Generalization of the Exact Solution of 1D Poisson Equation with Robin Boundary Conditions, Using the Finite Difference Method

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1 Journl of Electromnetc Anlyss nd Applctons Pulshed Onlne Octoer 4 n ScRes Generlzton of the Exct Soluton of D Posson Equton wth Ron Boundry Condtons Usn the Fnte Dfference Method Serne Br Gueye Khroun Tll Cheh Mow Déprtement de Physque Fculté des Scences et Technques Unversté Cheh Ant Dop Dr-Fnn Sénél Eml: sry@mlcom Receved 8 Auust 4; revsed Septemer 4; ccepted 6 Septemer 4 Copyrht 4 y uthors nd Scentfc Reserch Pulshn Inc Ths wor s lcensed under the Cretve Commons Attruton Interntonl Lcense (CC BY) Astrct A new nd nnovtve method for solvn the D Posson Equton s presented usn the fnte dfferences method wth Ron Boundry condtons The exct formul of the nverse of the dscretzton mtrx s determned Ths s the frst tme tht ths fmous mtrx s nverted explctly wthout usn the rht hnd sde Thus the soluton s determned n drect very ccurte (O(h )) nd very fst (O()) mnner Ths new pproch trets ll cses of oundry condtons: Drchlet eumnn nd mxed Therefore t cn serve s reference for solvn the Posson equton n one dmenson Keywords Ron Boundry Posson Equton Mtrx Inverson Introducton The Posson equton s n ellptc dfferentl equton well nown nd common to vrous scentfc nd techncl domns such s physcs mthemtcs chemstry oloy etc Its resoluton enertes lot of nterest to enneers techers nd reserchers Snce t llows nlyzn n quntttve mnner the studed phenomen: electrosttc mnetosttc wve propton or het dffuson n stedy stte Mny methods of resoluton of ths mportnt equton exst however usn the rht hnd sde (RHS) for exmple the method of Gussn elmnton or the Thoms lorthm How to cte ths pper: Gueye SB Tll K nd Mow C (4) Generlzton of the Exct Soluton of D Posson Equton wth Ron Boundry Condtons Usn the Fnte Dfference Method Journl of Electromnetc Anlyss nd Applctons

2 S B Gueye et l Recently new method s proposed [] [] deln wth ths equton n one dmensonl cse Ths new pproch usn the fnte dfferences method (FDM) determned the nverse of the mtrx otned from lerc equtons However only the cse of oundry condtons Drchlet-Drchlet () []; eumnn-drchlet (D) nd Drchlet-eumnn (D) [] were treted The present study enerlzes the soluton of the Posson equton nd determnes ts soluton for oundry condtons of thrd nd: Ron condtons These mxed oundry condtons present ret nterest n prctce ecuse of comnn two qunttes: the functon nd ts dervtve Here ret nnovton s tht the soluton s otned n very fst nd precse wy nd wthout usn the RHS Frst n nventory s mde for possle cses of oundry condtons () (D) (D) () (R) (R) (DR) nd (RD) Then the cses whch must e solved re dentfed consdern tht the frst three cses of oundry prolems hve een solved n [] nd [] Then the cses () (R) (R) were treted The fnl soluton s otned usn n dequte choce of dscretzton nd nvertn drectly nd exctly the dscretzton mtrx Thus numercl verfcton s done consdern oundry prolem of type () The senslty s determned shown the predcted ehvor of the truncton error nd the very ood ccurcy of ths new method Fnlly the cses of oundry condtons of type (DR) nd (RD) re dscussed nd solved Generl Prolem The follown oundry prolem s to e solved: ( x) f( x) x ] [ α β = α + = () β + = Here s sclr feld whch depends on the rel vrle x The functon f(x) s well nown exctton nd wll e clled the Rht Hnd Sde (RHS) of the Posson equton = = = nd = re unnown vlues of nd of ts frst dervtve t ponts nd respectvely The rels nd re ven constnts The coeffcents α α β nd β re lso nown The type of oundry prolem s ven y the nture of the qudruplet ( α α β β ) For exmple ( α β ) corresponds to oundry prolem of type Drchlet-Drchlet () whle the comnton ( α β ) s for oundry prolem of type eumnn-drchlet (D) etc Dependn on the vlue of the qudruplet of coeffcents nne (9) oundry prolems exst: () (D) (D) () () (R) (R) (DR) nd (RD) The frst three prolems () (D) nd (D) were solved n [] nd [] The prolem () leds to non-reulr dscretzton mtrx Ths study wll determne the solutons for ll the remnn cses; nd therefore cn serve s reference for ll tht wll nvolve the D Posson equton We dstnush three prts: α nd β We wll cll ths type of oundry prolem Ron-Ron () Ths prt lso covers Ron-eumnn (R) nd eumnn-ron (R) oundry prolems Ths s the most mportnt prt of ths study ecuse helpn to resolve the next two prts α = nd β Ths s the cse of the Drchlet-Ron (DR) oundry prolem α nd β = Ths s the cse of the Ron-Drchlet (RD) oundry prolem 3 D Posson Equton wth Ron-Ron () Boundry Condtons We consder the D Posson equton wth oundry condtons of type Ron-Ron () Ths prolem corresponds to the cse where α nd β At ponts nd we hve the follown reltons: α + = () β + = where α α = α β β = β = nd α = β We propose new method of resoluton of ths prolem The ltter requres comfortle nd dequte dscretzton s shown n Fure 373

