Mixed Integral Equation of Contact Problem in Position and Time

Transcription

1 Ieraoal Joural of Basc & Appled Sceces IJBAS-IJENS Vol: No: 3 ed Iegral Equao of Coac Problem Poso ad me. A. Abdou S. J. oaquel Deparme of ahemacs Faculy of Educao Aleadra Uversy Egyp Deparme of ahemacs Faculy of Scece Kg Abdul Azz Uversy Saud Araba abdella_777@yahoo.com;smoaquel@homal.com Absrac-- I hs work we cosder a med egral equao of he frs kd of ype Fredholm- Volerra poso ad me respecvely. he Fredholm egral erm s cosdered a varable poso he space L [ ] ad has a sgular kerel. Whle he Volerra egral erm s cosdered me he class C [ ]. Usg a umercal mehod we have a fe sysem of Fredholm egral equaos of he secod kd whch wll be solved umercally usg Chebyshev polyomals mehod. Numercal resuls are compued ad he error esmae s calculaed Ide erm-- ed sgular egral equao (SIE) lear algebrac sysem (LAS) Chebyshev polyomals (CPs) Fredholm-Volerra egral equao (F-VIE) CS: 45B5 45E.. INRODUCION ay problems mahemacal physcs coac problems he heory of elascy ad med boudary value problems lead o egral equaos of lear or olear case see [-3]. hese egral equaos wh couous or dscouous kerels have receved cosderable eres mahemacal applcaos dffere area of sceces for eample see [4-7]. he soluo of hese problems ca be obaed aalycally see [8-]. A he same me he sese of umercal mehods akes a mpora place solvg he egral equaos see [3-3]. he soluo of he SIE of ype F-VIE poso ad me oe wo ad hree dmesos was dscussed [9]. Also [9] a asympoc umercal mehod was used o dscuss he soluo of F-VIE of he secod kd. he same auhor [] appled he regular ad sgular asympoc mehod oe wo ad hree dmesoal o oba he soluo of F-VIE of he frs kd. Cosder he SIE y k F ( ) ( y ) dy d G ( ) ( ) d [ ( ) f r ( )] f ( ) ( ( ) G ( ) ( ) L [ ]) f r (.) L( u)cosu z u m y k( z) du L( u) z ( m ). (.) u u Uder he codo ( ) d P( ) [ ]. (.3) he SIE (.) wh s badly kerel of poso (.) ad he wo couous kerels of me F( τ); G( τ) uder he pressure codo (.3) s vesgaed from he coac problem of a elasc maeral of a srp G ) of hckess h ha occupes a rego y h ad les whou frco ( ( o a elasc surface G ) of equao f ( ) L [ r ]. Here G ; are called he dsplaceme magude ad Posso's rao for he upper ad lower surface respecvely IJBAS-IJENS Jue IJENS

2 Ieraoal Joural of Basc & Appled Sceces IJBAS-IJENS Vol: No: 3 Cosder a rgd recagular samp of legh a s mpressed o he boudary of he srp varable force ( ) [ ] P. hs varable force causes dsplaceme () y h by a agas he force of maeral of he coac rego F ( ). Also cosder he coac rego has he ressace force G ( ) for all [ ]. I order o guaraee he esece of a uque soluo of equao (.) uder he pressure codo (.3) we assume he followg codos y () he kerel of he poso k sasfes he space [ ] codo y s a cosa. () he wo kerels of me ) () class [ ] k ddy A A F( ad G( ) for [ ] C ad sasfes L Fredholm belog o he F B G D B Dare cosas. he gve fuco f( ) free erm fuco s couous wh s paral dervaves he space L C ad s orm s defed as: f ma f d ( (v) he ukow fuco ) poeal fuco sasfes Lpschz codo wh respec o s frs argume ad Hölder o he secod argume. I hs work we cosder uder he codos (-v) he esece ad uqueess soluo of a SIE (.). he SIE of he frs kd (.) s cosdered poso ad me. he Fredholm egral erm s cosdered poso L [ ] ad s kerel has a sgular erm ad wll be adaped o ake a logarhmc form. Whle he Volerra egral erm s cosdered me he space C [ ] ad s kerels are couous fucos wh s dervaves. Usg a suable umercal mehod wh respec o me he med egral equao s reduced o a ALS of FIEs of he secod kd. he usg CPs he soluo of he FIEs ca be obaed. Numercal resuls are compued ad he error esmae s calculaed.. HE KERNEL OF POSIION he fuco (u) L of equao (.) s couous ad posve for u ( ) sasfes he followg asympoc equales: L ( u) m ( m ) u O( u 3 ) u ad he m L ( u ) O ( u ) u m. (.) u y Whe m (.) ad (.) such ha he erm ( ) s very small we have from [6] ha cosuz du l y d u I hs case he kerel of poso akes a logarhmc fuco form. 4 d l. (.) IJBAS-IJENS Jue IJENS

