ed using hese equa ions, such as disloca ions in c ys als, physics, non- linea op ics, and mechanics. The main objec i e o his wo k is o de elop no el o m o wa e solu ions o ac ional Tzi zéica ype NLEs using gene alized ac ional de i a i e. The ac ional Tzi zeica- ype nonlinea e olu ion equa ions appea in such p oblems in which fluid flow is a ying o quan um heo y. Fu he mo e, hese equa ions pa icipa e in many a enas such as he ci cula ion o fluxons in Josephson junc ions be ween wo supe conduc o s, nonlinea op ics, he mo ion o inflexible weigh s a ached o a s e ched wi e, disloca ions in me als and solid s a e physics. To he bes o ou cu en knowledge, he EDAM has no ye been employed o gene alized ac ional Tzi zeica- ype nonlinea e olu ion equa ions o disco e soli on solu ions. The applica ion o EDAM ex ends o se e al fields o non-linea sciences. Howe e , his me hod is imp o ed and applied on a non-linea ac ional models. In his s udy, ou p ima y ocus is o es ablish ad anced and widely applicable soli on solu ions o gene alized ac ional Tzi zeica- ype nonlinea e olu ion equa ions using he sugges ed me hod. The es ablished solu ions show wa e-like beha io and a e s a ed in igonome ic, exponen ial and hype bolic o ms. Fu he mo e, he soli on solu ions a ained om his s udy will also subsidize o he in e p e a ion o complex phenomena associa ed wi h hese specific ac ional models. This a icle is s uc u ed as ollows: In Sec . 2, some p ope ies o gene alized ac ional de i a i e and na a i e o ex ended di ec algeb aic me hod a e p esen ed. In Sec . 3, he solu ions o ac ional Tzi zéica ype e olu ion equa ions ha e been acqui ed using he unde discussion echnique. In Sec . 4, g aphical illus a ion o some selec ed solu ions has been gi en. Las ly, findings o his s udy a e gi en in Sec . 5. 2. The gene alized ac ional de i a i e This sec ion deals wi h ew basic defini ions and concep s abou he gene alized ac ional-de i a i e (GFD) [4]. Defini ion 1. I :(0; ∞) ⟶ℜ hen he GFD o 𝑓o o de 0 <𝛼≤1is s a ed as 𝑡𝐷𝐺𝐹𝐷 𝛼𝑓(𝑡) = lim 𝜖→0 𝑓(𝑡+Γ(𝜚) Γ(𝜚−𝛼+1) 𝜖𝑡 1−𝛼)−𝑓(𝑡) 𝜖,𝜚>−1,𝜚∈ℜ+.