Numerical Simulation of One-Dimensional Wave Equation by Non-Polynomial Quintic Spline
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1 IOSR Journal of Matematcs (IOSR-JM) e-issn: , p-issn: X. Volume 14, Issue 6 Ver. I (Nov - Dec 018), PP Numercal Smulaton of One-Dmensonal Wave Equaton by Non-Polynomal Quntc Splne Md. Jon Alam 1, Laek Sazzad Andalla 1 Department of Matematcs, IUBAT-Internatonal Unversty of Busness Agrculture and Tecnology, Uttara, Daka, Banglades. Department of Matematcs, Jaangrnagar Unversty, Savar, Daka, Banglades. Correspondng Autor: Md. Jon Alam Abstract: Ts paper present a numercal algortm for te lnear one-dmensonal wave equaton. In ts metod a fnte dfference approac ad been used to descrbe te tme dervatve wle quntc splne s used as an nterpolaton functon n te space dmenson. We dscuss te accuracy of te metod by expandng te equaton based on Taylor seres and mnmzng te error. Te proposed metod was egt-order accuracy n space and fourt-order accuracy n tme varables. From te computatonal pont of vew, te soluton obtaned by ts metod s n excellent agreement wt tose obtaned by prevous works and also t s effcent to use. Numercal examples are gven to sow te applcablty and effcency of te metod Date of Submsson: Date of acceptance: I. Introducton One dmensonal wave equaton arse n many pyscal and engneerng applcatons suc as contnuum pyscs, mxed models of transonc flows, flud dynamcs, etc. Many autors ave studed te numercal solutons of lnear yperbolc wave equatons by usng varous tecnques. Consder one-dmensonal wave equaton of te form u t = u c, 0 x l, t 0 (1) x wt ntal condton u(x, 0) u x, 0 = f x, = f t 3 t, 0 x l () and boundary condtons u 0, t = p t, t 0 and u l, t = q t, t 0 (3) were c and l are postve fnte real constants and f 1 x, f x, f 3 x, p 1 t, p t, q 1 (t) and q (t) are real contnues functons. Some penomena, wc arse n many felds of scence suc as sold state pyscs, plasma pyscs, flud dynamcs, matematcal bology and cemcal knetcs, can be modeled by partal dfferental equatons. Te wave equaton s of prmary mportance n many pyscal systems suc as electro termal analogy, sgnal formaton, dranng flm, water transfer n sols, mecancs and pyscs, elastcty and etc. Tere are several numercal scemes tat ave been developed for te soluton of wave equaton. In ts paper te formaton of non-polynomal quntc splne as been developed and te consstency relaton obtaned s useful to dscretze wave equaton. We present dscretzaton of te equaton by a fnte dfference approxmaton to obtan te formulaton of proposed metod. Truncaton error and stablty analyss are dscussed. In ts study, we approxmate te functons based on Taylor seres to mnmze te error term and to obtan te class of metods. Numercal experments are conducted to demonstrate te vablty and te effcency of te proposed metod computatonally. II. Formulaton of Tenson Quntc Splne Functon In recent years, many scolars ave used non-polynomal splne for solvng dfferental equatons. Te splne functon s a pecewse polynomal or non-polynomal of degree n satsfyng te contnuty of (n 1)t dervatve. Tenson quntc splne s a non-polynomal functon tat as sx parameters to be determned ence t can satsfy te condtons of two end ponts of te nterval and contnuty of frst, second, trd and fourt dervatves. We ntroduce te set of grd ponts n te nterval [0, l] n space drecton. x =, = l, = 0,1,,, n + 1 n + 1 For eac segment quntc splne p (x) s defned by DOI: / Page