3 Ieraoal Joural of Basc & Appled Sceces IJBAS-IJENS Vol: No: 3 Assume (.) followg famous relaos [6] u. he cosder he frs ad he secod appromao of u cosvdv s he Durak fuco L ad he b b a y y dy (.3) [ h a h a] a b. a Hece he SIE (.) wll be reduced o he followg egral equao: ad H ( ) ( ) d F( )l y ( y ) dyd g( ) (.4) G H F ( ) m m f g d P ( ) F d m 4 m d l m. m From he hree formulas (.4) (.6) he mporace of he physcal meag bewee m s: - he Fredholm codo for he logarhmc kerel leads o (.5) (.6) l y d dy... (.7) m m - Also from (.5) we ca esablsh ha: for large values of m he oal ressace force H wll be equal o he ressace force F ( τ).e. m s o a avalable ad he oal ressace hs case s he ressace force of maeral oly. 3- Also he gve fuco of he free erm of (.4) wll deped he case he value of he pressure P() ad he ressace force of maeral. 3. SYSE OF FREDHOL INEGRAL EQUAIONS o oba he soluo of (.4) uder he codo (.3) we dvde he erval [] as... le N. he we appromae he Volerra o N egral erms afer usg he quadraure formula w... N Uder he codo see [3] o have H F l y ydy O g. (3.) Here d P N. (3.) ma h h ad are called he characersc pos ad s called N he quadrac coeffces. he values of ad G ( ). ad deped o he umber of dervaves of F ( ) IJBAS-IJENS Jue IJENS

4 Ieraoal Joural of Basc & Appled Sceces IJBAS-IJENS Vol: No: 3 3 ore formao for he characersc pos ad coeffces are foud [ 3]. he formula (3.) ca be adaped he followg form: y y dy m l (3.3) m g ( ) H F l y y dy ad H F. (3.4) he formula (3.3) represes LAS of FIEs of he secod kd wh logarhmc kerel. As a mpora specal case from (3.3) whe we oba l y dy g. (3.5) o o o o o Dffereag (3.5) wh respec o we have d y dg o dy h h (3.6) o o o d o y d Here (3.6) deoes egrao he sese of Cauchy prcpal value ad he ukow fuco wh s dervaves are couous L o. akg he rasformaos y u v he egro dffereal (3.6) o og he dfferece oaos becomes u d du z v. (3.7) dv v u hs equao has appeared boh combed frared gaseous radaos ad molecular coduco (3.7) s kow as he radao coduco umber for he large pah legh lm ad represes he sgle parameer of he dmesoless sysem. he formula (3.7) s cosdered ad dscussed wh s specal cases ad solved whe z v v uder he codos by Frakel [5] represes he ukow emperaure. 4. CHEBYSHEV POLYNOIALS o oba he soluo of (3.3) we use he CPs wh s famous relaos. For hs we wre he ukow fucos ( ) for each value N he form of he wegh fuco of CPs of he frs kd B mulplyg by ukow fucos B ( ) N. he we wre he CPs o ge IJBAS-IJENS Jue IJENS

5 Ieraoal Joural of Basc & Appled Sceces IJBAS-IJENS Vol: No: 3 4 a. (4.) Here (4.) he fuco s called CP of he frs kd ad order ad a are he ukow coeffces of wll be deermed. I s dffcul o oba he soluo of equao (3.3) umercally he form of equao (4.). For hs he formula (4.) mus be rucaed o he followg: a. (4.) Usg (4.) ad he followg famous relaoshps [] he formula (3.3) yelds l y y dy l y (4.3) a a a l a H F a l g a (4.4) g g d. (4.5) he formula (4.4) leads us o dscuss he followg wo cases: Case : a we have. (4.6) a g a a l H l F a Iegrag (4.6) wh respec o from - o we ge a [g H a l F a ] l ( l N ). (4.7) IJBAS-IJENS Jue IJENS