(4) Some basic esul s and o mulas o GFD a e po ed, as Theo em 1. I 𝛼∈(0, 1] and 𝑓, 𝑔be 𝛼-diffe en iable a a poin , hen ∙𝑡𝐷𝐺𝐹𝐷 𝛼(𝑎𝑓 +𝑏𝑔)=𝑎𝑡𝐷𝐺𝐹 𝐷 𝛼𝑓+𝑏𝑡𝐷𝐺𝐹𝐷 𝛼𝑔, ∀𝑎, 𝑏 ∈ℜ. ∙𝑡𝐷𝐺𝐹𝐷 𝛼(𝑡𝑣)=𝑣Γ(𝜚) Γ(𝜚−𝛼+1)𝑡𝑣−𝛼,∀𝑣>−1,𝑣∈ℜ. ∙𝑡𝐷𝐺𝐹𝐷 𝛼(𝑓𝑔)=𝑓𝑡𝐷𝐺𝐹𝐷 𝛼𝑔+𝑔𝑡𝐷𝐺𝐹𝐷 𝛼𝑓. ∙𝑡𝐷𝐺𝐹𝐷 𝛼(𝑓 𝑔)=𝑔𝑡𝐷𝐺𝐹𝐷 𝛼𝑓−𝑓𝑡𝐷𝐺𝐹𝐷 𝛼𝑔 𝑔2. I 𝑓is diffe en iable, hen ∙𝑡𝐷𝐺𝐹 𝐷 𝛼𝑓(𝑡) =𝑣Γ(𝜚) Γ(𝜚−𝛼+1) 𝑡𝑣−𝛼𝑑𝑓 𝑑𝑡 . 2.1. Na a i e o he me hod Assume he ollowing NFPDE 𝐻(𝑢, 𝑡𝐷𝐺𝐹𝐷 𝛼𝑢, 𝑥𝐷𝐺𝐹𝐷 2𝛽𝑢, 𝑡𝐷𝐺𝐹𝐷 2𝛼𝑢, 𝑥𝐷𝐺𝐹𝐷 2𝛽𝑢, 𝑡𝐷𝐺𝐹𝐷 𝛼 𝑥𝐷𝐺𝐹𝐷 𝛽𝑢, ....)=0,(5) Alexand ia Enginee ing Jou nal 92 (2024) 102–116 104 N. Ullah, H.U. Rehman, M.I. Asjad e al. whe e 0 <𝛼, 𝛽≤1and 𝑢 =𝑢(𝑥, 𝑡)is an unknown unc ion, 𝐷𝛼 𝑡𝑢and 𝐷𝛽 𝑥𝑢a e GFD o 𝑢and 𝐻is a polynomial in 𝑢and i s de i a i e. The s eps o ex ended di ec algeb aic me hod a e elabo a ed as ollows: S ep 1: Pu sue he wa e a iable o Eq. (5) o change FPDE in o ODE, as 𝑢(𝑥, 𝑡)=𝑢(𝜉),𝜉=𝑘Γ(𝜚−𝛽+1) 𝛽Γ(𝜚)𝑥𝛽−𝐿Γ(𝜚−𝛼+1) 𝛼Γ(𝜚)𝑡𝛼,(6) whe e 𝑘and 𝐿a e cons an s. Using Eq. (6) in o Eq. (5)and yield a NODE as ollows: 𝑅(𝑢, −𝐿𝑢′,𝑘𝑢 ′,𝐿 2𝑢′,𝑘 2𝑢′′,−𝑘𝐿𝑢′′, ...)=0,(7) he e p ime indica es he de i a i es. S ep 2: On in eg a ion Eq. (7)as pe need and keep he in eg a ion cons an s equal o ze o. An Eq. (7)adop he solu ions as 𝑢(𝜉)= 𝑁 ∑ 𝑖=0 𝑓𝑖Ψ𝑖(𝜉),(8) whe e 𝑓𝑖(0 ≤𝑖 ≤𝑁)a e eal numbe s. Ψ′(𝜉)=𝐿𝑛(𝑃)(𝛿+ΛΨ(𝜉)+𝜎Ψ2(𝜉)),𝑃≠0,1.(9) The amilies o solu ions o (9)a e gi en, like as Family 1: I Λ2−4𝛿𝜎 < 0𝑎𝑛𝑑 𝜎 ≠0, Ψ1(𝜉)=−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉), Ψ2(𝜉)=−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉), Ψ3(𝜉)=−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉)± √−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉), Ψ4(𝜉)=−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉)± √−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉), Ψ5(𝜉)=−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎 4𝜉)−√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝐴(√−(Λ2−4𝛿𝜎) 4𝜉). Family 2: I Λ2−4𝛿𝜎 > 0𝑎𝑛𝑑 𝜎 ≠0, Ψ6(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉), Ψ7(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜ℎ𝑡𝑃(√(Λ2−4𝛿𝜎) 