3 S B Gueye et l Fure Dscretzton for Ron-Ron oundry condtons The mesh s composed of + ponts of them re n the ntervl of nterton [] The other two re extr ddtonl ponts [3] [4] The mesh ponts ( x ) re defned y the follown relton: x = + ( ) x = + where ( ) x desntes the step sze: x = = h The pproxmte vlue of the desred sclr feld t pont x s denoted y wth ( x) And t ths pont the vlue of the rht-hnd sde functon s: f = f ( x) At pont x the frst nd second dervtve of the functon re = ( x) nd = ( x) respectvely Wth the centered dfference pproxmton (O(h )) []-[5] one ets the frst dervtve: nd the second dervtve s: + = + O h (3) h ++ = + O ( h ) = 3 (4) h The system of lner equtons ssocted to the Boundry Prolem of type Ron-Ron cn e wrtten: + = h f = (5) + At oundry ponts the dscretzton of the condtons ves: α + = h + β + = h The pproxmte vlues of the sclr feld t ddtonl mnry ponts nd + cn e elmnted Ths s done consdern the dscrete dfferentl equton t oundry ponts = x nd = x wth the Eq- = : uton (6) One otns the follown equton t pont f h + = h + h ( α ) : = α nd the follown equton t pont = : f ( hβ + ) + = h h : = β One remrs tht the ddtonl ponts re not ncluded n the clcultons They llowed the use of the centered dfference pproxmton; even t the oundry ponts Therefore the truncton error ehves le ( O( h )) [3] The vector F cn e defned wth ts components F : f f F = h + h F = h h nd F = h f = 3 Comnn the Equtons (5) (7) nd (8); one ets the follown mtrx equton: (6) (7) (8) (9) 374

4 S B Gueye et l f α h + h h f 3 h f 3 4 h f4 = 5 h f5 h f β f h h : = A : = : = F A s the dscretzton mtrx n the cse of oundry condtons of type Ron-Ron () Ths mtrx s wdespred n the lterture The Equton () cn e solved wth methods such s Gussn elmnton ( O( )) or the Thoms lorthm ( O( )) [6] We propose here new method of resoluton fster nd more ccurte thn tht of Thoms; s we hve lredy done for the oundry prolem of type () [] nd (D) or (D) [] Ths method s sed essentlly on the exct formulton of the nverse of the mtrx A The formul of the nverse of the dscretzton mtrx wll e determned explctly nd drectly We denote t B 4 Inverse of the Mtrx A The mtrx A ( j ) clculton of = s symmetrc nd ts determnnt A permts to verfy the reulrty of the mtrx sults of [] nd [] the formul of the determnnt of A () A depend on the coeffcents α nd β The A e ts nvertlty Explotn the re- A cn e deduced It holds: ( αβ α β )( ) ( αβ ) = () Ths determnnt s zero for the cse of oundry condtons of type eumnn-eumnn () whch corresponds to α = β = Aprt from ths cse A s dfferent from zero nd the nverse of the mtrx A exsts ow the nverse of the mtrx A = ( j ) wll e determned usn the mtrx ssocted to the prolem () whch s resolved n []: A = + ( α + ) + ( β + ) : = A Snce A nd ts nverse formul s nown One ets: I s the dentty mtrx of sze The mtrces (C) nd (D) re defned n follown mnner respec- where tvely: B re nown [] one cn multply the Equton () y ( α ) ( β ) + + B A = I + + : = ( C) : = ( D) () B whose exct (3) 375

