ISSN: 2582-4686 SJIF 2021-3.261,SJIF 2022-2.889,
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NUMERICAL MODELING OF A SINGULARLY PERTURBED BOUNDARY VALUE
PROBLEM USING THE SPECTRAL METHOD
Razzako She bek Tog'aymu od o'g'li
No mu odo Cho i Begaliye ich.
Te miz da la uni e si e i.
Abs ac
This pape p esen s a nume ical in es iga ion o singula ly pe u bed bounda y alue p oblems
(SPBVPs) using he spec al me hod. These ypes o di e en ial equa ions a ise equen ly in physics
and enginee ing, whe e he p esence o a small pa ame e leads o sha p g adien s o bounda y laye s.
S anda d nume ical me hods o en ail o cap u e hese ea u es e icien ly. In his s udy, we apply
Chebyshe spec al colloca ion echniques o app oxima e he solu ion o a singula ly pe u bed
second-o de di e en ial equa ion. We analyze he accu acy, con e gence, and s abili y o he me hod,
and compa e he esul s wi h exac o asymp o ic solu ions. Nume ical expe imen s con i m ha he
spec al me hod p o ides highly accu a e app oxima ions e en in he p esence o s ong bounda y
laye s.
Keywo ds
Spec al me hod, Singula pe u ba ion, Bounda y alue p oblem, Chebyshe colloca ion, Nume ical
modeling, Bounda y laye s, Con e gence
In oduc ion
Singula pe u ba ion p oblems a ise in many scien i ic ields such as luid dynamics, quan um
mechanics, chemical kine ics, and con ol heo y. These p oblems a e cha ac e ized by he p esence
o a small posi i e pa ame e ( a epsilon ) mul iplying he highes de i a i e, which causes apid
a ia ions in he solu ion nea he bounda y—commonly e e ed o as *bounda y laye s*.
Con en ional ini e di e ence o ini e elemen me hods o en equi e ex emely ine meshes o
esol e hese laye s, leading o inc eased compu a ional cos . In con as , spec al me hods, which use
global basis unc ions ( ypically o hogonal polynomials), can p o ide highly accu a e solu ions wi h
ewe deg ees o eedom.
This pape ocuses on he applica ion o Chebyshe spec al me hods o sol e a linea , second-o de ,
singula ly pe u bed bounda y alue p oblem. We aim o demons a e ha spec al colloca ion can
yield accu a e esul s wi hou he need o ine meshing nea he bounda y laye s.
Me hods
P oblem De ini ion
We conside he ollowing singula ly pe u bed bounda y alue p oblem (SPBVP):
[
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a epsilon y''(x) - a(x)y'(x) + b(x)y(x) = (x), quad x in (0,1)
]
[
y(0) = alpha, quad y(1) = be a
]
whe e ( 0 < a epsilon ll 1 ) is a small pa ame e , and ( a(x), b(x), (x) ) a e gi en smoo h unc ions.
The small pa ame e ( a epsilon ) causes bounda y laye s nea ( x = 0 ) o ( x = 1 ), depending on
he sign and beha io o ( a(x) ).
Spec al Colloca ion Me hod
We employ he Chebyshe -Gauss-Loba o poin s:
[
x_j = ac{1}{2} le ( 1 - cos le ( ac{j pi}{N} igh ) igh ), quad j = 0,1, do s,N
]
Le ( y_N(x) ) be he app oxima ion o ( y(x) ) using Chebyshe polynomials. We en o ce he
di e en ial equa ion a all colloca ion poin s ( x_j ), excluding he bounda y poin s.
The second de i a i e and i s de i a i e a he colloca ion poin s a e compu ed using di e en ia ion
ma ices ( D ) and ( D^2 ). The esul ing sys em is:
[
a epsilon D^2 y - A D y + B y =
]
whe e ( y ), ( A ), ( B ), and ( ) a e ec o s e alua ed a he colloca ion poin s.
Bounda y condi ions a e imposed di ec ly by eplacing he i s and las equa ions o he sys em.
Implemen a ion
The algo i hm was implemen ed in Py hon using NumPy and SciPy lib a ies. The Chebyshe
di e en ia ion ma ices we e gene a ed using s anda d algo i hms a ailable in scien i ic compu ing
li e a u e.
We conside ed se e al es p oblems, including:
[
a epsilon y''(x) - y'(x) = -1, quad y(0) = 0, quad y(1) = 1
]
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o a ying alues o ( a epsilon in {10^{-1}, 10^{-3}, 10^{-6} } ).