2𝜉), Ψ8(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎𝜉)±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√Λ2−4𝛿𝜎𝜉), Ψ9(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√Λ2−4𝛿𝜎𝜉)± √𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√Λ2−4𝛿𝜎𝜉), Ψ10(𝜉)=−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉)− √Λ2−4𝛿𝜎 4𝜎𝑐𝑜𝑡ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉). Family 3: I 𝛿𝜎 > 0𝑎𝑛𝑑 Λ =0, Ψ11(𝜉)=√𝛿 𝜎𝑡𝑎𝑛𝑃(√𝛿𝜎𝜉), Ψ12(𝜉)=− √𝛿 𝜎𝑐𝑜𝑡𝑃(√𝛿𝜎𝜉), Ψ13(𝜉)=√𝛿 𝜎𝑡𝑎𝑛𝑃(2√𝛿𝜎𝜉)±√𝑟𝑠 𝛿 𝜎𝑠𝑒𝑐𝑃(2√𝛿𝜎𝜉), Ψ14(𝜉)=− √𝛿 𝜎𝑐𝑜𝑡𝑃(2√𝛿𝜎𝜉)±√𝑟𝑠 𝛿 𝜎𝑐𝑠𝑐𝑃(2√𝛿𝜎𝜉), Ψ15(𝜉)= 1 2(√𝛿 𝜎𝑡𝑎𝑛𝐴(√𝛿𝜎 2𝜉)−√𝛿 𝜎𝑐𝑜𝑡𝑃(√𝛿𝜎 2𝜉)). Family 4: I 𝛿𝜎 < 0𝑎𝑛𝑑 Λ =0, Alexand ia Enginee ing Jou nal 92 (2024) 102–116 105 N. Ullah, H.U. Rehman, M.I. Asjad e al. Ψ16(𝜉)=− √−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(√−𝛿𝜎𝜉), Ψ17(𝜉)=− √−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(√−𝛿𝜎𝜉), Ψ18(𝜉)=− √−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(2√−𝛿𝜎𝜉)±𝜄√−𝑟𝑠 𝛿 𝜎𝑠𝑒𝑐ℎ𝑃(2√−𝛿𝜎𝜉), Ψ19(𝜉)=− √−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(2√−𝛿𝜎𝜉)±√−𝑟𝑠 𝛿 𝜎𝑐𝑠𝑐ℎ𝑃(2√−𝛿𝜎𝜉), Ψ20(𝜉)=−1 2(√−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(√−𝛿𝜎 2𝜉)+√−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(√−𝛿𝜎 2𝜉)). Family 5: I Λ =0𝑎𝑛𝑑 𝜎 =𝛿, Ψ21(𝜉)=𝑡𝑎𝑛𝑃(𝛿𝜉), Ψ22(𝜉)=−𝑐𝑜𝑡𝑃(𝛿𝜉), Ψ23(𝜉)=𝑡𝑎𝑛𝑃(2𝛿𝜉)±√𝑟𝑠𝑠𝑒𝑐 𝑃(2𝛿𝜉), Ψ24(𝜉)=−𝑐𝑜𝑡𝑃(2𝛿𝜉)±√𝑟𝑠𝑐𝑠𝑐 𝑃(2𝛿𝜉), Ψ25(𝜉)= 1 2(𝑡𝑎𝑛𝑃(𝛿 2𝜉)−𝑐𝑜𝑡𝑃(𝛿 2𝜉)). Family 6: I Λ =0𝑎𝑛𝑑 𝜎 =−𝛿, Ψ26(𝜉)=−𝑡𝑎𝑛ℎ𝑃(𝛿𝜉), Ψ27(𝜉)=−𝑐𝑜𝑡ℎ𝑃(𝛿𝜉), Ψ28(𝜉)=−𝑡𝑎𝑛ℎ𝑃(2𝛿𝜉)±𝜄√𝑟𝑠𝑠𝑒𝑐ℎ 𝑃(2𝛿𝜉), Ψ29(𝜉)=−𝑐𝑜𝑡ℎ𝐴(2𝛿𝜉)±√𝑟𝑠𝑐𝑠𝑐ℎ 𝑃(2𝛿𝜉), Ψ30(𝜉)=−1 2(𝑡𝑎𝑛ℎ𝑃(𝛿 2𝜉)+𝑐𝑜𝑡ℎ𝑃(𝛿 2𝜉)). Family 7: I Λ2=4𝛿𝜎, Ψ31(𝜉)= −2𝛿(Λ𝜉𝐿𝑛(𝑃)+2) Λ2𝜉𝐿𝑛(𝑃). Family 8: I Λ =𝑙, 𝛿 =𝑛𝑙 (𝑛 ≠0) 𝑎𝑛𝑑 𝜎 =0, Ψ32(𝜉)=𝑃𝑙𝜉 −𝑛. Family 9: I Λ =𝜎=0, Ψ33(𝜉)=𝛿𝜉𝐿𝑛(𝑃). Family 10: I Λ =𝛿=0, Ψ34(𝜉)= −1 𝜎𝜉𝐿𝑛(𝑃). Family 11: I 𝛿=0𝑎𝑛𝑑 Λ ≠0, Ψ35(𝜉)=− 𝑃Λ 𝜎(𝑐𝑜𝑠ℎ𝐴(Λ𝜉)−𝑠𝑖𝑛ℎ𝑃(Λ𝜉+𝑟), Ψ36(𝜉)=− Λ(𝑠𝑖𝑛ℎ𝑃(Λ𝜉)+𝑐𝑜𝑠ℎ𝑃(Λ𝜉)) 𝜎(𝑠𝑖𝑛ℎ𝑃(Λ𝜉)+𝑐𝑜𝑠ℎ𝑃(Λ𝜉)+𝑠). Family 12: I Λ =𝑙, 𝜎 =𝑛𝑙, (𝑛 ≠0) 𝑎𝑛𝑑 𝛿 =0, Ψ37(𝜉)= 𝑟𝑃 𝑙𝜉 𝑠−∓𝑃𝑙𝜉 . No e: The gene alized hype bolic and iangula unc ions a e desc ibed like as [40] 𝑠𝑖𝑛ℎ𝑃(𝜉)=𝑟𝑃 𝜉−𝑠𝑃 −𝜉 2,𝑐𝑜𝑠ℎ 𝑃(𝜉)= 𝑟𝑃 𝜉+𝑠𝑃 −𝜉 2, 𝑡𝑎𝑛ℎ𝑃(𝜉)=𝑟𝑃 𝜉−𝑠𝑃 −𝜉 𝑟𝑃 𝜉+𝑠𝑃 −𝜉,𝑐𝑜𝑡ℎ 𝑃(𝜉)=𝑟𝑃 𝜉+𝑠𝑃 −𝜉 𝑟𝑃 𝜉−𝑠𝑃 −𝜉, 𝑠𝑒𝑐ℎ𝑃(𝜉)= 2 𝑟𝑃 𝜉+𝑠𝑃 −𝜉,𝑐𝑠𝑐ℎ 𝑃(𝜉)= 2 𝑟𝑃 𝜉−𝑠𝑃 −𝜉, 𝑠𝑖𝑛𝑃(𝜉)=𝑟𝑃 𝜄𝜉 −𝑠𝑃 −𝜄𝜉 2𝜄,𝑐𝑜𝑠 𝑃(𝜉)=𝑟𝑃 𝜄𝜉 +𝑠𝑃 −𝜄𝜉 2, Alexand ia Enginee ing Jou nal 92 (2024) 102–116 106 N. Ullah, H.U. Rehman, M.I. Asjad e al. 