Numerical Solution of the First-Order Hyperbolic Partial Differential Equation with Point-Wise Advance

Transcription

1 Iteratioal oural of Sciece ad Research (ISR) ISSN (Olie): Ide Copericus Value (3): 4 Impact Factor (3): 4438 Numerical Solutio of the First-Order Hyperbolic Partial Differetial Equatio with Poit-Wise Advace Chhatra Pal, Viit Chauha A R S D College, Departmet of Mathematics, iversity of Delhi, Delhi Dr Bhim Rao Ambedkar iversity, Agra, ttar Pradesh-84, Idia Abstract: I this paper, we costruct a eplicit umerical scheme based o a-friedrichs fiite differece approimatio to fid the umerical solutio of first-order hyperbolic partial differetial equatio with poit-wise advace The differetial equatio ivolvig poit-wise delay ad advace models the distributio of the time itervals betwee successive euroal firigs We costruct higher order umerical approimatio ad discuss their cosistecy, stability ad covergece Aalysis shows that umerical scheme is coditioally stable, cosistet ad coverget i discrete orm We also eted our method to the higher space dimesios Some test eamples are icluded to illustrate our approach These eamples verify the theoretical results ad show the effect of poit-wise advace o the solutio Keywords: hyperbolic partial differetial equatio, trasport equatio, poit-wise advace, fiite differece method, a-friedrichs scheme Itroductio Hyperbolic partial differetial-differece equatios provide a tool to simulate several realistic physical ad biological pheomea Several biological pheomea ca be modeled by time depedet first-order partial differetial differece equatios of hyperbolic type which cotais poit-wise advace or shifts i space We cosider the followig first-order hyperbolic partial differetial equatio havig poit-wise advace with a iitial data u o the domai : (,X) I geeral it reads u au bu(, t),, t, t u u (,) ( ),, u(, t) ( t), for a, u( s, t) ( s, t), s [X,X ], for a, where a a(, t) ad b b(, t) are sufficietly smooth ad bouded fuctios of ad t i the etire domai ad does ot chage its sig i the etire domai, is the value of poit-wise advace which is ozero fied real umber et a(, t) A ad b(, t) B, (, t) The ukow fuctio u is defied i the uderlyig domai ad also i the iterval X, X due to the presece of poit-wise advace So our domai is, X X, X () ad t The coefficiets are sufficietly smooth fuctios i these itervals ad the ukow fuctio u is as smooth as the iitial data Due to the presece of poit-wise advace i equatio (), we eed a boudary-iterval coditio i the right side of domai, ie, i the iterval X, X The equatio () is first-order hyperbolic with advace terms, so it requires oe boudary coditio accordig to the directio of characteristics, see [8] Due to the presece of poit-wise advace ad ocostat coefficiets, it is ot difficult but impossible to fid aalytical solutio of such type of partial differetial equatios by usig the usual methods to fid the eact solutio of partial differetial equatios, see [4] If delay ad advace argumets are sufficietly small, the author used the Taylor series approimatio for the term cotaiig shift argumets The Taylor series approimatio may lead to a bad approimatio whe the size of shift argumets is large Therefore, the umerical scheme preseted i [9] does ot work for the problem with large shift argumet To overcome this difficulty, we geerate a special type of mesh so that the differece term lies o the odal poit i the discretize domai ad preset a umerical scheme that works icely i both the cases We costruct the umerical scheme to fid the approimate solutio of problem () i Sectio ad discuss the stability, cosistecy ad covergece I Sectio 3, we discuss the etesio of umerical scheme i higher space dimesios I Sectio 4, we iclude some test eamples for umerical illustratio Fially, i Sectio 5, we make coclusios that illustrate the effect of advace argumets o the solutio behavior Numerical Approimatio I this sectio, we costruct umerical scheme based o the fiite differece method [8] We discuss first ad secod order eplicit umerical approimatios for the give equatio () based o a-friedrichs fiite differece approimatios For space time approimatios based o fiite differeces, the ( t, ) plae is discretize by takig mesh width ad time step t, ad defiig the gird poits (, t ) by,,,,, ; Volume 4 Issue 4, April 5 wwwisret Paper ID: SB icesed der Creative Commos Attributio CC BY

