Alexand ia Enginee ing Jou nal 92 (2024) 102–116
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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
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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,
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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)

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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)
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−√(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)