𝑡𝑎𝑛𝑃(𝜉)=−𝜄𝑟𝑃 𝜄𝜉 −𝑠𝑃 −𝜄𝜉 𝑟𝑃 𝜄𝜉 +𝑠𝑃 −𝜄𝜉 ,𝑐𝑜𝑡 𝑃(𝜉)=𝜄𝑟𝑃 𝜄𝜉 +𝑠𝑃 −𝜄𝜉 𝑟𝑃 𝜄𝜉 −𝑠𝑃 −𝜄𝜉 , 𝑠𝑒𝑐𝑃(𝜉)= 2 𝑟𝑃 𝜄𝜉 +𝑠𝑃 −𝜄𝜉 ,𝑐𝑠𝑐 𝑃(𝜉)= 2𝜄 𝑟𝑃 𝜄𝜉 −𝑠𝑃 −𝜄𝜉 , whe e 𝑟, 𝑠 >0. S ep 3: The alue o 𝑁can be ound by applying balance p inciple echnique in (7). Subs i u ing (6) in o (5), gi es a se o algeb aic equa ions in ol ing powe s o Ψ𝑖(𝜉)(𝑖 =0, 1, 2, ...). Compa ing he coefficien s o Ψ(𝜉) o ze o yields a bundle o equa ions. S ep 4: On solu ions he se o equa ions using Maple so wa e, hen inse he esul s in o (9) o acqui e he solu ions o (1). 3. The Tzi zéica equa ion The ac ional Tzi zéica equa ion is s a ed as 𝑡𝐷𝐺𝐹𝐷 2𝛼𝑢−𝑥𝐷𝐺𝐹𝐷 2𝛽𝑢−𝑒𝑢+𝑒−2𝑢=0,(10) whe e 0 <𝛼, 𝛽≤1and he pa ame e s 𝛼and 𝛽indica ing he ac ional de i a i es in ime and space. Using he ollowing wa e ans o ma ions 𝑢(𝑥, 𝑡)=𝑢(𝜉),𝜉=𝑘Γ(𝜚−𝛽+1) 𝛽Γ(𝜚)𝑥𝛽−𝐿Γ(𝜚−𝛼+1) 𝛼Γ(𝜚)𝑡𝛼,(11) con e s Eq. (10)as ollows: (𝐿2−𝑘2)𝑢′′ −𝑒𝑢+𝑒−2𝑢=0,(12) whe e 𝑑𝑢 𝑑𝜉 =𝑢′. By choosing he ans o ma ion 𝑙𝑛𝑣 =𝑢o equi alen ly 𝑒𝑢=𝑣, Eq. (12), gi es (𝐿2−𝑘2)(𝑣𝑣′′ −(𝑣′)2)−𝑣3+1=0.(13) 3.1. Applica ions o he new EDAM Using he balance p inciple on he e ms 𝑣3and 𝑣𝑣′′, we ge 𝑁=2. So, Eq. (8)con e s o he ollowing o m 𝑢(𝜉)=𝑓0+𝑓1Ψ(𝜉)+𝑓2Ψ(𝜉)2,(14) whe e 𝑓0, 𝑓1and 𝑓2a e cons an s. Plugging (14)in (13)and equa ing he a bi a y cons an s o Ψ(𝜉) o ze o, a se o equa ions in Λ,𝜎,𝑓 0,𝑓 1,𝑓 2,𝐿 and 𝑘is achie ed. On solu ion i , we ge 𝑓0=−3 5,𝑓 1=4 5√𝜎 𝛿,𝑓 2=−8 5 𝜎 𝛿, 𝐿=√−1 5 −5𝛿𝜎𝐿𝑛(𝑃)2𝑘2+4 𝛿𝜎 𝐿𝑛(𝑃),𝑘=𝑘, Λ=−1 2√𝜎𝛿. (15) Eqns. (13), (14)and (15)gi e he amilies o solu ions o (10), like as: Family 1: I Λ2−4𝛿𝜎 < 0and 𝜎≠0, hen 𝑢1(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉))2],(16) 𝑢2(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉))2],(17) 𝑢3(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))2],(18) Alexand ia Enginee ing Jou nal 92 (2024) 102–116 107 N. Ullah, H.U. Rehman, M.I. Asjad e al. 