Application of Bernoulli wavelet method for numerical solution of fuzzy linear Volterra-Fredholm integral equations Abstract Keywords

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1 Applicaion o enoulli wavele mehod o numeical soluion o uzz linea Volea-edholm inegal equaions Mohamed A. Ramadan a and Mohamed R. Ali b a Depamen o Mahemaics acul o Science Menouia Univesi Egp mamadan@eun.eg; amadanmohamed3@ahoo.com b Depamen o Mahemaics acul o Engineeing enha Univesi Egp EPH - Inenaional mohamededa@bhi.bueud.eg; Jounal o Mahemaics and Saisics mohamededaabhi@ahoo.com Absac his wo enoulli wavele mehod is omed o solve nonlinea uzz Volea-edholm inegal equaions. enoulli waveles have been Ceaed b dilaion and anslaion o enoulli polnomials. is we inoduce popeies o enoulli waveles and enoulli polnomials and hen we used i o ansom he inegal equaions o he ssem o algebaic equaions. We compaed he esul o he poposed mehod wih he eac soluion o show he convegence and advanages o he new mehod. he esuls go b pesen wavele mehod ae compaed wih ha o b collocaion mehod based on adial basis uncions mehod. inall he numeical eamples eplain he accuac o his mehod. ewods: enoulli polnomials; enoulli waveles Volea-edholm uzz inegal equaions Inegaion o he coss poduc Poduc mai Coeicien mai.. Inoducion he sud o uzz inegal equaions begins wih he invesigaions o elva [] and Seiala [] o he uzz Volea inegal equaion ha is equivalen o he iniial value poblem o is ode uzz dieenial equaions whee he anach s ied poin heoem and he mehod o successive appoimaions ae applied in he poblem o he eisence and uniqueness he soluions. he main poblems ha aise o uzz inegal equaions ae: he eisence and uniqueness o he soluion and he consucion o numeical mehods o appoimae i. Man eseaches have ocused hei inees on his ield and published man aicles which ae available in lieaue. Man analical mehods lie Adomian decomposiion mehod [5] homoop analsis mehod [6] and homoop peubaion mehod [7] have been used o solve

2 uzz inegal equaions. hee ae available man numeical echniques o solve uzz inegal equaions. he mehod o successive appoimaions [89] quadaue ule [] Nsom mehod [] Lagange inepolaion [] ensein polnomials [3] Chebshev inepolaion [4] Legende wavele mehod [5] sinc uncion [6] esidual minimizaion mehod [7] uzz ansoms mehod [8] and Galein mehod [9] have been applied o solve uzz inegal equaions numeicall. we inoduce uzz linea Volea-edholm inegal equaion is inoduced. he es o he pape has been oganized as ollows: In secion we pesen some peliminaies and noaions useul o uzz inegal equaions. In secion 3 we discuss he popeies o enoulli waveles and uncion appoimaion. In secion 4 we esablish he mehod o solving Volea-edholm inegal equaion. Secion 5 deals wih he illusaive eample which show he eicienc o he pesened mehod.. Peliminaies o uzz inegal equaion Deiniion.. See Re. []. A uzz numbe u is epesened b an odeed pai o uncions u u ; which saising he ollowing popeies. I. u is a bounded monoonic inceasing le coninuous uncion. II. u is a bounded monoonic deceasing le coninuous uncion. III. u u. o abia u = u u v = v v and we deine addiion u v and scala muliplicaion b as: a. v u = u + v b. u v u + v = u u u c. u Rema.. See Re. []. I he uzz uncion is coninuous in he meic D is deinie inegal eiss. Also EPH - Inenaional Jounal o Mahemaics and Saisics

