Research Article Cubic B-Spline Collocation Method for One-Dimensional Heat and Advection-Diffusion Equations

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1 Appled Mathematcs Volume 22, Artcle ID 4587, 8 pages do:.55/22/4587 Research Artcle Cubc B-Splne Collocaton Method for One-Dmensonal Heat and Advecton-Dffuson Equatons Joan Goh, Ahmad Abd. Majd, and Ahmad Izan Md. Ismal Pusat Pengajan Sans Matematk, Unverst Sans Malaysa, 8 Pulau Pnang, Malaysa Correspondence should be addressed to Joan Goh, joangoh.usm@gmal.com Receved 2 January 22; Accepted 9 March 22 Academc Edtor: Ram N. Mohapatra Copyrght q 22 Joan Goh et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Numercal solutons of one-dmensonal heat and advecton-dffuson equatons are obtaned by collocaton method based on cubc B-splne. Usual fnte dfference scheme s used for tme and space ntegratons. Cubc B-splne s appled as nterpolaton functon. The stablty analyss of the scheme s examned by the Von Neumann approach. The effcency of the method s llustrated by some test problems. The numercal results are found to be n good agreement wth the exact soluton.. Introducton The combnaton of advecton and dffuson s mportant for mass transport n fluds. It s well known that the volumetrc concentraton of a pollutant, ux, t, atapontx a x b n a one-dmensonal movng flud wth a constant speed β and dffuson coeffcent α n x drecton at tme t t s gven by the one-dmensonal advecton-dffuson equaton, whch s n the form u t βu x αu xx, a x b, t,. subject to the ntal condton ux, φx, x a, b,.2

2 2 Appled Mathematcs and the boundary condtons ua, t g t,.3a ub, t g t, t, T,.3b where g and g are assumed to be smooth functons. It should be noted that, when β, the advecton-dffuson equaton wll be reduced to the one-dmensonal heat equaton n the case of thermal dffuson. Advecton-dffuson equaton arses very frequently n transferrng mass, heat, energy, and vortcty n chemstry and engneerng. Thus, t has been of nterest to many authors. A thrd-degree B-splne functon has been used by Caglar et al. for solvng one-dmensonal heat equaton wth a nonlocal ntal condton. Mohebb and Dehghan 2 have presented a fourth-order compact fnte dfference approxmaton and cubc C -splne collocaton method for the soluton wth fourth-order accuracy n both space and tme varables, Oh 4,k 4.In3, Dag and Saka concluded that collocaton scheme s easy to mplement compared to other numercal methods wth gvng a better result. In ths paper, a combnaton of fnte dfference approach and cubc B-splne method would be consdered to solve the one-dmensonal heat and advecton-dffuson equaton. Forward fnte dfference approach would be used for dscretzng the dervatve of tme, whle cubc B-splne would be appled to nterpolate the solutons at tme t. Von Neumann approach would be used to prove the uncondtonally stable property of the method. Fnally, the approxmated solutons and the numercal errors would be presented to demonstrate the effcency of the method. 2. Collocaton Method In ths paper, cubc B-splnes are used to construct the numercal solutons to solve the problems. Consder a partton of a, b that s equally dvded by knots x nto n subnterval x,x, where,,...,n such that a x <x < <x n b. Hence, an approxmaton Ux, t to the exact soluton ux, t based on collocaton approach can be expressed as 4 n Ux, t C tb 3, x, 3 2. where C t are tme-dependent quanttes to be determned and B 3, x are thrd-degree B- splne functons whch are defned by the relatonshp 5 x x 3, B 3, x h 3 3h 2 x x 3hx x 2 3x x 3, 6h 3 h 3 3h 2 x 3 x 3hx 3 x 2 3x 3 x 3, x 4 x 3, x x,x, x x,x 2, x x 2,x 3, x x 3,x 4, 2.2

