Solving the Linear Homogeneous One-Dimensional Wave Equation Using the Adomian Decomposition Method

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1 Applied Mahemaical Sciences, Vol. 13, 019, no. 5, 39-5 HIKARI Ld, hps://doi.org/ /ams Solving he Linear Homogeneous One-Dimensional Wave Equaion Using he Adomian Decomposiion Mehod Chrisian Kasumo Deparmen of Science and Mahemaics School of Science, Engineering and Technology Mulungushi Universiy, P.O. Box 80415, Kabwe, Zambia This aricle is disribued under he Creaive Commons by-nc-nd Aribuion License. Copyrigh c 019 Hikari Ld. Absrac The use of he sandard Adomian decomposiion mehod for obaining he approximae soluion of he linear homogeneous one-dimensional wave equaion is invesigaed. The resuls are compared wih he exac soluions obained using d Alember s formula. The resuls obained show ha he mehod has a high degree of efficiency, validiy and accuracy as i leads o he exac soluion. Mahemaics Subjec Classificaion: 35J05, 35L05, Keywords: Wave equaion, Adomian decomposiion mehod, Iniial value problem, Homogeneous parial differenial equaion, D Alember s formula 1 Inroducion The wave equaion is a non-rivial parial differenial equaion ha seems o be everywhere in several places a he same ime. Wave propagaion represens one of he mos common physical phenomena experienced in everyday life. Waves occur mos frequenly hrough sigh and hearing, as well as hrough elecommunicaion, radar, medical imaging, ec. Indusrial applicaions range from aero acousics o music (acousic waves), from oil prospecing o non-desrucive esing (elasic waves), from opics o sealh echnology (elecromagneic waves) and sabilizaion of ships and offshore plaforms [1].

2 40 Chrisian Kasumo Waves possess several ineresing characerisics. Typically, a wave is a disurbance ha propagaes in a medium. However, wo imporan kinds of waves ha have been sudied exensively are ligh waves (special relaiviy) and probabiliy waves (quanum mechanics), neiher of which propagaes in a convenional medium. Waves undergo inerference and diffracion (e.g., bending around corners). The engine for hese phenomena is superposiion. Furhermore, waves end o be spread ou in space or in he medium in which hey propagae. Waves saisfy he wave equaion. Being a parial differenial equaion, he wave equaion is a somewha non-inuiive mahemaical absracion and encapsulaes many properies of waves in a very convenien form. In his paper, we limi ourselves o linear waves in one-dimensional space. This is a simple prooype example of all oher kinds of wave propagaion models. There is a wide variey of numerical schemes for approximaing he soluion of he linear one-dimensional wave equaion. The uniqueness of he soluion of a wave equaion is obained by imposing addiional condiions, viz. iniial and/or boundary condiions. Recenly, he Adomian decomposiion mehod (ADM) [] has proved o be an effecive mahemaical ool for obaining approximae and analyical soluions o many ypes of ordinary and parial differenial equaions and yields resuls ha are exac or very close o exac in many siuaions. The ADM is a well-known sysemaic mehod for pracical soluion of linear or non-linear and deerminisic or sochasic operaor equaions, including ordinary differenial equaions (ODEs), parial differenial equaions (PDEs), inegral equaions, inegro-differenial equaions, ec. (Duan e al. [3]). The efficiency of he ADM for soluion of hese differen kinds of equaions is well known. The ADM is a grea mehod in ha i provides he soluion as an infinie series in which each erm can be easily deermined. The mehod accuraely compues a rapidly convergen series soluion. Oher advanages of he ADM have been poined ou in he lieraure. I has been argued ha he ADM mainains a high degree of accuracy of he numerical soluions while a he same ime reducing he amoun of compuaional work compared o radiional approaches (Bulu e al. [4]). Oher advanages include is abiliy o solve non-linear problems wihou linearizaion, he wide applicabiliy o several ypes of problems and scienific fields, and he developmen of a reliable, analyic soluion. Addiionally, his mehod does no linearize he problem, nor does i use assumpions of weak non-lineariy [5]. For his reason he ADM can handle fairly general non-lineariies and generaes soluions ha may be more realisiic han hose achieved by simplifying he model...o achieve condiions required for oher echniques. These advanages provide jusificaion for he ADM s wide-ranging applicabiliy in such fields as engineering, chemisry, biology and physics [6, 7, 8, 9]. I has recenly been poined ou ha he mehod has proved o be highly ap-