2 p x = a + b x x + c (x x ) + d (x x ) 3 + e e w x x e w x x + f e w x x e w x x, = 0,1,,, n (4) Were a, b, c, d, e and f are te unknown coeffcents to be determned also ω s free parameter. If ω 0 ten p (x) reduces to quntc splne n te nterval [0, l]. To derve te unknown coeffcents, we defne p x = u, p x = u, p x = M, p x = M 4, p x = S, 4 p x = S (5) From Eq. (4) and Eq.(5) we can determne te unknown coeffcents a = u s ω 4, b = u u + S ω 4 + 3ω + S 6ω 1 ω 4 3 M 6 M c = M S ω, d = 1 M 6 M + S ω S ω S e = ω 4 (e θ e θ ) S e θ + e θ ω 4 e θ e θ, f = S ω 4 Were θ = ω and = 0, 1, n Fnally usng te contnuty of frst dervatve at te support ponts for =, 3,.., n 1, we ave u u + u 1 6 M 1 3 M 6 M = S 1 ω 4 6ω eθ + e θ ω 3 e θ e θ + eθ e θ ω 3 (6) and from contnuty of trd dervatves, we ave M M + M 1 = S 1 ω eθ + e θ ω e θ e θ + eθ e θ +S ω + eθ + e θ ω e θ e θ + S 1 ω + ω e θ e θ (7) From Eq. (6) and Eq. (7), after elmnatng S we ave te followng useful relaton for =,3,, n. u + + u 6u + u 1 + u = 0 αm + + βm + γm + βm 1 + αm (8) Were α = θ 4 θ 3 e θ e θ 1 3θ e θ e θ, β = 4θ eθ + e θ θ 3 e θ e θ + eθ + e θ 4 3θ e θ e θ, γ = 6θ 4 + 4eθ + 4e θ 3θ e θ e θ 4 + 4eθ + 4e θ θ 3 e θ e θ Wen ω 0so θ 0, ten (α, β, γ) (1, 6, 66) and te relaton defned by Eq.(8) reduce nto ordnary quntc splne u + + u 6u + u 1 + u = 0 M M + 6M 1 + M (9) III. Numercal Tecnque By usng Eq.(9) for ( + 1)t, ()t and ( 1)t tme level we ave u u +1 6u +1 + u u = 0 αm βm +1 + γm +1 + βm αm ω (10) u + + u 6u + u 1 + u = 0 αm + + βm + γm + βm 1 + αm (11) u u 1 6u 1 + u u 0 αm βm 1 + γm 1 + βm αm tat we wll use tese equatons to dscretze wave equaton. = DOI: / Page (1)
3 Wave Equaton Fnte dfference approxmaton for second order tme dervatve s u tt = u +1 u 1 + u k = u tt + O k As we consder te space dervatve s approxmated by non-polynomal tenson splne (13) u xx = p x, t = M (14) By usng Eq.(13) and Eq.(14), we can develop a new approxmaton for te soluton of wave equaton, so tat te Eq.(1) can be replaced by ηm η M + η M +1 = u +1 u 1 + u c k were 0 η 1 s a free varable (15) Agan we multply Eq.(11) by 1 η and add ts to Eq.(10) and Eq.(1) multpled by η and elmnate M, ten we obtan te followng relaton for wave Eq(1) η u +, 1 + u, η u +, + u, + η u +, u, η u + 1, 1 + u 1, η u + 1, + u 1, 6η u, η u, 6η u, c α u +, + 1 u +, + u +, 1 + u, + 1 u, k + u, 1 40c β u + 1, + 1 u + 1, + u + 1, 1 + u 1, + 1 u 1, k + u 1, 1 40c γ u, + 1 u, + u, 1 = 0, k = 1,,3 =,.. N (16) Error Estmate n Splne Approxmaton To estmate te error for wave equaton we expand Eq.(16) n Taylor seres about u(x, t ) and ten we fnd te optmal value for α, β and γ. Error Estmate for Wave Equaton For wave equaton, we expand Eq.(16) n Taylor seres and replace te dervatves nvolvng t by te relaton + u x t = + u c (17) x+ ten we derve te local truncaton error. Te prncpal part of te local truncaton error of te proposed metod for wave equaton s T = α 10 γ 0 β D x,x U 0, 0 + α 5 β α 10 γ 40 β η k c D x,x,x,x U 0, 0 + α 15 β α 60 β η 4 k c D x,x,x,x,x,x U 0, 0 + α 3600 γ 700 β η k 4 c D x,x,x,x,x,x U 0, 0 + α 5 β D x,x,x,x,x,x,x,x U 0, 0 + (18) If we coose α = 7, β = 76 and γ = 67 n Eq.(18) we obtan a new sceme of order O( k 4 ), 6 3 furtermore by coosng η = 1 we can optmze our sceme, too. 1 Stablty Analyss In ts secton, we dscuss stablty of te proposed metod for te numercal soluton of wave equaton. We assume tat te soluton of Eq. (16) at te grd pont (l, k) s u l = ξ e lθ (19) Were = 1, θ s a real number and ξ s a complex number. DOI: / Page