5 S B Gueye et l j ( j ) δ C : = c = + j = nd (4) ( j ) j D : = d = δ j = (5) The mtrces (C) nd (D) cn e fctorzed nd put n the form ( C) ( E)( A ) ( D) ( F)( A ) ( C) = One otns for the mtrx (C): A = A = nd A ( )( β + ) ( j)( β + ) β ( ) ( )( β + ) ( ) ( j)( β + ) ( ) β ( ) ( ) ( )( β + ) ( ) ( j)( β + ) ( ) β ( ) ( )( β + ) 3 ( j) ( β + ) 3β 3 ( )( β + ) ( j)( β + ) β ( )( β + ) ( j)( β + ) β ( ) ( )( β ) ( ) ( j)( β ) ( ) β ( ) :=( E) nd for the mtrx (D) one ets: ( j ) ( ) ( j ) ( ) ( j ) ( ) α α + α + α α + α + 3 3α 3 α + 3 α + D = α ( j )( α ) ( )( α ) + + A A ( ) ( ) α ( ) ( j )( α + ) ( ) ( )( α + ) ( ) ( ) α ( ) ( j ) ( α + ) ( ) ( )( α + ) α ( j )( α + ) ( )( α + ) :=( F ) The Equton (6) shows tht the mtrx (E) s defned n the follown mnner: ( β ) E : = ej = + j + j = (8) From the Equton (7) t s noted tht the mtrx (F) s defned n the follown mnner: ( α ) Thus the mtrx Equton (3) ecomes: F : = fj = j + j = (9) ( α ) ( β ) + + B A = I + ( E)( A ) + F A + A + A And therefore one hs the relton tht provdes the nverse of the mtrx ( A ) : A (6) (7) () 376

6 S B Gueye et l ( α ) ( β ) + + ( B ) ( E) ( F) ( A ) = I + A + A : = B So we et B nverse of the mtrx A Ths fmous mtrx s nverted exctly nd n elent mnner [5] [7]-[] We ve to the mtrx B the nme Br_ Mtrx It holds: wth ( B ) = ( j ) = ( j ) ( α )( ) ( j)( β ) ( β ) ( j )( α ) ( + ) A ( ) () () j j + j = j = (3) ( j ) j + < The Equton () s very mportnt n the feld of numercl resoluton of dfferentl equton n one dmenson Becuse t presents the exct formul of the fmous mtrx A It llows to solve the Equton () n extremely fst wy very precsely nd ndependently of the RHS It provdes n nnovtve soluton to the Posson equton for the cse of oundry condtons of type Ron-Ron A smple mtrx-vector multplcton ves the soluton of the dfferentl equton But we re not lmtn there Becuse y explotn the propertes of the mtrx B we cn retly reduce the cost n tme nd memory spce A further nlyss of the mtrx B ves the soluton n ny mesh pont x : = + ( α + + β + + β + α + ) F (4) + A = where ( B ) ( ) = F : = ( mx ( ) + ) mn ( ) F + (5) = Ths ltter equton s the sme s the follown []: = = = + ( ) F ( ) F (6) The nlyss of ths soluton shows tht one loop s suffcent to otn the soluton t pont x Ths soluton ven y the Equton (4) hs een otned ndependently from the RHS So t s strhtforwrd soluton tht does not use the RHS of the dfferentl equton In ddton prormmer does not need to declre rrys to store mtrces or vectors Ths soluton n Equton (4) comned wth Equton (5) corresponds to n lorthmc complexty of O( ) It s stle roust nd very economcl wth respect to memory occupton The Equton (4) s soluton for oundry prolems stsfyn α nd β () It s lso soluton of oundry prolems of type Ron-eumnn (R) or eumnn-ron (R) 5 Verfcton wth Ron-Ron () Boundry Prolem We consder sclr feld ( x) defned n [ ] whch stsfes n ] [ The feld ( x) ( x) x = = f x = π sn πx x fulflls the follown Ron-Ron oundry condtons s n (7): 377

