Finite Difference Approximation for Transport Equation with Shifts Arising in Neuronal Variability

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1 Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Ide Copericus Value (3): 64 Impact Factor (3): 4438 Fiite Differece Approimatio for Trasport Equatio with Shifts Arisig i Neuroal Variability Chhatra Pal, Agam Dwivedi A R S D College, Departmet of Mathematics, iversity of Delhi, Delhi, Idia A R S D College, Departmet of Mathematics, iversity of Delhi, Delhi, Idia Abstract: This paper studies some fiite differece approimatios to fid the umerical solutio of first-order hyperbolic partial differetial equatio of mied type, ie, trasport equatio with poit-wise delay ad advace We are iterested i the challegig issues i euroal sciece stemmig from the modelig of euroal variability based o Stei s Model [8] The resultig mathematical model is a first order hyperbolic partial differetial equatio havig poit-wise delay ad advace which models the distributio of time itervals betwee successive euroal firigs We costruct, aalyze ad implemet eplicit umerical scheme for solvig such type of iitial ad boudary-iterval problems Aalysis shows that umerical scheme is coditioally stable, cosistet ad coverget i discrete orm Some umerical tests are reported to validate the computatioal efficiecy of the umerical approimatio Keywords: hyperbolic partial differetial equatio, euroal firig, poit-wise delay ad advace, fiite differece method, a- Friedrichs scheme Itroductio Partial differetial-differece equatios or more geerally partial fuctioal differetial equatios are of great importace, sice they arise i may mathematical models of cotrol theory, mathematical biology, climate models, mathematical ecoomics, meteorology ad may other areas, see [4] Some differetial models from populatio dyamics are give The time depedet first-order hyperbolic partial differetial equatio (trasport equatio) is a geeral partial equatio that describes the trasport pheomea such as heat trasfer, mass trasfer, mometum trasfer, etc Some trasport equatios have multiple poit-wise delay ad advace which are classified as hyperbolic partial differetial differece equatios These equatios provide a tool to simulate several realistic physical ad biological pheomea For istace, i Stei s model [8], the distributio of euroal firig itervals satisfies a trasport equatio of mied type with appropriate iitial-boudary coditio give by F F (, t) ( / ) (, t) pe[ F(, t) F(, t)] t pi[ F(, t) F(, t)], () F(,) F ( ), where t V equal the depolarizatio at time t ; F(, t) equal the probability that Vt at time t F is iitial data Here, it assume that ecitatory ad ihibitory impulse occur radomly with a frequecy p e /sec ad p i /sec, respectively After each euroal firig there is a refractory period of duratio t, durig which the impulse have o effect ad the membrae depolarizatio V t is reset to be zero At times t t, a ecitatory impulse produces uit depolarizatio while ihibitory impulse produces uit repolarizatio, ad if the depolarizatio reaches a threshold of r uits, the euro fires For sub- threshold levels, the depolarizatio decays epoetially betwee impulses with the time Volume 4 Issue 3, March 5 wwwisret costat To study the euro variability i quatitative terms, Stei trasformed this equatio ad obtai the characteristic fuctio of the distributio ad aalyzed the mea ad variace of the distributio We refer to [8] for more detailed iformatio about the assumptios of the model We are iterested to fid the value of ukow F Formulatio of the Problem The equatio () give i the previous sectio is a first-order partial differetial-differece equatio Motivated from the model described i the previous sectio, we cosider the followig geeral trasport equatio with poit-wise delay ad advace: u u (, t) a(, t) (, t) b(, t)[ u(, t) u(, t)] t c(, t)[ u(, t) u(, t)] () with iitial coditio u(,) u ( ) (3) where a(, t), b(, t ) ad c(, t ) are sufficietly smooth fuctios of ad t ad are small positive costats et us suppose the regio i which we wat to fid the solutio of equatio () is X Sice the partial differetial equatio () is first order hyperbolic type with differece terms, accordig to the directio of characteristics, we require oly oe boudary coditio If a, we eed a boudary coditio at left side of the domai ie at ad if a, we eed a boudary coditio at right side of the domai ie at X, Morto [6] Therefore the boudary iterval coditios for this equatio are give by u( s, t) ( s, t), s [,]; for a, Paper ID: SB icesed der Creative Commos Attributio CC BY u( s, t) ( s, t), s [X,X ]; for a, (4) The Taylor series approimatio for the delay ad advace term i equatio (), yields