2 t t,,,, Now we look for discrete solutio Iteratioal oural of Sciece ad Research (ISR) ISSN (Olie): Ide Copericus Value (3): 4 Impact Factor (3): 4438 that approimate u(, t),, We write the closure of as ad (,,,,, ) Costructio of the umerical scheme I this approimatio, we approimate the time derivative by forward differece ad space derivative by cetered differece ad the we replace by the mea value betwee ad for stability purpose Numerical scheme is give by a b u(, t) () t To tackle the poit-wise advace i the umerical scheme (), we discretize the domai i such a way that ( ) is a grid poit,,,, ; ie, we choose such that m, m ad we take total umber of poits i directio st X matissa( lx ), l, where matissa of ay real umber is defied as positive fractioal part of that umber The term cotaiig poit-wise delay (,,, ) ca be writte as u(, t ) u( m, t ) u(( m ), t ) m Therefore the umerical approimatio is give by, a b m, t,,, (3) together with iitial ad boudary-iterval coditios are give by u ( ),,,, (, t ),,, (s, t ), s [ X, X ],, Stability Aalysis Defiitio: The fiite differece method is called stable i the certai orm if there eists costat C, idepedet of the space step ad possibly depedig o the time step such that C,,, ow cosider the fiite differece scheme as give equatio (3) ie t t a a b t m (4) Volume 4 Issue 4, April 5 wwwisret t t a a takig the sup orm, we get sup t b Paper ID: SB534 5 icesed der Creative Commos Attributio CC BY m t t sup a sup a t sup b m t sig CF coditio A, (where A is the boud of a(, t), (,t) ), first two terms i the above iequality ca be combied ad we get ( B t) where b(, t) B, (, t) The term B t ca be cotrolled by t from which we ca predict that the effect of the term B t is of the form O( t) sig these values, we get ie, ( O( t)) C which implies the stability of the umerical scheme where stability costat C is of the form C ( O( t)) 3 Cosistecy of umerical scheme The cosistecy error of the umerical scheme is the differece betwee both sides of the equatio whe the approimate solutio is replaced by eact solutio u(, t ) i the umerical scheme If u is sufficietly smooth, the cosistecy error scheme is give by T T of this fiite differece u(, t) u(, t) u(, t ) t u(, t) u(, t) a b u m ow usig Taylor series approimatio for the term u(, t ) wrt to t ad for the terms u(, t) ad u(, t ) wrt to, we get 4 T ut tutt u O( t ) O( ) t t 4 ( ) ( ) m a u u O b u u au b u tu u t [ ] t m tt

3 Iteratioal oural of Sciece ad Research (ISR) ISSN (Olie): Ide Copericus Value (3): 4 Impact Factor (3): 4438 a 4 4 u O ( t t ) Sice u is eact solutio, we get [ ut au ] b u m hece T tu u a u t tt 4 4 O( t t ) Therefore, T while (, t) (,), which shows that the umerical scheme is cosistet of order i space ad of order i time as log as t 4 Covergece of the scheme Defiitio: A fiite differece scheme is said to be coverget if for ay fied poit (, t ) i a give domai (,X) (, t ),, t t u t (, ) the error i the approimatio is give by e u(, t ) Now satisfies the fiite differece scheme (3) eactly, while u(, t ) leaves the remaider T error i the approimatio is give by t t e a e a e b te tt ad e m et E ma e,,,, t Hece for a, E ma e E b te t ma T ( Bt) E t ma T usig the give iitial value for, so t Therefore the E ad if we suppose that the cosistecy error is bouded ie T T, ma the by iductio method E tt, ma ttma which shows that the method has first-order coverget provided that the solutio has bouded derivatives up to secod order 3 Etesio to Higher Spatial Dimesios Now we cosider the etesios of the umerical schemes to the higher spatial dimesios For the sake of simplicity, we Volume 4 Issue 4, April 5 wwwisret cosider the problem i two spatial dimesios The etesio to three spatial dimesios ca be doe i similar fashio The atural geeralizatio of the oe dimesioal model problem () is the followig equatio together with the iitial data ad boudary iterval coditios i the rectagular domai (, X) (, Y) u au bu cu(,y, t), (5) t y where abc,, are fuctios of y, ad t ad are the values of poit-wise delay i ad y -directio respectively For umerical approimatios, we discretize the domai by takig uiform gird poits with the spacig i the -directio ad y i the y -directio The gird poits (, yk, t ) are defied as follows,,,, ; Paper ID: SB534 5 icesed der Creative Commos Attributio CC BY y ky, k,,, ; k y t t,,, Now we write the etesio of a-friedrichs scheme (5) The approimate solutio at the gird poit (, y, t ) is k deoted by k, Thus the umerical scheme is give by 4 t t a b y, k, k, k, k, k, k, k, k, k, k, k tc (), k m, k q, together with appropriate iitial ad boudary-iterval coditios We take the grid poits i the both directios ( ad y ) i such a way that the term cotaiig poit-wise advace is also belog to discrete set of grid poits which ca be doe as we did i the oe dimesioal case We take total umber of poits i both ad y directio such that correspodig delays are o m ad q odal poits ad total umber of poits i the both directios is give by X matissa( ) lx, l, Y matissa() y ry, r, y Table : The maimum absolute error for eample t / / /4 /8 / / / / Table : The maimum absolute error for eample t / / /4 /8 / / / /