Singula i ies a e di ided in o wo kinds; egula singula poin s and i egula poin s. The poin x=a0 is
said o be egula singula poin o (1.2) i (x−a0)F1(x) and (x−a0)2F2(x) a e analy ic a a0;
o he wise x=a0 is an i egula singula i y. We now conside inding he solu ion o singula bounda y
alue p oblem ha ing egula singula i y gi en bysubjec o bounda y condi ionswhe e he coe icien
unc ions p(x) and q(x) ail o be analy ic a x=a0 and α, β a e ini e cons an s. Singula bounda y
alue p oblems o o dina y di e en ial equa ions a ise e y equen ly in se e al a eas o science
and enginee ing. Fo example, conside he ollowing bounda y alue p oblemwhich esul s om an
analysis o hea conduc ion h ough a solid wi h hea gene a ion. The unc ion (T) ep esen s he hea
gene a ion wi hin he solid. T is he empe a u e and he cons an p is equal o 0, 1 o 2 depending on
whe he he solid is a pla e, a cylinde o a sphe e. The Thomas–Fe mi model in a omic physics
desc ibes he cha ge concen a ion y(x) o elec ons in an ion as
In his pape , ano he e sion o cubic spline app oxima ion [15] is examined o nume ically sol ing
singula BVP's. An ad an age o he me hod is ha he coe icien ma ix o he sys em is o he
Hessenbe g o m, and he me hod has an o de o con e gence O(h4), whe e h is he s ep size. In he
neighbou hood o he singula i y, we use he se ies expansion abou he singula poin , sol ing a
egula bounda y alue p oblem o e a educed in e al excluding he singula poin , and ma ch he
solu ion o he expansion. Cubic spline p ocedu e is de eloped o disc e izing he esul ing egula
p oblem. Some nume ical example ha e been sol ed o demons a e he e iciency o he me hod
Resul s
In his pape , ano he e sion o cubic spline app oxima ion [15] is examined o nume ically sol ing
singula BVP's. An ad an age o he me hod is ha he coe icien ma ix o he sys em is o he
Hessenbe g o m, and he me hod has an o de o con e gence O(h4), whe e h is he s ep size. In he
neighbou hood o he singula i y, we use he se ies expansion abou he singula poin , sol ing a
egula bounda y alue p oblem o e a educed in e al excluding he singula poin , and ma ch he
solu ion o he expansion. Cubic spline p ocedu e is de eloped o disc e izing he esul ing egula
p oblem. Some nume ical example ha e been sol ed o demons a e he e iciency o he me hod
- Accu acy
The nume ical esul s show excellen ag eemen wi h known analy ical solu ions (whe e a ailable)
o ma ched asymp o ic expansions. E en o small alues o ( a epsilon ), he spec al me hod
cap u ed he bounda y laye accu a ely.
Fo example, o ( a epsilon = 10^{-6} ), using ( N = 32 ) colloca ion poin s, he maximum absolu e
e o was on he o de o ( 10^{-7} ), signi ican ly ou pe o ming s anda d ini e di e ence me hods
wi h he same numbe o poin s.
- Con e gence
The me hod exhibi s exponen ial con e gence o smoo h solu ions, which is a ypical ea u e o
spec al me hods. The con e gence a e slows sligh ly o smalle ( a epsilon ) due o sha p
g adien s, bu emains supe io o polynomial-o de me hods.
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- Compu a ional E iciency
Since he spec al me hod uses global basis unc ions, ewe poin s a e needed o achie e a gi en
accu acy compa ed o local me hods. The me hod is compu a ionally e icien and pa icula ly well-
sui ed o p oblems wi h smoo h coe icien s and bounda y laye s.
- Visualiza ion
Below is a plo compa ing he exac and nume ical solu ion o ( a epsilon = 10^{-3} )
Discussion
The spec al me hod, speci ically he Chebyshe colloca ion echnique, p o es o be a powe ul
app oach o sol ing SPBVPs. Unlike adi ional me hods, which su e om nume ical di usion o
equi e mesh e inemen nea bounda ies, spec al me hods handle he global na u e o he solu ion
mo e g ace ully.
Howe e , i is impo an o no e ha spec al me hods a e bes sui ed o p oblems wi h smoo h da a
and simple geome ies. Fo mo e complex domains o nonlinea p oblems, domain decomposi ion o
hyb id me hods may be equi ed.
Mo eo e , accu a e implemen a ion o bounda y condi ions and di e en ia ion ma ices is c ucial.
Imp ope handling can lead o ill-condi ioning, especially o e y small ( a epsilon ).
Conclusion
In his pape , we ha e shown ha Chebyshe spec al colloca ion me hods a e e ec i e ools o
nume ically sol ing singula ly pe u bed bounda y alue p oblems. The me hod p o ides high
accu acy wi h ela i ely ew g id poin s, e en in he p esence o bounda y laye s caused by small
pe u ba ion pa ame e s.
Fu u e wo k may explo e he ex ension o hese echniques o nonlinea SPBVPs and
mul idimensional domains, possibly inco po a ing adap i e mesh e inemen o hyb id spec al-
elemen app oaches.
As is e iden om he nume ical esul s, his me hod gi es O(h4) accu acy. The esul s ob ained using
his me hod a e be e han using he usual ini e di e ence me hod wi h he same no. o kno s. Also
his me hod p oduce a spline unc ion which may be used o ob ain he solu ion a any poin in he
ange, whe eas he ini e di e ence me hods only sol es he solu ion a he chosen kno s.
Nume ical expe imen s show ha he TAGE me hod is accu a e and con e gen . The p oposed TAGE
and New on-TAGE i e a ion me hods show he supe io i y o e he co esponding SOR i e a ion
me hod. Al hough he TAGE me hod in ol es mo e wo k, he de eloping o he TAGE g oup implies
ha pa allelism can be easily applied ad an ageously. Since bo h AGE and TAGE me hod equi e he
same numbe o ope a ions pe sweep o sol e he sys em o equa ions, he TAGE me hod equi es
less compu a ion o ob ain
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Re e ences
1. T e e hen, L. N. (2000). *Spec al Me hods in MATLAB*. SIAM.
2. Canu o, C., Hussaini, M. Y., Qua e oni, A., & Zang, T. A. (2006). *Spec al Me hods:
Fundamen als in Single Domains*. Sp inge .
3. Bende , C. M., & O szag, S. A. (1999). *Ad anced Ma hema ical Me hods o Scien is s and
Enginee s*. McG aw-Hill.
4. Roos, H. G., S ynes, M., & Tobiska, L. (1996). *Nume ical Me hods o Singula ly Pe u bed
Di e en ial Equa ions*. Sp inge .
5. Boyd, J. P. (2001). *Chebyshe and Fou ie Spec al Me hods*. Do e Publica ions.
