Numerical solution of fractional elliptic PDE's by the collocation method
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1 Avlle t Appl Appl Mth ISS: Vol Issue (June 07) pp Applctons nd Appled Mthemtcs: An Interntonl Journl (AAM) umercl soluton of frctonl ellptc PE's y the collocton method Fut Ust eprtment of Mthemtcs Fculty of Scence nd Arts üzce Unversty üzce-turey futust@duzceedutr Receved: ecemer 06; Accepted: My 3 07 Astrct In ths presentton numercl soluton for the soluton of frctonl order of ellptc prtl dfferentl equton n R s proposed In ths method we use the Rdl ss functons (RBFs) method to eneft the desred propertes of mesh free technques such s no need to generte ny mesh nd esly ppled to mult dmensons In the numercl soluton pproch the RBF collocton method s used to dscrete frctonl dervtve terms wth the Gussn ss functon Two dmensonl numercl emples re presented nd dscussed whch conform well wth the correspondng ect solutons Keywords: Conformle frctonl clculus; rdl ss functon; collocton method MSC 00 o: 65L60 6A33 Introducton In numer of prctces dt s produced wth no nowledge of functon from whch t ws derved Therefore n ppromton model s needed However mny physcl systems could only e modelled y usng the non-nteger order of dervtves nd ntegrls For nstnce nonnteger order of models re studed n control theory computtonl nlyss nd engneerng Kls (06) Smo (993) Thus numer of new defntons hve een ntroduced n cdem to provde the est method for frctonl clculus Here ll frctonl dervtves do not provde some propertes such s Product Rule Quotent Rule Chn Rule Roll s Theorem nd Men Vlue Theorem To overcome some of these nd other dffcultes Adeljwd (05) Khll (04) Ktugmpol (04) cme up wth n nterestng de tht etends the fmlr lmt defnton of the dervtve of functon In ths wor we focus on numercl soluton of prtl dfferentl equtons whch re modelled wth Ktugmpol dervtves Ktugmpol (04) Posson equton s one of the most populr ellptc dfferentl equtons wth rod utlty n theoretcl physcs mechncl engneerng nd electrosttcs However numer of physcl systems could only e modelled y usng the non-nteger order of dervtves nd ntegrls A lot 470
2 AAM: InternJ Vol Issue (June 07) 47 of nlytcl nd numercl methods of such systems hve een proposed n cdem such s vrton terton method Khn (0) frctonl fnte dfference method Borhnfr (0) Meerschert (06) homotopy perturton method L (009) Song (007) Adomn decomposton method Yng (00) Grg (0) nd ny other res Eslm (06) Eslm (06) Ec (06) In ddton to ths rdl ss functons method s one of the more prctcl wys of solvng frctonl order of models The most mportnt property of n RBF technque s tht there s no need to generte ny mesh nd so t s clled the mesh-free method One only requres the prwse dstnce etween ponts for n RBF ppromton Buhmnn (003) Cheney (999) Ths method s esy to mplement n mult dmensonl cses due to the nture of RBF On the other hnd n order to solve prtl dfferentl equtons (PEs) Kns proposed RBF collocton method whch s mesh-free nd esy-to-hndle n comprson wth the other methods Kns (990) Kns (990) Frne (998) Ths prospectve study ws desgned to nvestgte the use of RBF methods to solve the conformle frctonl ellptc prtl dfferentl equtons v Kns's collocton method The remnder of ths wor s orgnzed s follows: In Secton the relted defntons of RBFs s summrsed Then non nteger order of Posson equton s summrzed n Secton 3 In Secton 4 the conformle frctonl dervtve nd ntegrls hve een revewed The