4 Numerical solutio Numerical solutio Numerical solutio Iteratioal oural of Sciece ad Research (ISR) ISSN (Olie): Ide Copericus Value (3): 4 Impact Factor (3): 4438 Most of the aalysis of the umerical approimatio i oe dimesio is easily eteded to the two dimesioal case [8] Trucatio error of this approimatio () will remai as i the oe dimesioal case ecept some additios due to the presece of y variable, see [8] sual aalysis will give the CF coditio for stability i the followig form At Bt, where A ad B are the bouds of a ad b respectively The proof of covergece follows i similar way, leadig to error i the approimatio E tt, ma ttma provided that the CF coditio is satisfied ad u has bouded derivatives up secod order 4 Numerical Illustratio The purpose of this sectio is to iclude some umerical eamples to validate the predicted results established i the paper ad to illustrate the effect of poit-wise advace o the solutio behavior We perform umerical computatios usig MATAB The maimum absolute errors for the cosidered eamples are calculated usig half mesh priciple as the eact solutio for the cosidered eamples are ot available [5] We calculate the errors by refiig the grid poits The error i the umerical approimatio is give by t t/ E(, t) ma (, ) (, ),Nt / We cosider (,), t for eample ad The umerical solutio is plotted for differet values of at the time t 5 i figure ad 3 ad for various values of time t i figure ad 4 Eample Cosider the problem () with the followig coefficiets ad iitial- boudary coditios: a(, t) ; 4 t b(, t) 5; u(,) ep[ (4 ) ]; u( s, t), s [, ] Eample Cosider the problem () with the followig coefficiets ad iitial- boudary coditios: a(, t) ; b(, t) ; 4 t 4 t u(,) ep[ (4 ) ]; u( s, t), s [, ] We cosider the two dimesioal problem (5) with variable coefficiets We cosider (,) (,), y ad time step t The approimate solutio is plotted with 5 ad 5 at time t 5 i figure 5 Eample3 Cosider the -D problem (5) with the followig coefficiets ad iitial- boudary coditios: y a(, y, t) ; 4 ( y) t ( y ) b(, y, t) ; 4 ( y ) t u(, y,) ep[ (4 4y ) ]; Volume 4 Issue 4, April 5 wwwisret u( s, s, t), s [, ] ad s [, ] 4 8 -directio Paper ID: SB534 5 icesed der Creative Commos Attributio CC BY = = 5 = Figure : The effect of the poit-wise advace o solutio at t 5 for eample t = t = 4 t = directio Figure : The umerical solutio of Eample for differet time levels for = = 5 = directio Figure 3: The effect of the poit-wise advace o solutio at 5 t for eample

5 Numerical Solutio Numerical solutio t = t = 5 t = 7 Iteratioal oural of Sciece ad Research (ISR) ISSN (Olie): Ide Copericus Value (3): 4 Impact Factor (3): directio Figure 4: The umerical solutio of Eample at differet time levels for y-directio a-friedrichs 5 -directio Figure 5: The umerical solutio of Eample 3 for 5 ad 5 at t 5 5 Coclusio I this paper we propose a umerical scheme based o a- Friedrichs fiite differece approimatios of order greater tha oe i space to solve hyperbolic partial differetial equatio with poit-wise advace The cosistecy, stability ad covergece aalysis prove that the proposed umerical schemes are cosistet, stable with CF coditio ad coverget i both space ad time This secod order umerical scheme i space maitais the height ad width better tha a first-order scheme as author discussed i paper [9] The effect of poit-wise advace o the solutio behavior is show by the some test eamples Error tables illustrate the fact that the methods are coverget i space ad time The solutios are plotted i graphs which show i figures -5 Also we eted our ideas i higher space dimesios ad iclude umerical eperimet to show the behavior of solutio i two space dimesio Refereces [] Belle, A ad Zearo, M, Numerical Methods for Delay Differetial Equatios, Oford iversity Press, New York (3) [] Causo, D M, ad Migham, CG, Itroductory fiite differece methods for PDEs, Veturs Publishig ApS, K () [3] Eccles, C, The Physiology of Syapses, Spriger, Berli, (94) [4] Evas, C, Partial Differetial Equatios, Secod editio, AMS, Pro-videce, () [5] Doola, E P, Miller, H ad Schilders, W H A, iform Numerical Methods for Problems with Iitial ad Boudary ayers, Boole Press, Dubli (98) [] Hale, K, ad Verduy uel, S M, Itroductio to Fuctioal Differetial Equatios, Spriger-Verlag, New York, (993) [7] Katz, B, Depolarizatio of Sesory Termials ad the Iitiatio of Impulses i the Muscle Spidle, Physiol, (95), -8 [8] Morto, KW ad Mayers, DF, Numerical solutio of partial differetial equatio, Cambridge iv-press, Cambridge, (994) [9] Sharma, Kapil K ad Sigh, Parameet, Hyperbolic Partial Differetial Differece Equatio i the Mathematical Modelig of Neuroal Firig ad its Numerical Solutio, Appl Math Comput, (8), 9-38 [] Stei, RB, A theoretical aalysis of euroal variability, Biophysical, 5 (95), [] Strikwerda, C, Fiite differece schemes ad Partial Differetial Equatios, Secod Editio, SIAM, Philadelphia (4) Author Profile Chhatra Pal received the MPhil degree (Fiite differece methods for trasport equatio) from iversity of Delhi, Delhi i 5 Volume 4 Issue 4, April 5 wwwisret Paper ID: SB icesed der Creative Commos Attributio CC BY