2 u(, t) u(, t) u (, t) R u(, t) u(, t) u (, t) R Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Ide Copericus Value (3): 64 Impact Factor (3): 4438 where R ad R are the remaiders i the Taylor series epasios st, R u(, t)! ad R u(, t)! By substitutig the epasios u(, t) ad u(, t) values i (), we get u (, t) a(, t) u (, t) b(, t)[ u(, t) u (, t) R u(, t)] t c(, t)[ u(, t) u(, t) R u(, t)] For sufficietly small values of poit-wise delay ad advace, remaider terms R ad R are egligible Therefore ut [ a(, t) b(, t) c(, t)] u (5) with iitial ad boudary-iterval coditios, u(,) u ( ), u(, t) (, t), for ( a b c) (6) u( X, t) ( X, t), for ( a b c) I the followig sectios, a umerical method based o fiite differece is developed to solve such type of iitial ad boudary value problems The proposed method is aalyzed for stability ad covergece Some test eamples are give to validate covergece ad computatioal efficiecy of the proposed umerical algorithm 3 Numerical Approimatio I this sectio, we costruct umerical scheme based o the fiite differece method [6] We discuss first ad secod order eplicit umerical approimatios for the give equatio () based o a-friedrichs fiite differece approimatios The differetial equatio () is hyperbolic ad first-order with differece terms 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,,,, J, J; t t,,,, Now we look for discrete solutio u(, t ),, u that approimate A t solvig for, we get t ( ) A ( ),,,, J (7) t where A together with iitial ad boudary-iterval coditios are give by u ( ),,, J, (, t ),,, ( X, t ),,, J 3 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 time step such that C,,, ow cosider the fiite differece scheme as give equatio (7) ie t t A A t t A A takig the orm, we get sup t t A sup A sup t t A A t If A, the above iequality reduces to which implies the stability of the umerical scheme t provided A 3 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 33 Error Aalysis The local trucatio error of the umerical scheme is obtaied by replacig the approimate solutio by eact solutio u(, t ) i the umerical scheme If u is sufficietly smooth, the trucatio error differece scheme is give by T of this fiite Volume 4 Issue 3, March 5 wwwisret Paper ID: SB icesed der Creative Commos Attributio CC BY

3 Numerical solutio T u u u u u A t Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Ide Copericus Value (3): 64 Impact Factor (3): ut tutt u O( t ) O( ) t t 4 ( ) ( ) A u u O 6 [ u Au ] tu u Au t 6 t tt O t t 4 4 ( ) Sice u is eact solutio, we get t [ u Au ] hece T tu u Au t 6 tt 4 4 O( t t ) which shows that the umerical scheme is cosistet of order i space ad of order i time as log as t 34 Covergece of the scheme Defiitio: A fiite differece scheme (Numerical method) 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 (7) eactly, while u(, t ) leaves the remaider T error is give by t t e A e A e tt ad e et E ma e,,,, J t Hece for A, E ma e E t ma T ad t Therefore the E If we suppose that the trucatio error is bouded ie T the by iductio method T ma, E tt, ma ttma which shows that the method has first-order coverget provided that the solutio has bouded derivatives up to secod order 4 Numerical Eperimets I this sectio, we preset some umerical eamples to validate the predicted results established i the paper 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 [3] We calculate the errors by refiig the grid poits The error i the umerical approimatio is give by t t/ E(, t) ma (, ) (, ) J,Nt / I the followig eamples the domai of cosideratio is [,] [, 7] Eample Cosider the problem () with the followig coefficiets ad iitial- boudary coditios: a(, t) ; b(, t) 5; c( t, ) ; 4 t u(,) ep[ (4 ) ]; u( s, t), s [,] Eample Cosider the problem () with the followig coefficiets ad iitial- boudary coditios: 4 a(, t) ; b(, t) t ; 4 t c(, t) t; u u( s, t), s [,] (,) ep[ (4 ) ]; Eample3 Cosider the problem () with the followig coefficiets ad iitial- boudary coditios: a(, t) ; b(, t) ; 4 t 4 t c( t, ) ; t u(,) ep[ (4 ) ]; u( s, t), s [,] Table : The maimum absolute error for eample = = = 5 = directio Figure: The umerical solutio for Eample at t 5 Volume 4 Issue 3, March 5 wwwisret Paper ID: SB576 4 icesed der Creative Commos Attributio CC BY