computtonl scheme s gven n Secton 5 whle some numercl eperments re presented n Secton 6 Fnlly we hve summrsed the current study n Secton 7 Rdl Bss Functon Interpolton In ths prt the fundmentl concept of the mesh-free rdl ss functon nterpolton re eplned Consder functon u : R d R rel vlued functon wth d vrles whch s to e ppromted y I : R d R for gven vlues u( ) : n where : n s set of dstnct ponts n Then the ppromton to the functon u s of the form: X d R nmed the center set X I X ( ) ( ) where R d R s unvrte rdl ss functon ow the nterpolton condton cn e : constructed s I ) u( ) m mely the nterpolton condton s X( m m ( ) u( ) m m In other words the system of mtr for nterpolton condton cn e wrtten s [ A]{ } { u} where the entres of the mtr A re A ( ) such tht m m m m { } T T nd u { u u u } Ths scheme s lso clled RBF collocton method The nterpolnt of u () s unque f nd only f the mtr X s non-sngulr It hs een dscussed out suffcent condtons for (r) to gurntee non-sngulrty of the mtr
3 47 Fut Ust Buhmnn (003) Cheney (999) Commonly used rdl ss functons re ( r) r e r r r r log( r) r Gussn M ultqudrc Inverse M ultqudrc Inverse Qudrtc Thn - Plte Splne Lner Splne In ddton to these RBFs cn e stted wth the help of sclng prmeter clled the shpe prmeter Ths cn e done n the mnner tht (r) s replced y ( r) In generl shpe prmeters hve een chosen rtrrly snce there re no ect results out how to choose the est shpe prmeter nd so t cn e decded y the user 3 Posson Equtons The generl form of Posson equton on fnte domn {( ( [0] [0]} s u( f ( where s the Lplce opertor In two dmensonl Crtesn coordntes the Posson equton tes the form y u( f ( In the cse of f ( 0 Posson equton convert to Lplce's equton Here we egn y refly revewng the frctonl Posson equton The frctonl order of Posson equton cn e gven s follows: u ( u( f ( () y wth rchlet oundry condtons In order to provde mesh-free numercl soluton of Equton () we wll use the rdl ss functon method whch wll e summrzed elow 4 Conformle Frctonl Clculus In ths pper we wll present nd test conformle frctonl verson of ordnry dfferentl equtons wth the help of the Ktugmpol conformle frctonl clculus In detl
4 AAM: InternJ Vol Issue (June 07) 473 Ktugmpol conformle dervtves or -dervtves for 0 nd t [ 0 ) gven y t u te ut u t lm 0 u0 lm ut t0 () provded the lmts est (for detl see Ktugmpol (04)) If u s fully dfferentle t t then du dt ut t t A functon u s -dfferentle t pont t 0 f the lmt n () ests nd s fnte Ths defnton yelds the followng results; Theorem nd u v e dfferentle t pont t 0 Then Let 0 u v u v for ll R 0 for ll constnt functons f t uv u v v u v u u v v v v u v t n nt n for ll R n u vt uvt vt v for u s dfferentle t v (t) efnton Let (0] nd 0 A functon u : [ ] R s -frctonl ntegrle on [ ] f the ntegrl d : u d u ests nd s fnte All -frctonl ntegrle on [ ] s ndcted y L Remr I t u u t I t u d
5 474 Fut Ust where the ntegrl s the usul Remnn mproper ntegrl nd (0] We wll lso use the followng mportnt results whch cn e derved from the results ove Lemm Let the conformle dfferentl opertor e gven s n () where (0] nd t 0 nd ssume the functons u nd v re -dfferentle s needed Then ln t t for t 0 t d s u( t t) ut s t ut s d s u vd uv v u d In ths study we ntroduced numercl soluton of Ktugmpol type conformle frctonl ordnry dfferentl equton v rdl ss functon collocton method 5 Computtonl Scheme In ths secton we present numercl scheme to solve frctonl ellptc prtl dfferentl equton v non-symmetrc method