𝑢4(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))2],(19) 𝑢5(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 4𝜉) −√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 4𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 4𝜉) −√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 4𝜉))2].(20) Family 2: I Λ2−4𝛿𝜎 > 0. and 𝜎≠0, hen 𝑢6(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉))2],(21) 𝑢7(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉))2],(22) 𝑢8(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))2],(23) 𝑢9(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))2],(24) 𝑢10(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉) −√(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 4𝜉)) −8 5 𝜎 𝛿(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉) Alexand ia Enginee ing Jou nal 92 (2024) 102–116 108 N. Ullah, H.U. Rehman, M.I. Asjad e al. −√(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 4𝜉))2].(25) Family 3: I Λ2=4𝛿𝜎, hen 𝑢11(𝑥, 𝑡)=ln[−3 5+4 5√𝜎 𝛿(−2 𝛿(Λ 𝜉𝐿𝑛(𝑃)+2 Λ2𝜉𝐿𝑛(𝑃))−8 5 𝜎 𝛿(−2 𝛿(Λ 𝜉𝐿𝑛(𝑃)+2 Λ2𝜉𝐿𝑛(𝑃))2].(26) 3.2. The DBM equa ion The ac ional DBM equa ion is s a ed as 𝑡𝐷𝐺𝐹𝐷 𝛼 𝑥𝐷𝐺𝐹𝐷 𝛽𝑢+𝑒𝑢+𝑒−2𝑢=0.(27) Using he Eq. (11), we con e Eq. (27) o −𝐿𝑘𝑢′′ +𝑒𝑢+𝑒−2𝑢=0,(28) whe e 𝑑𝑢 𝑑𝜉 =𝑢′. By u ilizing he ans o ma ion 𝑢 =𝑙𝑛𝑣 o equi alen ly 𝑣 =𝑒𝑢, Eq. (28), educes o 𝑘𝐿(−𝑣𝑣′′ +(𝑣′)2)+𝑣3+1=0.(29) 3.3. Applica ion o he new EDAM He e, we u ilize he EDAM o he solu ions o DBM equa ion. U ilizing balancing ule on (29), yields 𝑁=2, hus (8)con e s o 𝑢(𝜉)=𝑓0+𝑓1Ψ(𝜉)+𝑓2Ψ(𝜉)2,(30) whe e 𝑓0, 𝑓1and 𝑓2a e cons an s. Plugging (30)in (29)and equa ing he a bi a y cons an s o Ψ(𝜉) o ze o, a sys em o equa ions in 𝑓0,𝑓 1,𝑓 2,Λ,𝑘, and 𝛿is achie ed. On solu ion i , we ge 𝑓0=1 2+1 2kL 𝐿𝑛(𝑃)2Λ2,𝑓 1=2kL Λ𝜎𝐿𝑛(𝑃)2, 𝑓2=2kL 𝜎2𝐿𝑛(𝑃)2,Λ=Λ ,𝐿=𝐿, 𝑘 =𝑘, 𝛿=3+kL 𝐿𝑛(𝑃)2Λ2 4kL 𝜎𝐿𝑛(𝑃)2.(31) Eqns. (29), (30)and (31)gi e he amilies o solu ions o (28), like as: Family 1: I Λ2−4𝛿𝜎 < 0and 𝜎≠0, hen 𝑢1(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉)) +4𝜎2(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 2𝜉))2))],(32) 𝑢2(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉)) +4𝜎2(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 