3 Appled Mathematcs 3 Table : Values of B, B,andB. x x x 2 x 3 x 4 4 B B B 2 h 2 h 2 h 2 where h b a/n. The approxmaton U k at the pont x,t k over the subnterval x,x can be smplfed nto U k C k j B 3,jx, j where,,...,n. To obtan the approxmatons of the solutons, the values of B 3, x and ts dervatves at the knots are needed. Snce the values vansh at all other knots, they are omtted from Table. The approxmatons of the solutons of. at t j th tme level can be consdered by 6: U t k θf k θf k, 2.4 where f k βu x k αu xx k and the superscrpts k and k are successve tme levels, k,, 2,... Now, dscretzng the tme dervatve by a frst-order accurate forward dfference scheme and rearrangng the equaton, we obtan U k θδtf k U k θδtf k, 2.5 where Δt s the tme step. Note that the system becomes an explct scheme when θ, a fully mplct scheme when θ, and a mxed scheme of Crank-Ncolson when θ.5 6. Here, Crank-Ncolson approach s used. Hence, 2.5 takes the form U k.5δtf k U k.5δtf k 2.6 for,,...,nat each level of tme. Therefore, a lnear system of order n s obtaned wth n 3 unknowns C k C k 3,Ck 2,...,Ck n at the level tme t t k. To solve the system, two addtonal lnear equatons are needed. Thus, 2.3 s appled to the boundary condtons.3a-.3b to obtan U k g t k, 2.7a U k n g t k. 2.7b

4 4 Appled Mathematcs Equatons 2.6, 2.7a-2.7b lead to a n 3 n 3 trdagonal matrx system, whch can be solved by the Thomas algorthm. Once the ntal vector C has been calculated from the ntal condtons 7, the approxmaton soluton U k at each level of tme t k can be determned by the vector C k whch s found by solvng the recurrence relaton repeatedly. The ntal vector C can be obtaned from the ntal condton and boundary values of the dervatves of the ntal condton as the followng expressons 6: U x φ x,, 2 U φx,,,...,n, 3 U x φ x, n. Ths yelds a n 3 n 3 matrx system where the soluton can be found by Thomas algorthms. 3. Stablty Analyss Von Neumann stablty method s appled for analyzng the stablty of the proposed scheme. Ths type of stablty analyss had been used by many researchers 3, 8. Consder the tral soluton one Fourer mode out of the full soluton at a gven pont x m C k m δ k exp ( ηmh ), 3. where andη s the mode number. By substtutng 2.3 nto 2.5 and rearrangng the equaton, t leads to p C k m 3 p 2C k m 2 p 3C k m p 4C k m 3 p 5C k m 2 p 6C k m, 3.2 where p 6 θδtβ 2θΔtα p p 3 6 θδtβ p 4 6 p p 6 6 θδtα θδtα θδtβ 2 θδtα θδtβ θδtα θδtα h Insertng the tral soluton 3. nto 3.2 and smplfyng the equaton gve δ A B C D, 3.4

5 Appled Mathematcs 5 where A ( ) 2 θδtα( ) 2 cos ηh cos ηh, 3 h 2 θδtβ B sn ηh, h C ( ) 2θΔtα( ) 2 cos ηh cos ηh, 3 h 2 D θδtβ h sn ηh. 3.5 If the amplfcaton factor δ, then the proposed scheme s stable, or else the approxmatons grow n ampltude and become unstable. As θ.5 s used n the proposed scheme, thus substtute the θ value nto 3.4 and after some algebrac manpulaton, t can be notced that a 2 b 2 c 2 d 2 or δ 2 a2 b 2 c 2 d Thus, ths had been proved that the presented numercal scheme for the advecton-dffuson equaton s uncondtonally stable. 4. Numercal Results 4.. Problem Suppose the heat equaton s as follows : u t u xx, <x<, t >, 4. wth ntal and boundary condtons ux, snπx, u,t u,t. 4.2 The exact soluton s known to be ux, t exp π 2 t snπx. Ths problem s tested by dfferent values of h and Δt to show the capablty of the presented method for solvng one-dmensonal heat equaton. The fnal tme s chosen as T. The maxmum absolute errors of the method are compared wth those obtaned by Crank-Ncolson CN scheme and compact boundary value method CBVM n. The numercal errors are presented n Table 2. Although the fourth-order compact boundary value method gves a much more better soluton, the present method s stll well compared wth the Crank-Ncolson scheme.