3 b b ; d ; d a a b b d ; d a a Deiniion.3. [] A uzz numbe is a uncion such as u : R [;] saising he ollowing popeies: i u is nomal i.e. R wih u ii u is a conve uzz se i.e. u min{ u u } R [ ] iii u is uppe semi-coninuous on R iv R : u is compac whee A denoes he closue o A. he se o all uzz eal numbes is denoed b E. Obviousl R E. Hee R E is R { : is usualeal numbe}. o i is [ u] { R; u } and [ u ] { R; u }. hen i is well-nown ha o an [] [ u] is a bounded closed ineval. o u ~ v ~ E and and R whee sum u ~ v ~ and he means he convenional addiion o wo inevals subses o [ u] { : [ u] } means he convenional poduc beween a scala and a subse o R. Deiniion.4. [] Suppose u ~ is a uzz numbe and []. hen he -cu epesenaion o u ~ is he pai o uncions L and R boh om [;] o R deined especivel b L in{ [ u] }; i ;] and EPH - Inenaional Jounal o Mahemaics and Saisics in{ sup p u ~ }; i R sup{ [ u] }; i ;] sup{ sup p u ~ }; i

4 Deiniion.5. [] ~ A uzz numbe veco X ~... ~ n given b ~ [ ~... i i i ] i n n is called he soluion o Volea-edholm inegal equaion i n j aij j n j aij j bi n n aij j aij j j Deiniion.6 EPH - Inenaional Jounal o Mahemaics and Saisics Le : R E be a uzz uncion whee E is a subse o a anach space and R. he deivaive ' o a a poin is deined b ' h lim h h povided ha his limi aen wih espec o he meic D eiss and h be suicienl small paamee. he elemens h and in he above equaion ae in anach space C[ ] C[]. hus i h a a and b b hen h a b a b. Cleal j bi [ h ]/ h ma no be a uzz numbe o all h. Howeve i i appoaches ' in and ' a is also a uzz numbe in E his numbe is he uzz deivaive o ' ' ' ' '.In his case i whee ae classic deivaive o especivel and R. 3. Waveles and enoulli waveles Waveles consiue a amil o uncions consuced om dilaion and anslaion o a single uncion called mohe wavele. When he dilaion paamee a and he anslaion paamee b va coninuousl we have he ollowing amil o coninuous waveles as b a b a a a b R a I we esic he paamees a and b o discee values as a a b nb a a b and n and ae posiive ineges we have he ollowing amil o discee waveles:

5 n a a nb n Z whee n oms a wavele basis o L R. In paicula when a b hen n om an ohonomal basis. enoulli waveles n m n m have ou agumens whee n... Z m is he ode o enoulli polnomials and is nomalized ime. he ae deined on he ineval [ as [4]. wih ~ m ~ n n m n n m oheewise EPH - Inenaional Jounal o Mahemaics and Saisics m m m! m m! m m whee m... M and n.... he coeicien is o he ohonomali he dilaion paamee is m m! m m! a and anslaion paamee is b n. h Hee m ae he well-nown m ode enoulli polnomials which ae deined on he ineval [ ] and can be deemined wih he aid o he ollowing eplici omula [5] m m i m = mi 5 i i whee ae i. i... m ae enoulli numbes. he is ou such polnomials especivel ae

6 enoulli polnomials sais he ollowing omula [4]. n m! n! n m d mn m n. 6 m n! 3.. Popeies o enoulli s polnomial Popeies o enoulli polnomials ae given as ollows [4]: m. m m m Z / EPH - Inenaional Jounal o Mahemaics and Saisics. m m m m Z m m! n! 3. m n d mn m n. m! n! m! 4. m d 6. mn m m m m a 5. m d. m a 6. sup m m. [] m 7. sup m m. [] Popeies o enoulli numbe he sequence o enoulli numbes m m N saising he ollowing popeies [4]:. m m m. m. m/ m. m m 3. m. m 3.3. uncion appoimaion b using enoulli wavele mehod An uncion which is squae inegable in he ineval [ can be epanded in a enoulli wavele mehod WM as: M n mbn m 7 n m