wth rdl ss functons Let te the Posson equton of the form u y f y y ( ) ( ) ( ) n u( g( ( on wth rchlet oundry condtons where R Thus we re tryng to compute u whle f nd g re fed We cn now use Kns's RBF collocton method Kns (990) Kns (990) We uld n smple one-dmensonl model Let us propose n ppromton soluton u of the form u ( ) (3) where X re the set of nodes n Then the collocton mtr whch constructed y usng Posson equton nd oundry condton to the collocton ponts X wll e of the form [ A ] where the two locs re consttuted of entres: [ ]
6 AAM: InternJ Vol Issue (June 07) 475 X B X I j j j j ) ( ) ( ] [ where I nd B represent set of nteror nd set of oundry ponts of the set of X collocton ponts respectvely (e B I X ) The prolem descred ove s clled well-posed (or correctly-set) f the lner mtr system F A where F s composed of )] ( [ f f I nd )] ( [ g g B hs unque soluton The outstndng propertes of multqudrcs n terms of certnty nd complety mde Kns to prtculrly suggest ts use The mn dfference etween numercl soluton of nteger nd non-nteger order of ellptc PE's s clculton of RBF dervtves In other words one need to compute conformle frctonl dervtves of ny rdl ss functons sy multqudrc An emple s Ktugmpol frctonl dervtve of multqudrc whch s gven elow t dt d t t These results re used strght-forwrdly n the collocton rdl ss functons for solvng frctonl PEs Although there ppers nfnte sums n the prevous formuls one cn truncte the terms once they re smller thn the mchne precson Consequently we hve the followng mtr system: g f f f As result the mtr system of equtons wth unnowns s vlle Then we must solve ths system to clculte the unnown coeffcents Hence we hve used the Guss elmnton method wth totl pvotng to solve such system Consequently ) ( u gven n equton (3) cn e computed 6 umercl Eperments In order to verfy the proposed method n the prevous secton we wll gve some numercl eperments results of some frctonl Posson equtons In ll our numercl eperments the numercl soluton of PEs re evluted t eqully spced ponts (these re unformly dstruted rndom ponts) n the domn 0] [ R Ths rnge cn e generlzed to wder rnge of possle solutons In these eperments we use the multqudrc ss functon nd te the 4 The mplementton of the method n ll our eperments hs een done n Mtl Fnlly n order to show the pplclty of the proposed technque under ll crcumstnces we
7 476 Fut Ust hve used dfferent non-nteger order of PEs Eperment Let us consder the followng conformle frctonl Posson equton 4 / 3 4 / 3 y 3/ 3/ 8 u( 7 / 3 y 8 4 / 3 y 8( y ) y / 3 / 3 3 y 4 / 3 / 3 y / 3 on fnte domn ( [0] wth the oundry condton u ( 0 ( The ect soluton s gven y u( ( ) y( Fgure Appromte soluton of trget functon u( (left) nd mmum error for RBF soluton (rght) Eperment Let us consder the Conformle frctonl Posson equton 3/ 4/3 cos( ) 4 u( cos 3/ 4/3 y / sn( ) y 4 on fnte domn ( 3 sn( ) sn( y/ ) y 3 y 3 [0] wth the oundry condton cos( y/ )
![AAM: InternJ Vol Issue (June 07) 477 u( sn( ) ( u( 0 ( where {( [0] y 0} nd \ The ect soluton s gven y u( sn( )cos( y / ) F Fgure Appromte soluton of trget functon u( (left) nd mmum error for RBF](/docs-images/87/95539590/images/8-0.jpg "soluton (rght) In Fgure nd Fgure we present the multqudrc soluton conformle frctonl Posson equtons long wth ts mmum error respectvely These fgures show tht the RBF method hs een successfully ppled to")