4 By substtutng Eq.(19) n Eq.(16) we obtan a quadratc equaton as follow Qξ + ξ + φ = 0 (0) For 1D wave equaton we ave Q = cos θ η α 10ck + cos θ 4η β 10ck 6η γ 0ck = cos θ 4η α 5ck + cos θ 4 8η β 5ck 6 + 1η + γ 10ck φ = cos θ η 10ck + cos θ 4η β 10ck 6η γ 0ck Tus we ave Q + φ ξ + Q + + φ = 0. In order to ξ < 1, we must ave < 0 and Q + φ > 0. Obvously Q + φ > 0 for eac η, f η > 1 + equaton. 7 10ck ten < 0, terefore our sceme wll be stable for wave Numercal Example We appled te presented metod to te followng wave equaton. For ts purpose we two examples for wave equaton We appled proposed metod wt α, β, γ = 1, 6, 66 (metod I) wc s ordnary quntc splne of wt order O 6 + k 4 and f we select α, β, γ = 7 6, 76 3, 67 we obtan a new metod wc s of order O 8 + 4k4(metod II). Example 1: We consder Eq.(1) wt c = 1, f x = 0, f 3 x = π cos xπ, p t = sn πt, q x = sn πt. Te exact soluton for te problem s u x, t = π cos πx sn(πt) Absolute Error for Example 1. x t Metod I Metod II Ts problem s solved by dfferent values of step sze n te x-drecton and te tme step sze t = Te soluton by proposed metod are compared wt te exact soluton at te grd ponts and te maxmum absolute errors are tabulated n above table. Te space-tme of te estmated soluton s gven n te fgure. Te maxmum absolute error of ts example by metod II s and by metod I s DOI: / Page
5 Example. Now we consder Eq.(1) wt c = 1, f x = cos πx, f 3 x = 0, p t = cos πt, q x = cos πt. Te exact soluton for te problem s Absolute Error for Example. u x, t = 1 cos π x + t + 1 cos π x t x t Metod I Metod II Ts problem s solved by dfferent values of step sze n te x-drecton and te tme step sze t = Te soluton by proposed metod are compared wt te exact soluton at te grd ponts and te maxmum absolute errors are tabulated n above table. Te space-tme of te estmated soluton s gven n te fgure. Te maxmum absolute error of ts example by metod II s and by metod I s Reference [1]. Saadatmand and M. Degan, Numercal Soluton of te One-Dmensonal Wave Equaton wt An ntegral Condton, Numercal Metods for Partal Dfferental Equatons, Vol, 3, no., pp. 8-9, 007. []. J. Rasdna, R. Jallan, and V. Kazem, Splne metods for te solutons of yperbolc equatons, Appled Matematcs and Computaton, vol. 190, no. 1, pp , 007. [3]. R. K. Moanty and V. Gopal, Hg accuracy cubc splne fnte dfference approxmaton for te soluton of one-space dmensonal non- lnear wave equatons, Appled Matematcs and Computaton, vol. 18, no. 8, pp , 011. [4]. Ryo, Fast decay of solutons for lnear wave equatons wt dsspaton localzed near nfnty n an exteror doman, J. Dfferental Equatons, 188 () (003) [5]. Rasdna J. and Mosenyzadea, Numercal Soluton of One-Dmensonal Heat and Wave Equaton by Non-Polynomal Quntc Splne, Internatonal Journal of Matematcal Modellng & Computatons Vol. 05, No. 04, ( 015), [6]. Won Y. Yang, Wenwu Cao, Tae-Sang Cung, Jon Morrs, Appled Numercal metods usng MATLAB, Wley student Edton, 013. [7]. M. Degan, On te Soluton of An ntal-boundary Value Problem ta Combnes Neumann and Integral Condton for te Wave Equaton, Numercal Metods for Partal Dfferental Equatons, Vol. 1, no. 1, pp:4-40 [8]. P. Karageorgs and Walter A. Strauss, Instablty of steady states for nonlnear wave and eat equatons, J. Dfferental Equatons, 41 (1) (007) Md. Jon Alam" Numercal Smulaton of One-Dmensonal Wave Equaton by Non-Polynomal Quntc Splne." IOSR Journal of Matematcs (IOSR-JM) 14.6 (018): DOI: / Page
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