4 Numerical solutio Numerical solutio Numerical solutio Numerical solutio Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Ide Copericus Value (3): 64 Impact Factor (3): = = = 5 = 8 = = = = directio Figure: The umerical solutio for Eample at t directio Figure5: The umerical solutio for Eample 3 at t 5 8 = = t = 3 t = 5 t = 7 Table : The maimum absolute error for eample 6 4 Table 3: The maimum absolute error for eample directio Figure3: The umerical solutio of Eample for differet values of t = = = = directio Figure4: The umerical solutio for Eample 3 at t 5 5 Coclusio I this paper, a first-order hyperbolic partial differetial differece equatio for the distributio of euroal firig based o the Stei s Model [8] For fidig the umerical solutio of the iitial ad boudary value problem, a umerical scheme based o upwid fiite differece is developed The maimum absolute error are computed ad tabulated i tables -3 for the cosidered eamples with ad The error table illustrates that the method is first order coverget i temporal ad secod order spatial directios Basically i this paper, we compare the results as already discussed by Sharma ad Sigh [7] Our results are better due to secod order covergece of scheme The graphs of the solutio of the cosidered eamples for differet values of poit-wise delay ad advace are plotted i Figures -5 to eamie the effect of poit-wise delay as well as advace o the solutio behavior of the problem We observe that if we fi ad icrease the value of, impulse moves towards left see (fig ad 5) while fiig ad icrease the value of, impulse moves towards right see (fig ad 4) Now fiig both ad, the impulse moves towards right with the time see (fig3) Volume 4 Issue 3, March 5 wwwisret Paper ID: SB576 4 icesed der Creative Commos Attributio CC BY

5 Refereces Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): Ide Copericus Value (3): 64 Impact Factor (3): 4438 [] Causo, D M, ad Migham, CG, Itroductory fiite differece methods for PDEs, Veturs Publishig ApS, K () [] Eccles, J C, The Physiology of Syapses, Spriger, Berli, (964) [3] Doola, E P, Miller, J J H ad Schilders, W H A, iform Numerical Methods for Problems with Iitial ad Boudary ayers, Boole Press, Dubli (98) [4] Hale, J K, ad Verduy uel, S M, Itroductio to Fuctioal Differetial Equatios, Spriger-Verlag, New York, (993) [5] Katz, B, Depolarizatio of Sesory Termials ad the Iitiatio of Impulses i the Muscle Spidle, J Physiol, (95), 6-8 [6] Morto, KW ad Mayers, DF, Numerical solutio of partial differetial equatio, Cambridge iv-press, Cambridge, (994) [7] 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 [8] Stei, RB, A theoretical aalysis of euroal variability, Biophysical J, 5 (965), Author Profile Chhatra Pal received the MPhil degree (Fiite differece methods for trasport equatio) from iversity of Delhi, Delhi i 5 Volume 4 Issue 3, March 5 wwwisret Paper ID: SB576 4 icesed der Creative Commos Attributio CC BY