THERMAL SCIENCE, Year 07, Vol., No. 4, pp. 595-599 595 NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS by Yula WANG *, Da TIAN, ad Zhiyua LI Departmet of Mathematics, Ier Mogolia Uiversity of Techology, Hohhot, Chia Itroductio Origial scietific paper http://doi.org/0.98/tsci6065040w The barycetric iterpolatio collocatio method is discussed i this paper, which is ot valid for sigularly perturbed delay partial differetial equatios. A modified versio is proposed to overcome this disadvatage. Two umerical examples are provided to show the effectiveess of the preset method. Key words: barycetric iterpolatio, sigularly perturbed, delay parameter, Chebyshev odes, Taylor s series expasio Sigularly perturbed delay partial differetial equatios arise from thermal sciece ad mechaics systems which are characterized by both spatial ad temporal variables, ad exhibit various spatio-temporal patters ad provide more realistic models for thermal sciece where time-lag or after-effect has to be cosidered. A characteristic example is []: = + τ + τ u u u vgu { [ ( xt, )]} cfu { [ ( xt, )] u( xt, )} t which models a furace used to process a metal sheet. Here u is the temperature distributio i a metal sheet, movig at a velocity, v, ad heated by a source ad specified by the fuctio, f. Both v ad f are dyamically adapted by a cotrollig device moitorig the curret temperature distributio. The fiite speed of the cotroller, however, itroduces a fixed delay of legth. Whe τ = 0, eq. () becomes a thermal problem without time delay. Whe we select D = (0, ) (0, T), the problem cosidered is the followig sigularly perturbed delay parabolic equatio with Dirichlet boudary coditios: (, ) uxt (, ) + axuxt ( ) (, ) = f( xt, ),( xt, ) [0,] [0, T] t uxt (, ) = ψ( xt, ),( xt, ) [0,] [ γ,0] u(0, t) = ϕt u(, t) = ϕr where 0 < e is sigular perturbed parameter, f(x, t), y(x, t), φ T(t), ad φ R(t) are sufficietly smooth ad bouded fuctios. The termial time, T, is assumed to satisfy the coditio * Correspodig author, e-mail: wylei@63.com () ()
596 THERMAL SCIENCE, Year 07, Vol., No. 4, pp. 595-599 T = K T, where K is a positive iteger. Uder the previous assumptios ad coditios, problem () with the iitial data ad boudary coditios has a uique solutio []. There are may methods to solve this problem [-9], for example, the variatioal iteratio method, the homotopy perturbatio method, ad others. I the past, the barycetric iterpolatio collocatio method (BICM) has bee preseted ad applied to may fields [, 3]. However, the direct use of the method ca ot solve sigularly perturbed delay partial differetial equatios, if igore the delay parameter, it ca ot always get good result. For this kid of sigularly perturbed delay partial differetial equatios, based o barycetric iterpolatio collocatio method, by Taylor s series expasio, we give a modified BICM to solve them. Two umerical examples are give to demostrate the efficiecy of the preset method. Modified BICM Expadig the delay term u(x, t d) aroud x by Taylor s series expasio, we obtai u(x, t d) u(x, t) δ[ u(x, t)/ t], ad eq. () ca be approximated by the followig sigularly perturbed problem: (, ) uxt (, ) Lu= [ ax ( )] + axuxt ( ) (, ) = f( xt, ),( xt, ) [0,] [0, T] t uxt (, ) = ψ( xt, ),( xt, ) [0,] [ τ,0] u(0, t) = ϕt u(, t) = ϕr The differetial matrix of barycetric iterpolatio is []: (3) () ij j i D = L ( x ), () ij j i ( m ) D ( m ) () ij Dij = m Dii Dij, i j xi xj Dii = Dij j=, j i D = L ( x ) (4) I view of eq. (3), let iterval [0, ] be dispersed as 0 = x < x < < x, =, iterval [0, T] dispersed as 0 = t < t < < t = T, let u, u, u as the values of fuctio u(x) at disperse odes x, x, x,, respectively. The barycetric iterpolatio collocatio is adopted to obtai a approximate solutio of u(x, t) i the form: u( x, t) = L ( xu ) (6) where u i(t) is expressed: j= u() t L () tu i k ik k = j j (5) = (7) By the assumptio give i eqs. (6) ad (7), we ca obtai a matrix equatio i the form LU = F, from eq. (3), where:
THERMAL SCIENCE, Year 07, Vol., No. 4, pp. 595-599 597 Numerical experimet () () L= EI ( D ) ( C I) + A U = [u, u, u ] A = diag[a i] I = I Ä I E = I A F = [f, f, f ] T kj D = [ D ( t )] kj C = [ C ( t)], k, j =,,... I this sectio, two umerical examples are studied to demostrate the accuracy of the preset method. Example. Cosider the followig equatio [4, 5]: (, ) uxt (, ) = e uxt (, ), ( xt, ) (0, ) (0, ] t x t+ uxt (, ) e =,( xt, ) [0,] [,0] t u(0, t) = e, t [0, ] t u(, t) e =, t [0, ] The exact solutio is: u( xt, ) = e T x t+ the error compared with fitted differece method i classical uiform meshes (CUM) ad i fitted piecewise uiform meshes (FPUM) are show i tab.. Table. Compariso of absolute errors for Example Preset method CUM [5] CUM [5] FPUM [5] FPUM [5] N = 64 N = 64 N = 56 N = 64 N = 56 0 0 3.87 0 4 4.505 0 3 6.696 0 4 4.505 0 3 6.696 0 4 0 3.89 0 6.44 0.6 0 3 4.78 0 3 8. 0 4 0 4 3.89 0 8.64 0 3.00 0 3 4.78 0 3 8. 0 4 0 6 3.89 0 0.6 0.07 0 4.78 0 3 8. 0 4 0 8 3.89 0.0 0.607 0 4.78 0 3 8. 0 4 0 0 3.89 0 4.664 0 3.640 0 4.78 0 3 8. 0 4
598 THERMAL SCIENCE, Year 07, Vol., No. 4, pp. 595-599 Example. Cosider the followig equatio [6]: (, ) uxt (, ) = f( xt, ) ( + x) uxt (, ),( xt, ) [0,] [0, ] t x x t+ t+ uxt (, ) ( x) e e = + +,( xt, ) [0,] [ τ,0] t t u(0, t) e e = +, t [0, ] 3 t t u(, t) = e + e, t [0, ] I this example: f( xt, ) = ( x )e ( x + )e the exact solutio is: x x t+ t+ ut ( xt, ) = ( + x) e + e x x t+ t+ the error compared with fitted operator fiite differece method (FODFDM) ad stadard fitted differece method (SFDM) are show i tab. Table. Compariso of absolute errors for Example Preset method FOFDM [6] FOFDM [6] FOFDM [6] SFDM [6] Coclusios ad remarks N = 6 N = 6 N = 3 N = 64 N = 5 0 8 6.558 0 4.30 0 6.370 0 3.40 0.739 0 3 0 0 6.557 0 6.30 0 6.370 0 3.40 0.75 0 5 0 6.557 0 8.30 0 6.370 0 3.40 0.75 0 7 0 4 6.557 0 0.30 0 6.370 0 3.40 0.75 0 9 0 6 6.557 0.30 0 6.370 0 3.40 0.75 0 0 8 6.557 0 4.30 0 6.370 0 3.40 0.75 0 3 I this paper, a modified BICM is proposed for solvig sigularly perturbed delay partial differetial equatios. Numerical results compared with other methods show that the preset method is simple ad accurate, ad it is effective for solvig sigularly perturbed delay partial differetial equatios. It is worthy to ote that our method expads the applicatio of BICM, ad provides a ew ad efficiet method for sigularly perturbed delay partial differetial equatios. All computatios are performed by the MATLABR03A software package.
THERMAL SCIENCE, Year 07, Vol., No. 4, pp. 595-599 599 Ackowledgmet The authors would like to express their thaks to the ukow referees for their careful readig ad helpful commets. This paper is supported by the Natioal Natural Sciece Foudatio of Chia (No. 36037), the Natural Sciece Foudatio of Ier Mogolia (No. 07MS003, 05MS08), ad Numerical aalysis of graduate course costructio project of Ier Mogolia Uiversity of Techology (KC0400). Refereces [] Asari, A. R., A Parameter-Robust Fiite Differece Method for Sigularly Perturbed Delay Parabolic Partial Differetial Equatios, Applied Mathematics ad Computatio, 05 (007),, pp. 55-566 [] Berrut, J. P., Trefethe, L. N., Barycetric Lagrage Iterpolatio, SIAM Review, 46 (004), 3, pp. 50-57 [3] Berrut, J. P., Liear Ratioal Iterpolatio ad Its Applicatios i Approximate ad Boudary Value Problems, Joural of Mathematics, 3 (00),, pp. 997-00 [4] Kumar, S., Kumar, M., High Order Parameter-Uiform Discretizatio for Sigularly Perturbed Parabolic Partial Differetial Equatios with Time Delay, Applied Mathematics ad Computatio, 68 (04), 0, pp. 355-367 [5] Asari, A. R., A Parameter-Robust Fiite Differece Method for Sigularly Perturbed Delay Parabolic Partial Differetial Equatios, Joural of Computatioal ad Applied Mathematics, 05 (007),, pp. 55-566 [6] Bashier, E. B. M., Patidar, K. C. A Novel fitted Operator Fiite Differece Method for a Sigularly Perturbed Delay Parabolic Partial Differetial Equatio, Applied Mathematics ad Computatio, 7 (0), 9, pp. 478-4739 [7] Salkuyeh, D. K., Tavakoli, A., Iterpolated Variatioal Iteratio Method for Iitial Value Problems, Applied Mathematical Modellig, 40 (06), 5, pp. 3979-3990 [8] Siddiqi, S., Iftikhar, M., Variatioal Iteratio Method for the Solutio of Seveth Order Boudary Value Problems Usig He s Polyomials, Joural of the Associatio of Arab Uiversities for Basic ad Applied Scieces, 8 (05),, pp. 60-65 [9] Nazari, G. A., A Modified Homotopy Perturbatio Method Coupled with the Fourier Trasform for Noliear ad Sigular Lae-Emde Equatios, Applied Mathematics Letters, 6 (03), 0, pp. 08-05 Paper submitted: Jue 5, 06 Paper revised: August 4, 06 Paper accepted: September 4, 06 07 Society of Thermal Egieers of Serbia. Published by the Viča Istitute of Nuclear Scieces, Belgrade, Serbia. This is a ope access article distributed uder the CC BY-NC-ND 4.0 terms ad coditios.