A modified new Homotopy Perturbation Method for solving linear integral equations differential
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1 Amercn Jornl of Appled Mhemc 4; 3: Plhed onlne Jne 4 hp:// do:.648/j.jm.43. A modfed new Homoop Perron Mehod for olvn lner nerl eqon dfferenl An Khojeh * Mhmod Prpor M. A. Appled Mhemc Scence nd Reerch Brnch Ilmc Azd Unver Brojerd Irn An Profeor of Appled Mhemc Hmedn Unver of echnolo Hmedn Irn Eml ddre: n.hojeh@hoo.com A. Khojeh Mhmod_ prpor@hoo.com M. Prpor o ce h rcle: An Khojeh Mhmod Prpor. A Modfed New Homoop Perron Mehod for Solvn Lner Inerl Eqon Dfferenl. Amercn Jornl of Appled Mhemc. Vol. No. 3 4 pp do:.648/j.jm.43. Arc: Mhemcl modeln of rel-lfe prolem ll rel n fnconl eqon ch ordnr or prl dfferenl eqon nerl nd nerl-dfferenl eqon ec. he heor of nerl eqon one of he mjor opc of ppled mhemc. In h pper new Homoop Perron Mehod HPM nrodced o on exc olon of he em of nerl eqon-dfferenl nd provded exmple for he ccrc of h mehod. h pper preen n nrodcon o new mehod of HPM hen nrodce he em of nerl - dfferenl lner eqon nd lo nrodce pplcon nd lerre. In econd econ we wll nrodce ceorzon of vern nerl - dfferenl nd everl mehod o olve h nd of chevemen. he hrd econ nrodce new mehod of HPM. Forh econ deermne qrer of nerl - dfferenl eqon n HPM. herefore we provde Conclon nd ome exmple h llre he effecvene nd convenence of he propoed mehod. Keword: New Homoop Perron Mehod Sem of Inerl Eqon - Dfferenl. Inrodcon he homoop nl mehod HAM w propoed Lo. In h mehod he olon condered he mmon of n nfne ere whch ll convere rpdl o he exc olon. he HAM ed on homoop fndmenl concep n opolo nd dfferenl eomer. Brefl pen men of he HAM one conrc conno mppn of n nl e pproxmon o he exc olon of condered eqon. An xlr lner operor choen o conrc ch nd of conno mppn nd n xlr prmeer ed o enre he converence of olon ere. he mehod enjo re freedom n choon nl pproxmon nd xlr lner operor. he pproxmon oned he HAM re nforml vld no onl foll prmeer lo for ver lre prmeer. Unl recenl he pplcon of he homoop nl mehod n nonlner prolem h een devoed cen nd enneer.. Sem of Dfferenl Eqon Newon' econd lw of moon for pon prcle one of he fr deferenl eqon o ever e wren. Even h erl exmple of deferenl eqon con no of nle eqon of em of hree eqon on hree nnown. he nnown fncon re he prcle hree coordne n pce fncon of me. One mporn dffcl o olve deferenl em h he eqon n em re ll copled. One cnno olve for one nnown fncon who nown he oher nnown. In h Chper we d how o olve he em n he prclr ce h he eqon cn e ncopled. We cll ch em donlzle. Explc forml for he olon cn e wren n h ce. Ler we enerlze h de o em h cnno e ncopled... Lner Dfferenl Sem We nrodce lner deferenl em wh vrle coeffcen. We preen n nl vle prolem for ch em nd we e h nl vle prolem lw hve nqe olon. he proof ed on enerlzon of he Pcrd-Lndel of eron ed n Secon.6. We hen nrodce he concep of fndmenl olon enerl olon fndmenl mrx nd Wronn of olon o lner em.