2𝜉))2))],(33) 𝑢3(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))+4𝜎2(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))2))],(34) 𝑢4(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))+4𝜎2(−Λ 2𝜎−√−(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎)𝜉) ±√−𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐𝑃(√−(Λ2−4𝛿𝜎)𝜉))2))],(35) Alexand ia Enginee ing Jou nal 92 (2024) 102–116 109 N. Ullah, H.U. Rehman, M.I. Asjad e al. 𝑢5(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 4𝜉) −√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 4𝜉))+4𝜎2(−Λ 2𝜎+√−(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛𝑃(√−(Λ2−4𝛿𝜎) 4𝜉) −√−(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡𝑃(√−(Λ2−4𝛿𝜎) 4𝜉))2))].(36) Family 2: I Λ2−4𝛿𝜎 > 0. and 𝜎≠0, hen 𝑢6(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉)) +4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉))2))],(37) 𝑢7(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉)) +4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 2𝜉))2))],(38) 𝑢8(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))+4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑡𝑎𝑛ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±𝜄√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑠𝑒𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))2))],(39) 𝑢9(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))+4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 2𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉) ±√𝑟𝑠(Λ2−4𝛿𝜎) 2𝜎𝑐𝑠𝑐ℎ𝑃(√(Λ2−4𝛿𝜎)𝜉))2))],(40) 𝑢10(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉) −√(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 4𝜉))+4𝜎2(−Λ 2𝜎−√(Λ2−4𝛿𝜎) 4𝜎𝑡𝑎𝑛ℎ𝑃(√Λ2−4𝛿𝜎 4𝜉) −√(Λ2−4𝛿𝜎) 4𝜎𝑐𝑜𝑡ℎ𝑃(√(Λ2−4𝛿𝜎) 4𝜉))2))].(41) Family 3: I 𝛿𝜎 > 0𝑎𝑛𝑑 Λ =0, hen 𝑢11(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√𝛿 𝜎𝑡𝑎𝑛𝑃(√𝛿𝜎𝜉))2],(42) 𝑢12(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√𝛿 𝜎𝑐𝑜𝑡𝑃(√𝛿𝜎𝜉))2],(43) 𝑢13(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(√𝛿 𝜎𝑡𝑎𝑛𝑃(2√𝛿𝜎𝜉)±√𝑟𝑠 𝛿 𝜎𝑠𝑒𝑐𝑃(2√𝛿𝜎𝜉))2],(44) 𝑢14(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√𝛿 𝜎𝑐𝑜𝑡𝑃(2√𝛿𝜎𝜉)±√𝑟𝑠 𝛿 𝜎𝑐𝑠𝑐𝑃(2√𝛿𝜎𝜉))2],(45) 𝑢15(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(1 2(√𝛿 𝜎𝑡𝑎𝑛𝑃(√𝛿𝜎 2𝜉)−√𝛿 𝜎𝑐𝑜𝑡𝑃(√𝛿𝜎 2)𝜉))2].