7 bn m 8 bn m bn m In 8.. denoes he inne poduc. I he ininie seies in 7 is uncaed hen 7 can be ewien as: [... M... M ] M [ b b... b M b... b M... b... b ] M heeoe we have u hen D u whee D D D. d. 9 D M hen b using 7 D i i... M is deined as ollows: Dn i j i EPH - Inenaional Jounal o Mahemaics and Saisics i n i j n i d i i n j n d We can also appoimae he uncion L[ ] as ollows: whee is an M mai ha we can obain as ollows: D D

8 3.4. Inegaion o enoulli wavele uncions In enoulli wavele uncions analsis o a dnamic ssem all uncions need o be ansomed ino WM uncions. he inegaion o WM uncions should be epandable ino WM uncions wih he coeicien mai P. hese ideas come om papes o Chen e al. [5]. We can appoimae uncion wih his base. o eample o EPH - Inenaional Jounal o Mahemaics and Saisics oheewise 64 oheewise 4 [ ] 64 3 oheewise and M oheewise d P [ 3 M M M M whee he M -squae mai P is called he opeaional mai o inegaion and is deined in Eq. 3. A subscip M M o P denoes is dimension and M P is he opeaional mai o inegaion and can be obained as: P = -3 - M M he inegaion o he coss poduc o wo WM uncion vecos can be obained as L D M d M whee L is an M diagonal mai given b L L -3-4

9 D 5 Eqs. 7-5 ae ve impoan o solving Volea- edholm inegal equaion o he second ind poblems because he D and P mai can incease he calculaing speed as well as save he memo soage. 4. Soluion o Volea- edholm inegal equaion via enoulli wavele mehod Conside he ollowing Volea- edholm inegal equaion o he om: d d 6 whee and ae uzz uncions. Le ] [ ] [ Eq. 9. in cisp sense conveed ino a ssem as d d d d 7 whee and Appoimae and as ollows: 8 wih subsiuing above equaions ino in Eq. 7. EPH - Inenaional Jounal o Mahemaics and Saisics

10 d d d d 9 d d d d Appling Eqs. and o Eq. and Eq. becomes P D P D In ode o ind we collocae Eq. in M nodal poins o Newon-Coes [6] as i M i om Eqs. we have a ssem o M linea equaions and M unnowns. Ae solving above linea ssem we can achieve he unnown vecos. he equied appoimaed soluion o Volea edholm inegal Eq. 6 can be obained b using Eqs. as ollows: P D 3 P D 4 5. Illusaive Numeical Eample We applied he pesened schemes o he ollowing Volea- edholm inegal equaion o second ind. o his pupose we conside he ollowing eample. Conside he ollowing linea Volea- edholm inegal equaion d d ] 5.4 [3.6 ].5.5. [ ] [ 3 5 I we solve 5 o diecl he analic soluion can be shown o be ].5.5. [.8 ] [ ]. [ ] [ EPH - Inenaional Jounal o Mahemaics and Saisics

11 he above poblem has been solved b enoulli wavele mehod. he compaison beween he WM soluion and he analic soluion o [ and [ is shown in ig. o M 4 and 3 hen he esuls ae compaed wih ha o obained b collocaion mehod based on adial basis uncions mehod. We ae and = and calculae he absolue eos as e. his compaison is pesened in he able. he appoimaion soluions o and o ae shown in ig. and EPH - Inenaional Jounal o Mahemaics and Saisics ig.. ee appoimaion is epeced b inceasing he ode o he enoulli polnomials. able.. Compaison o numeical soluions o in Eample a = in [3] e a Absolue eo using WM poposed mehod e a Absolue eo using collocaion mehod based on adial basis uncions e a e a

12 EPH - Inenaional Jounal o Mahemaics and Saisics ig.. Appoimae soluion o o o Eample. ig.. Appoimae soluion o o o Eample. 6. Conclusion In his pape we poposed an appoimaion echnique o solve uzz linea Volea-edholm inegal equaions. he mehod is based upon educing he ssem ino a se o algebaic equaions. he geneaion o his ssem needs jus sampling o uncions muliplicaion and addiion o maices and needs no inegaion. he mai D and P ae spase; hence ae much ase han ohe uncions and educes he CPU ime and he compue memo a he same ime