2 8 An Khojeh nd Mhmod Prpor: A Modfed New Homoop Perron Mehod for Solvn Lner Inerl Eqon Dfferenl 3. Nonlner Dfferenl nd Inerl Eqon Whn recen er nere n nonlner eqon h rown enormol. he re exremel mporn c eqon n mn re of mhemcl phc nd he hve receved renewed enon ece of prore n her olon mchne. h volme ndere defnon of he feld ndcn dvnce h hve een mde p hroh 96. he hor' poon h whle he dven of mchne h reled n mch new nowlede one hold no drerd nlcl mehod nce he olon of nonlner eqon poee nlre whch onl he nlcl mehod ed pon he wor of Poncre Lponoff Pn eve nd orl cn dcover. Afer enerl rve of he prolem preened nonlner eqon he hor dce he dfferenl eqon of he fr order follown h chper on he Rcc eqon rde eween lner nd nonlner eqon nd exence heorem wh pecl reference o Cch' mehod. Second order eqon re nrodced v Volerr' prolem nd he prolem of pr nd cceedn chper cover ellpc nerl nd fncon nd he fncon; dfferenl eqon of he econd order; nd econd order dfferenl eqon of he polnoml cl wh pecl reference o Pnleve rncenden. he echnqe of conno nlcl connon hown whle phenomen of he phe plne re ded n nrodcon o nonlneechnc. Nonlner echnc hen dced wh vro clcl eqon le Vn der Pol' eqon Emden' eqon nd he Dffn prolem. he remnn chper re concerned wh nonlner nerl eqon prolem from he clcl of vron nd nmercl neron of nonlner eqon. hroho he oo he rel of dnhed nl of he p nd modern mchne compon re oh en no ccon. Depe he horohne of covere h ver fne nrodcon o h mporn re of mhemc nd cn el e followed he mhemcll ophced reder who now ver lle o nonlner eqon for exmple: x x n d n x d 4. Mehod for Ordnr Dfferenl Eqon h pr wll nrodce he reder o he ermnolo nd noon of dfferenl eqon. Sden wll lo e remnded of ome of he elemenr olon mehod he re med o hve enconered n n nderrde core on he jec. A he conclon of h revew one hold hve n de of wh men o olve dfferenl eqon nd ome confdence h he cold conrc olon o ome mple nd pecl pe of dfferenl eqon for exmple: n x x d x d n A he core of ech of K 3 he eqon he eqon h ech nd ever one qre homepe. Ovol ech of he ove eqon he ernel nd he fncon re nown ch : K 3 K Kh h 4 K x x In ce ernel h nd conno hen: KL l L d L d L K d L K herefore we conder he follown dfferenl enerl form of n eqon nerl: P Q R d C r D r r r r... r r m m r r... r r...
3 Amercn Jornl of Appled Mhemc 4; 3: he Concep of he Inerl Eqon Inerl-dfferenl eqon pper n mn cenfc pplcon epecll when we conver nl vle prolem or ondr vle prolem o nerl eqon. he nerl-dfferenl eqon conn oh nerl nd dfferenl operor. he dervve of he nnown fncon m pper o n order. In clfn nerl-dfferenl eqon we wll follow he me ceor ed efore. Inerl eqon occr n vre of pplcon ofen en oned from dfferenl eqon. he reon for don h h m me olon of he prolem eer or omeme enle o prove fndmenl rel on he exence nd nqene of he olon. 6. Fredholm Inerl Eqon For Fredholm nerl eqon he lm of neron re fxed. Moreover he nnown fncon x m pper onl nde nerl eqon. P Q R d C r D r r r r... r r m m r r... r r... If o hve eqon. we conder he follown mpon: P Q R C D r e We hve he follown eqon: d 3 Accordn o he mlr of he mehod for olvn prolem of lner ordnr dfferenl eqon fne dfference pproxmon o he dervve o replce olon well replcn he nerl wh e of nmercl neron wh ppropre ccrc. B nrodcn f d he eqon 3 cn e wren follow. f 4 If he wo de of eqon.3 n he nervl of neron we wll hve: [ ] f d herefore wh replcemen we hve: f d 5 nern of e re [ ] we hve: d d f dd In whch ce he follown relon wll e cheved: f dd 6 Condern h we hve: herefore A B f d d A B f d 7 B f d We hve wh replcemen vle of A nd B n.6 f d For exmple: f d o S S dd 8 d d d