(46) Family 4: I 𝛿𝜎 < 0𝑎𝑛𝑑 Λ =0, hen Alexand ia Enginee ing Jou nal 92 (2024) 102–116 110 N. Ullah, H.U. Rehman, M.I. Asjad e al. 𝑢16(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(√−𝛿𝜎𝜉))2],(47) 𝑢17(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(√−𝛿𝜎𝜉))2],(48) 𝑢18(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(2√−𝛿𝜎𝜉)±𝜄√−𝑟𝑠 𝛿 𝜎𝑠𝑒𝑐ℎ𝑃(2√−𝛿𝜎𝜉))2],(49) 𝑢19(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−√−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(2√−𝛿𝜎𝜉 ±√−𝑟𝑠 𝛿 𝜎𝑐𝑠𝑐ℎ𝑃(2√−𝛿𝜎𝜉))2],(50) 𝑢20(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−1 2(√−𝛿 𝜎𝑡𝑎𝑛ℎ𝑃(√−𝛿𝜎 2𝜉)+√−𝛿 𝜎𝑐𝑜𝑡ℎ𝑃(√−𝛿𝜎 2𝜉)))2].(51) Family 5: I Λ =0𝑎𝑛𝑑 𝜎 =𝛿, hen 𝑢21(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(𝑡𝑎𝑛𝑃(𝛿𝜉))2],(52) 𝑢22(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑐𝑜𝑡𝑃(𝛿𝜉))2],(53) 𝑢23(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(𝑡𝑎𝑛𝑃(2𝛿𝜉)±√𝑟𝑠𝑠𝑒𝑐 𝑃(2𝛿𝜉))2],(54) 𝑢24(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑐𝑜𝑡𝑃(2𝛿𝜉)±√𝑝𝑞 𝑐𝑠𝑐𝑃(2𝛿𝜉))2],(55) 𝑢25(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(1 2(𝑡𝑎𝑛𝑃(𝛿 2𝜉)−𝑐𝑜𝑡𝑃(𝛿 2𝜉)))2].(56) Family 6: I Λ =0𝑎𝑛𝑑 𝜎 =−𝛿, hen 𝑢26(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑡𝑎𝑛ℎ𝑃(𝛿𝜉))2],(57) 𝑢27(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑐𝑜𝑡ℎ𝑃(𝛿𝜉))2],(58) 𝑢28(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑡𝑎𝑛ℎ𝑃(2𝛿𝜉)±𝜄√𝑟𝑠𝑠𝑒𝑐ℎ 𝑃(2𝛿𝜉))2],(59) 𝑢29(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−𝑐𝑜𝑡ℎ𝑃(2𝛿𝜉)±√𝑟𝑠𝑐𝑠𝑐ℎ 𝑃(2𝛿𝜉))2],(60) 𝑢30(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−1 2(𝑡𝑎𝑛ℎ𝑃(𝛿 2𝜉)+𝑐𝑜𝑡ℎ𝑃(𝛿 2𝜉)))2].(61) Family 7: I Λ2=4𝛿𝜎, hen 𝑢31(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(−2 𝛿(Λ 𝜉𝐿𝑛(𝑃)+2 Λ2𝜉𝐿𝑛(𝑃)) +4𝜎2(−2 𝛿(Λ 𝜉𝐿𝑛(𝑃)+2 Λ2𝜉𝐿𝑛(𝑃))2))].(62) Family 8: I Λ =𝛿=0, hen 𝑢32(𝑥, 𝑡)=ln[1 2+2 kL 𝜎2𝐿𝑛(𝑃)2(−1 𝜎𝜉𝐿𝑛(𝑃))2]. Family 9: I Λ =𝑙, 𝜎 =𝑛𝑙, (𝑛 ≠0) 𝑎𝑛𝑑 𝛿 =0, hen 𝑢33(𝑥, 𝑡)=ln[1 2(1+kL 𝐿𝑛(𝑃)2(Λ2+4Λ𝜎(𝑟𝑃 𝑙𝜉 𝑠−∓𝑃𝑙𝜉 )+4𝜎2(𝑟𝑃 𝑙𝜉 𝑠−∓𝑃𝑙𝜉 )2))].(63)