6 Ieraoal Joural of Basc & Appled Sceces IJBAS-IJENS Vol: No: 3 5 Case : For he formula (4.4) yelds a a g a H F a. ulplyg boh sdes of (4.8) by he erm m d ad egrag from o he usg he followg famous relaos [4] m m m we oba he followg LAS: (4.8) d 4... (4.9) A a a C m N m (4.) m m A m m eve m m m odd (4.) ad Am C g H a F a. (4.) m m heorem (4.): For he fe LAS of (4.) are bouded ad have a uque soluo. Proof: Cosder he merc space of real bouded se k s defed as p p sup { } p. (4.3) Ad a operaor K k such ha y K y { y } { }. (4.4) IJBAS-IJENS Jue IJENS

7 Ieraoal Joural of Basc & Appled Sceces IJBAS-IJENS Vol: No: 3 6 C { C } k we assume he bouded ad couous he followg fe LAS see Also for [5] y C K s a cosa. (4.5) Hece uder he codo sup K he operaor K sasfes : (4.5) has a uque soluo. K k k.e. he sysem So he same way whe we rewre (4.) o be a R a L m m m Am Lm C m R m. (4.6) he covergece of he LAS of equao (4.6) ca be obaed afer applyg Cauchy kovsk equaly. herefore we follow S R A. (4.7) m m m Hece we ge S. (4.8) m ( A m ) Usg he values of he covergece seres.39 for all values of N. [ ( Am ) ] as m we oba 5. NUERICAL RESULS we assume For he aalycal soluo of equao (3.4) 3 m F G ; k y l y (5.) Hece we have ad 3 H (5.) IJBAS-IJENS Jue IJENS

8 Ieraoal Joural of Basc & Appled Sceces IJBAS-IJENS Vol: No: g I (5.3) 5 4 I l l 3. he eac soluo ad he correspodg umercal soluo of he LAS (4.) for he prevous daa are obaed for he mes =. =.4 ad = hrough he followg able: able(.) E E = E E E E E- = E E E E E-9 = E E E E E-6 N We oe ha from he above able ad oher umercal resuls ha: - As he me creases he error decreases. Also as m creases he error decreases. - he aalycal soluo ad he umercal oe s very small whch assure he accuracy of he umercal ad he aalycal echques cosdered hs paper. REFERENCES []. Dago P. Lma Super covergece of colleco mehods for a class of weakly sgular Volerra egral equao J. Cam. Appl. ah. 8 (8) [] C. Zhag Y He he eeded oe leg mehod for olear eural delay- egro dffereal equaos Appl. Num. ah. 59 (8) [3] [3]. A. Abdou.. El- Bora. K. El-Kook oeplz mar mehod for solvg he olear egral equao of Hammerse ype J. Cam. Appl. ah. 3(9) [4] [4] C. Cosaa Iegral equao of he frs kd plae elascy J. Quar. Appl. ah. L (4) (995) [5] [5] J. L. Frakel A Galerk soluo o regularzed Cauchy sgular egro dffereal equaos J. Quar. Appl. ah. L () (995) [6] V.. Aleksadrov E. V. Kovaleko Problems echacs eda wh ed Boudary Codos Nauka oscow IJBAS-IJENS Jue IJENS

9 Ieraoal Joural of Basc & Appled Sceces IJBAS-IJENS Vol: No: 3 8 [7]. A. Abdou Fredholm Volerra egral equao of he frs kd ad coac problem Appl. ah. Compu. 5() [8]. A. Golberg Numercal Soluo of Iegral Equaos Pleum press New York 99. [9]. A. Abdou O asympoc mehod for Fredholm Volerra egral equao of he secod kd coac problem J. Comp. Appl. ah. 54(3) []. A. Abdou Iegral equao of med ype ad egrals of orhogoal polyomals J. Comp. Appl. ah. 38 () []. A. Abdou Fredholm Volerra egral equao wh sgular kerel Appl. ah. Compu. 37 (3) 3-4. [] L.. Delves J. L. ohamed Compuaoal ehods for Iegral Equaos Cambrdge Cambrdge Uversy 985 [3] K. E. Akso he Numercal Soluos of Iegral Equao of he Secod Kd Cambrdge Cambrdge Uversy 997 [4] I. S. Gradsheye ad I.. Ryzhk ables of Iegrals Seres ad Producs Eglad Academc Press ffh Ed. 994 [5]. A. Abdou K. I. ohamed A. S. Ismal o he umercal soluos of Fredholm Volerra egral equao Appl. ah. Compu. 46 (3) IJBAS-IJENS Jue IJENS