7 S B Gueye et l α + = (7) β + = where the prmeters α β nd re well nown rel constnts The exct soluton cn e expressed s follows: of course for β α β + α π β + + β + π exct ( x) = ( x) + x+ β β + α β β + α sn π β β + α The mesh s ten s specfed n Fure : h = x = x = = ( ) h ; = To verfy our new method of resoluton we choose: = α = nd β = 4 It follows: hα = α = 3 nd ( hβ + ) = β = 5 Also we choose: = nd = 5 The computed results re shown n Tle The otned soluton wth our new method usn the fnte dfference method s very ccurte; s Tle ε the reltve error n ech pont x : ove shows We denotes ε = FDM exct ( ) From Tle one cn deduce the vere reltve error whch enerl expresson s ven s follown: ε = exct ε (8) (9) = (3) 5 It hols: ε The senslty of our new method cn e determned y plottn the vere reltve error ε ( ) for dfferent vlues of Then we ot the curve shown n Fure whch s hyperol tht cn e ssumed to e Tle umercl results nd reltve error x FDM exct ε () E E E E 5 E E E E 4 3 E E E E 4 4 3E E E E 4 5 4E E E E 5 6 5E E E E 5 7 6E E E E 5 8 7E E E E 5 9 8E E E E 5 9E E E E E E E E E E E E E E E E E E E E E E E E E E E E 5 99E E E E 5 E E E E 5 378

8 S B Gueye et l proportonl to h Fure Senslty for oundry condtons = = x ( ) Thus curve fttn of the senslty cn e ven wth: Trunc( ) = α h = α ( ) α The two curves re shown n Fure The vere reltve error ( ) ( 4 h ) ( c) ( 4 ) ( c) where (3) ε e- hves le truncton error tht cn e expressed n the follown mnner desntes the fourth order dervtve of the exct functon exct n pont (here C) whch elons to the ntervl [ ] Comnn the ven functon f ( x ) the exct functon exct nd the results from the fttn one ets [6]: h π ε α h = α < (3) So we hve demonstrted tht ths new method s very fst very effcent In ddton t s very economcl n terms of occupton of memory spce It s lso very ccurte The mtrx A s nverted nlytclly nd n ccurte mnner; ndependently of the RHS The ehvor of the truncton error s very nterestn s shown n the senslty curve The resoluton of the remnn cses Drchlet-Ron (DR) nd Ron-Drchlet (RD) presents no dffculty It s esy nd cn e done on the ss of the foreon 6 D Posson Equton wth Ron-Drchlet nd Drchlet-Ron Boundry Condtons 6 D Posson Equton wth Drchlet-Ron (DR) Boundry Condtons We consder now Drchlet-Ron prolem s one where the feld t pont ( ) s nown nd the oundry condton t pont s of type Ron: β + = (33) where β The coeffcents β nd re ven For ths Drchlet-Ron oundry prolem we propose n pproprte dscretzton s shown n Fure 3 Here the mesh ponts ( x ) re defned y the follown relton: x = + h = + The step sze chnes nd ecomes: h = x = The only mnry pont s x + Its sclr feld + s elm- 379