13 eeping he accuac o he soluion. he numeical eample suppos his claim. he numeical esuls obained b pesen mehod is compaed wih he esuls obained b a combinaion o collocaion mehod and adial basis uncionsrs mehod. om he above able i maniess ha he pesen enoulli wavele mehod gives moe accuae esuls han a combinaion o collocaion mehod and adial basis uncions Rs esuls. Addiionall he compuaional ime o pesen mehod is much smalle han ha o obained b a combinaion o collocaion mehod and adial EPH - Inenaional basis Jounal o uncions Mahemaics and Saisics Rs. Moeove he absolue eo impoves b inceasing he ode o he enoulli polnomials. Illusaive eample is included o demonsae he validi and applicabili o he poposed echnique. his eample also ehibis he accuac and eicienc o he pesen mehod. Reeences [] O. aleva uzz dieenial equaions uzz Ses Ss [] S. Seiala On he uzz iniial value poblem uzz Ses Ss [3] R. Ezzai. Mohai Numeical soluion o edholm inegal equaions o he second ind b using uzz ansoms In. J. Phs. Sci [4] M. Oadi Numeical soluion o uzz inegal equaions using ensein polnomials Aus. J. asic Appl. Sci [5] E. abolian H. Sadeghi Gogha S. Abbasband Numeical soluion o linea edholm uzz inegal equaions o he second ind b Adomian mehod Appl. Mah. Compu [6] A. Molabahami A. Shida A. Ghasi An analical mehod o solving linea edholm uzz inegal equaions o he second ind Compu. Mah. Appl [7]. Allahvianloo S. Hashemzehi he homoop peubaion mehod o uzz edholm inegal equaions J. Appl. Mah. Islamic Azad Univesi o Lahijan [8] A.M. ica One-sided uzz numbes and applicaions o inegal equaions om epidemiolog uzz Ses Ss

14 [9] A.M. ica C. Popescu Appoimaing he soluion o nonlinea Hammesein uzz inegal equaions uzz Ses Ss [] P. Salehi M. Nejaian Numeical mehod o nonlinea uzz Volea inegal equaions o he second ind In. J. Ind. Mah [] S. Abbasband E. abolian M. Alavi Numeical mehod o solving linea edholm uzz inegal equaions o he second ind Chaos Solions acals [] M.A. abozi Aaghi N. Paandin Numeical soluion o uzz edholm inegal equaions b he Lagange inepolaion based on he eension pinciple So Compu [3] R. Ezzai S. Ziai Numeical soluion and eo esimaion o uzz edholm inegal equaion using uzz ensein polnomials Aus. J. asic Appl. Sci [4] M. ahodai Ahmadi M. hezeloo uzz bivaiae Chebshev mehod o solving uzz Volea edholm inegal equaions In. J. Ind. Mah [5] P.. Sahu S. Saha Ra wo dimensional Legende wavele mehod o he numeical soluions o uzz inego-dieenial equaions J. Inell. uzz Ss EPH - Inenaional Jounal o Mahemaics and Saisics [6] M. eanpou. Abaian New appoach o solving o linea edholm uzz inegal equaions using sinc uncion J. Mah. Compu. Sci [7] M. Jahanigh. Allahvianloo M. Oadi Numeical soluion o uzz inegal equaions Appl. Mah. Sci [9]. Loi. Mahdiani uzz Galein mehod o solving edholm inegal equaions wih eo analsis In. J. Ind. Mah [] R. Goeschel W. Voman Elemena uzz calculus uzz Ses Ss [] C. Wu Z. Gong On Hensoc inegal o uzz-numbe-valued uncions I uzz Ses Ss

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