6 6 Appled Mathematcs Table 2: Maxmum absolute error obtaned for problem. h Δt CN CBVM Present method / / / / / x.5 t.5 2 Fgure : Spatal-tme approxmatons for problem 2 wth h.5 and Δt.5h Problem 2 Consder the advecton-dffuson equaton n. wth β, α., as follows 2: u t u x.u xx, <x<, t >, 4.3 where the ntal condton s gven by [ ( π ) ( π )] ux, exp5x cos 2 x.25 sn 2 x 4.4 and the exact soluton ( ( ux, t exp 5 x t )) ( ) [ exp π2 ( π ) ( π )] 2 4 t cos 2 x.25 sn 2 x. 4.5 The boundary condtons at x andx can be obtaned from the exact soluton. Table 3 shows the absolute errors of the approxmatons at the grd ponts when T 2. It can be notced that the present method s comparable wth cubc C -splne collocaton method. The approxmatons of the solutons over a tme perod t, 2 along x s depcted n Fgure.

7 Appled Mathematcs 7 Table 3: Absolute error obtaned wth Δt at T 2 for problem 2. Grd pont h.2 h. C -splne 2 Present method C -splne 2 Present method Conclusons A numercal method based on collocaton of cubc B-splne had been descrbed n the prevous secton for solvng one-dmensonal heat and advecton-dffuson equatons. A fnte dfference scheme had been used for dscretzng tme dervatves and cubc B-splne for nterpolatng the solutons at each tme level. From the test problems, the obtaned results show that the presented method s capable for solvng one-dmensonal heat and advectondffuson equatons accurately wth a promsed stablty. Acknowledgment The authors would lke to acknowledge wth thanks the fnancal support from Malaysan Government n the form of Fundamental Research Grant Scheme FRGS of number 23/PMATHS/675. References H. Caglar, M. Ozer, and N. Caglar, The numercal soluton of the one-dmensonal heat equaton by usng thrd degree B-splne functons, Chaos, Soltons & Fractals, vol. 38, no. 4, pp. 97 2, A. Mohebb and M. Dehghan, Hgh-order compact soluton of the one-dmensonal heat and advecton-dffuson equatons, Appled Mathematcal Modellng, vol. 34, no., pp , 2. 3 I. Dag and B. Saka, A cubc B-splne collocaton method for the EW equaton, Mathematcal & Computatonal Applcatons, vol. 9, no. 3, pp , P. M. Prenter, Splnes and Varatonal Methods, Wley Classcs Lbrary, John Wley & Sons, New York, NY, USA, C. de Boor, A Practcal Gude to Splnes, vol. 27 of Appled Mathematcal Scences, Sprnger, New York, NY, USA, I. Dag, D. Irk, and B. Saka, A numercal soluton of the Burgers equaton usng cubc B-splnes, Appled Mathematcs and Computaton, vol. 63, no., pp. 99 2, J. Goh, A. A. Majd, and A. I. M. Ismal, A numercal soluton of one-dmensonal wave equaton usng cubc B-splnes, n Proceedngs of the Natonal Conference of Applcaton Scence and Mathematcs (SKASM ), pp. 7 22, December 2. 8 H. Fazal, I. Sraj, and I. A. Trmz, A numercal technque for soluton of the MRLW equaton usng quartc B-splnes, Appled Mathematcal Modellng, vol. 34, no. 2, pp , 2. 9 S. M. Hassan and D. G. Alamery, B-splnes collocaton algorthms for solvng numercally the MRLW equaton, Internatonal Nonlnear Scence, vol. 8, no. 2, pp. 3 4, 29.

8 8 Appled Mathematcs A. K. Khalfa, K. R. Raslan, and H. M. Alzubad, A collocaton method wth cubc B-splnes for solvng the MRLW equaton, Computatonal and Appled Mathematcs, vol. 22, no. 2, pp , 28. H. Sun and J. Zhang, A hgh-order compact boundary value method for solvng one-dmensonal heat equatons, Numercal Methods for Partal Dfferental Equatons, vol. 9, no. 6, pp , 23.

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