8 AAM: InternJ Vol Issue (June 07) 477 u( sn( ) ( u( 0 ( where {( [0] y 0} nd \ The ect soluton s gven y u( sn( )cos( y / ) F Fgure Appromte soluton of trget functon u( (left) nd mmum error for RBF soluton (rght) In Fgure nd Fgure we present the multqudrc soluton conformle frctonl Posson equtons long wth ts mmum error respectvely These fgures show tht the RBF method hs een successfully ppled to the numercl soluton prolem of frctonl order Posson equton n R wth encourgng performnce These results confrm the superor performnce of RBF methods for numercl soluton of frctonl PEs 7 Concluson The purpose of the current study ws to propose numercl scheme to solve conformle frctonl ordnry dfferentl equton wth the help of rdl ss functon collocton technque The contruton of ths study hs een to confrm y numerclly The eperments verfed tht the numercl solutons re comptle wth the ect solutons REFERECES Adeljwd T (05) On conformle frctonl clculus Journl of Computtonl nd Appled Mthemtcs Borhnfr A nd Vlzdeh S (0) umercl soluton for frctonl prtl dfferentl equtons usng Crn-colson method wth Shfted Grünwld estmte Wll Journl of Scence nd Technology
9 478 Fut Ust Buhmnn M (003) Rdl Bss Functons: Theory nd Implementtons Cmrdge Unversty Press Cheney W nd Lght W (999) A Course n Appromton Theory Wllm Alln ew Yor Eslm M nd Rezzdeh H (06) The frst ntegrl method for WuZhng system wth conformle tme-frctonl dervtve Clcolo 53 (3) Eslm M (06) Ect trvelng wve solutons to the frctonl coupled nonlner Schrodnger equtons Appled Mthemtcs nd Computton Ec M Mrzzdeh M Eslm M Zhou Q Moshoo S P Bsws A nd Belc M (06) Optcl solton perturton wth frctonl temporl evoluton y frst ntegrl method wth conformle frctonl dervtves Opt-Interntonl Journl for Lght nd Electron Optcs 7() Frne C nd Schc R (998) Solvng prtl dfferentl equtons y collocton usng rdl ss functons Appled Mthemtcs nd Computton Grg M nd Shrm A (0) Soluton of spce-tme Frctonl Telegrph equton y Adomn decomposton method Journl of Inequltes nd Specl Functons -7 Kns E J (990) Multqudrcs scttered dt ppromton scheme wth pplctons to computtonl flud-dynmcs I Surfce ppromtons nd prtl dervtve estmtes Computers nd Mthemtcs wth Applctons 9(8-9) 7-45 Kns E J (990) Multqudrcs scttered dt ppromton scheme wth pplctons to computtonl flud-dynmcs II Solutons to prolc hyperolc nd ellptc prtl dfferentl equtons Computers nd Mthemtcs wth Applctons 9(8-9) 47-6 Ktugmpol U (04) A new frctonl dervtve wth clsscl propertes ArXv:406535v Khll R Al horn M Yousef A nd Sheh M (04) A new defnton of frctonl dervtve Journl of Computtonl Appled Mthemtcs Khn Y Frz Yıldırım A nd Wu Q (0) Frctonl vrtonl terton method for frctonl ntl-oundry vlue prolems rsng n the pplcton of nonlner scence Computers nd Mthemtcs wth Applctons Kls A A Srvstv H M nd Trujllo J J (006) Theory nd Applctons of Frctonl fferentl Equtons Elsever BV Amsterdm etherlnds L X Xu M nd Jng X (009) Homotopy perturton method to tme-frctonl dffuson equton wth movng oundry condton Journl of Computtonl nd Appled Mthemtcs Smo S G Kls AA nd Mrchev O I (993) Frctonl Integrls nd ervtves: Theory nd Applctons Gordonnd Brech Yverdon et l Meerschert M M nd Tdjern C (06) Fnte dfference ppromtons for two-sded spce-frctonl prtl dfferentl equtons Appled umercl Mthemtcs Song L nd Zhng H (007) Applcton of homotopy nlyss method to frctonl KV- Burgers-Kurmoto equton Physcs Letters Yng Q Lu F nd Turner I (00) umercl methods for frctonl prtl dfferentl equtons wth Resz spce frctonl dervtves Appled Mthemtcl Modellng
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