9 S B Gueye et l Fure 3 Dscretzton for Drchlet-Ron oundry condtons nted s done n Equton (8) Thus consdern the Equtons (3)-(8) nd dptn them to the (DR) prolem one ets the follown mtrx equton: h f h f h f 3 3 h f4 4 h f = (34) 5 5 h f β f h h DR : = A : = : = F One remrs tht wth respect to the Ron-Ron oundry prolem there s only two chnes The frst component of the vector F ecomes F = h f The second one chne reltvely to the () prolem s DR tht ther mtrces re relted y: A = A ( α = β) Therefore the nverse of the mtrx A DR s: DR B = B ( α = β) Then the soluton of the Drchlet-Ron oundry prolem s nown consdern the equtons ()-(6) nd the chne of F nd replcn α y 6 D Posson Equton wth Ron-Drchlet (RD) Boundry Condtons We consder now Ron-Drchlet prolem s one where the feld t pont ( ) s nown nd the oundry condton t pont s of type Ron: α + = (35) where α The coeffcents α nd re ven For ths (RD) oundry prolem we propose the follown pproprte dscretzton s shown n Fure 4 Here the mesh ponts ( x ) re defned y the follown relton: x = + ( ) h = + The step sze chnes nd ecomes: h = x = The mnry pont s x Its sclr feld s elmnted s done n Equton (7) Thus consdern the Equtons (3)-(8) nd dptn them to the (RD) prolem one ets the follown mtrx equton: f α h + h h f 3 h f 3 4 h f 4 = (36) 5 h f5 h f h f RD : = A : = := F 38

10 S B Gueye et l Fure 4 Dscretzton for Ron-Drchlet oundry condtons One remrs tht wth respect to the Ron-Ron oundry prolem there s only two chnes The lst component of the vector F ecomes F = h f The second one reltvely to the () prolem s tht the RD RD mtrces of the two prolems re relted y: A = A ( αβ = ) Therefore the nverse of the mtrx A RD s: B = B ( αβ = ) Then the soluton of the Ron-Drchlet oundry prolem s nown consdern the Equtons ()-(6) nd the chne of F nd replcn β y 7 Concluson Ths study reltes to the resoluton of D Posson equton wth the enerl cse of Ron oundry condtons The mjor nnovton tht s presented s the exct formulton of the nverse of the dscretzton mtrx whch s otned from usn the fnte dfference method Ths remrle nd drect nverson of ths mtrx provdes n elent nd effcent soluton to ths very mportnt dfferentl equton Ths new proposed method of resoluton s precse economc n memory occupncy nd extremely fst It cn serve s reference for solvn numerclly the D Posson equton nd the sttonry wve or het equton References [] Gueye SB (4) The Exct Formulton of the Inverse of the Trdonl Mtrx for Solvn the D Posson Equton wth the Fnte Dfference Method Journl of Electromnetc Anlyss nd Applctons [] Gueye SB Tll K nd Mow C (4) Soluton of D Posson Equton wth eumnn-drchlet nd Drchlet-eumnn Boundry Condtons Usn the Fnte Dfference Method Journl of Electromnetc Anlyss nd Applctons [3] Kress HO (97) Dfference Approxmtons for Boundry nd Eenvlue Prolems for Ordnry Dfferentl Equtons Mthemtcs of Computton [4] Eneln-Muelles G nd Reutter F (99) Formelsmmlun zur umerschen Mthemt mt QucBsc-Prormmen Drtte Aufle BI-Wssenchftsverl Mnnhem [5] LeVeque RJ (7) Fnte Dfference Method for Ordnry nd Prtl Dfferentl Equtons Stedy Stte nd Tme Dependent Prolems SIAM [6] Conte SD nd de Boor C (98) Elementry umercl Anlyss: An Alorthmc Approch 3rd Edton McGrw- Hll [7] Mthews JH nd Fn KK (4) umercl Methods Usn Mtl 4th Edton Prentce-Hll Inc ew Jersey [8] Sdu Mtthew O () umercl Technques n Electromnetcs nd Edton CRC Press Boc Rton [9] Gustfson K (988) Domn Decomposton Opertor Tronometry Ron Condton Contemporry Mthemtcs [] Renhold P (8) Anlyss of Electromnetc Felds nd Wves: The Method of Lnes John Wley Chchester 9 38