4 8 An Khojeh nd Mhmod Prpor: A Modfed New Homoop Perron Mehod for Solvn Lner Inerl Eqon Dfferenl dd dd d d d 9 dd dd dd d d d d d lo: d d d We hve wh replcemen n: d dd dd dd dd dd 3 dd dd We hve wh replcemen n 4 dd d 4 dd Or dd d 5 herefore d dd We wre he Fredholm Inerl Eqon V d H d H 6 d V 7. Conclon In h pper he dvded dfference mehod ppled o olve he lner Fredholm nerl eqon of he econd nd. In h mehod he coecn of dvded dfference re ven olvn em of eqon. In he compron he propoed mehod eer hn he Adomn' decompoon mehod o pproxme he exc olon. he dvne of he propoed mehod over oheehod h he nerl eqon olved hvn ppor pon of he olon of nerl eqon. he propoed mehod powerfl procedre for olvn lner nerl eqon. he exmple nlzed llre he l nd rell of he mehod preened n h pper nd revel h h one veple nd effecve. he oned olon n compron wh exc olon dm remrle ccrc. Rel ndce h he converence re ver f nd lower pproxmon cn cheve hh ccrc
5 Amercn Jornl of Appled Mhemc 4; 3: Exmple: dx x x d 3 3 S x x x dx x x d S x x d x For olve he em wh new Homoop devon Homop he follown ld: dx x d 4 3 p x x x dx x d 3 3 S x x d 4 p x 3 x 4 3 For exmple: In fc we hve: X X S x x d x X α p n n n X β Pn n n P X X αn n αn n P n n X τ τ τ X τ ddτ n n 4 3 X S S S X d 5 βn n βn n 3 P n n 5 n n τ S τ X τ τ S τ X τ ddτ Reference [] A nd S. 6. Nmercl olon of he nerl eqon: homoop perron mehod nd Adomn decompoon mehod. Appled Mhemc nd Compon [] Belendez A. Belendez. Mrqez A. Nepp C. 8. Applcon of He homoop perron mehod o conervve rl nonlner ocllor. Cho Solon nd Frcl [3] Bzr J. hzvn H. 7. Exc olon for nonlner Schrodner eqon He homoop perron mehod. Phc Leer A [4] Bzr J. hzvn H. 8. Nmercl olon for pecl nonlner Fredholm nerl eqon HPM. Appled Mhemc nd Compon [5] Bzr J. hzvn H. 9. He homoop perron mehod for olvn em of Volerr nerl eqon of he econd Knd. Cho Solon nd Frcl [6] Bo.L. Xe L. Zhen X.J. 7. Nmercl pproch o wnd rpple n deer Inernonl Jornl of Nonlner Scence nd Nmercl Smlon [7] nj D.D. 6. he pplcon of He homoop perron mehod o nonlner eqon rn n he rnfer. Phc Leer A [8] nj D.D. 6. he pplcon of He homoop perron nd vronl eron mehod o nonlner he rnfer nd poro med eqon. Jornl of Componl nd Appled Mhemc [9] ol A. Jvd M. 7. Applcon of He homoop perron mehod for olvn ehh-order ondr vle prolem. Mhemc nd Compon [] ol A. Kerm B. 8. Modfed homoop perron mehod for olvn Fredholm nerl eqon. Cho Solon nd Frcl [] He J.H Homoop perron echnqe Comper mehod n Appled Mechnc nd Enneern [] He J.H.. A copln mehod of homoop perron echnqe for nonlner prolem. Inernonl Jornl of Non-lner Mechnc [3] He J.H. 3. Homoop perron mehod: new nonlner nlcl echnqe. Appled Mhemc nd Compon [4] He J.H. 4. Compron homoop perron mehod nd homoop nl mehod. Appled Mhemc nd Compon [5] He J.H. 4. he homoop perron mehod for nonlner ocllor wh dconne. Appled Mhemc nd Compon [6] He J.H. 5. Lm ccle nd frcon of nonlner prolem. Cho Solon nd Frcl
6 84 An Khojeh nd Mhmod Prpor: A Modfed New Homoop Perron Mehod for Solvn Lner Inerl Eqon Dfferenl [7] He J.H. 5. Applcon of homoop perron mehod o nonlner wve eqon. Cho Solon nd Frcl [8] He J.H. 6. Homoop perron mehod for olvn ondr vle prolem. Phc Leer A [9] Mohd-Dn S.. Yldrm A. Demrl.. rveln wve olon of Whhm-Broor-Kp eqon homoop perron mehod. Jornl of Kn Sd Unver-Scence [] Od Z. Momn S. 8. Modfed homoop perron mehod: pplcon o qdrc Rcc dfferenl eqon of frconl order. Cho Solon nd Frcl [] Rfr B. Yldrm A.. he pplcon of homoop perron mehod for MHD flow of UCM fld ove poro rechn hee. Comper & Mhemc wh Applcon [] Sher Femeh Dehhn Mehd 8. Solon of del dfferenl eqon v homoop perron mehod Mhemcl nd Comper Modeln [3] Sddq A.M. Mhmood R. hor Q.K. 8. Homoop perron mehod for hn flow of hrd rde fld down n nclned plne. Cho Solon nd Frcl [4] Sn F.Z. o M. Le S.H. e l. 7. he frcl dmenon of he frcl model of drop-we condenon nd expermenl d. Inernonl Jornl of Nonlner Scence nd Nmercl Smlon 8 -. [5] Wn H. F H.M. Zhn H.F. e l. 7. A prccl hermo-dnmc mehod o clcle he e l-formn compoon for l mellc le. Inernonl Jornl of Nonlner Scence nd Nmercl Smlon [6] X L. H J.H. L Y. 7. Elecropn nno-poro phere wh Chne dr. Inernonl Jornl of Nonlner Scence nd Nmercl Smlon [7] Yldrm A. ln Y.. Anlcl pproch o frconl Zhrov-Kzneov eqon He homoop perron mehod. Commncon n heorecl Phc [8] Yldrm A. Mohd-Dn S.. Zhn D.H.. Anlcl olon o he pled Klen-ordon eqon n Modfed Vronl Ieron Mehod MVIM nd Boer Polnoml Expnon Scheme BPES. Comper & Mhemc wh Applcon
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