3 Solving he linear homogeneous one-dimensional wave equaion 41 plicable o such diverse areas as non-linear opics, paricle ranspor, mass and/or hea ransfer and chaos heory [10]. Over agains hese advanages, he mehod also suffers from cerain disadvanages. One major concern lies in he region and rae of convergence of he series produced by he ADM. While he series can be rapidly convergen in very small regions, i has a very slow convergence rae in he wider regions [11]. Also, because of runcaion he series soluion is inaccurae in ha region and his grealy resrics he applicaion area of he mehod. The res of his paper is organized as follows: Secion derives he model o be solved and oulines he proposed mehods of soluion. In Secion 3 some resuls are presened and discussed based on seleced examples and he paper closes wih conclusions and possible exensions in Secion 4. Mehods and Maerials If u = φ(x v) describes a wave, he quesion ha arises is, Wha ype of equaion does i saisfy? We can wrie u = φ(α) = φ(x v), i.e., α = x v. We noe ha u = dφ α = dφ α (since = 1). So x dα x dα x Similarly, u = dφ α dα v d φ α = dα v d φ and so dα = v dφ dα 1 u v From (1) and (), we have ha u x = d φ (1) dα α (since = v). Thus, u = v ( dφ dα) = = d φ dα () u x = 1 u (3) v or, in compac form, u xx = v u, which is he linear second-order homogeneous wave equaion ha describes he propagaion of waves wih respec o space and ime. Here, v > 0 is a consan represening he wave velociy which is deermined by he physical properies of he maerial hrough which he wave propagaes. The iniial condiions associaed wih his wave equaion are u(0, ) = f(), u x (0, ) = g(). (3) admis soluions of he form u(x, ) = φ(x v) + ψ(x + v) where φ and ψ are arbirary funcions. No maer he shape of he funcion φ, he wave u = φ(x v) saisfies he wave equaion, since φ x = φ, φ x = φ, u x = 1 u v φ = vφ, = φ φ 1 v (v φ ) = 0 = ( v) φ = v φ. Thus,

4 4 Chrisian Kasumo Similarly, any funcion of he form u = φ(x + v) also saisfies he wave equaion, i.e., φ = x φ, φ = φ φ, = x vφ, φ = v φ, giving, again, u x = 1 u v = φ 1 v (v φ ) = 0 I follows ha any funcion of he form u = φ(x v)+ψ(x+v) is a soluion of he wave equaion. We can wrie (3) as u = v u xx. In he sandard form, his is L (u(x, )) = L x ( v u(x, ) ) (4) where he differenial operaors are defined, respecively, as L = and L x =. Suppose he inverse operaor L 1 x exiss. Then L 1 is he wo-fold inegral operaor from 0 o defined by L 1 = 0 0 ( )dsds Applying his inverse operaor on boh sides of (4) gives L 1 [ ( (L (u(x, ))) = L 1 Lx v u(x, ) )] Since v is a consan, i can be facored ou and he above equaion can be wrien as L 1 (L (u(x, ))) = v L 1 [L x (u(x, ))] (5) From (5), i follows, by applicaion of he iniial condiions, ha u(x, ) = f() + xg() + v L 1 [L x (u(x, ))] (6) where he unknown funcion u is decomposed ino a sum of componens defined by he decomposiion series u(x, ) = u n (x, ) n=0 wih u 0 idenified as u(x, 0) and he componens u n (x, ) obained from he recursive formula n=0 u n (x, ) = f() + xg() + v L 1 [ ( )] L x u n (x, ) n=0 (7)

5 Solving he linear homogeneous one-dimensional wave equaion 43 or i.e., u 0 (x, ) = f() + xg() u n+1 (x, ) = v L 1 [L x (u n (x, ))], n 0 u 0 = f() + xg() u 1 = v L 1 [L x (u 0 )] u = v L 1 [L x (u 1 )] (8). u n+1 = v L 1 [L x (u n )] If he PDE conains a non-linear erm, his erm is represened as Nu and decomposed ino a series Nu = where he A n, depending on u 0, u 1, u,..., u n, are called Adomian polynomials obained, for he non-lineariy Nu = β(u), by he formula [ ( A n = 1 )] n! L λ β u i λ i, n = 0, 1,,... i=0 n=0 where λ is a grouping parameer of convenience and L λ = n. In his way, λ n he componens u 0, u 1, u,... are idenified and he series soluion o he wave equaion is compleely deermined. The exac soluion may be deermined using he approximaion A n λ=0 u(x, ) = lim n Φ n where Φ n = n 1 k=0 u k. I is imporan o noe ha he recursive relaionship (7) is consruced on he basis ha he zeroh componen u 0 (x, ) is defined by all he erms ha arise from he iniial condiions and from inegraing he source erm. However, since we are dealing wih a homogeneous equaion, he source erm is zero. The remaining componens u n (x, ), n 1 are compleely deermined such ha each erm is compued by using he immediaely preceding erm. Accordingly, considering he firs few erms only, he recursive relaion (8) gives u 0 (x, ), u 1 (x, ), u (x, ),.... Remark.1. Adomian and Rach [1] and Wazwaz [13] invesigaed he phenomenon of self-cancelling noise erms where some erms in he series vanish

6 44 Chrisian Kasumo in he limi. These noise erms do no appear for homogeneous equaions bu only for specific ypes of non-homogeneous equaions. Furhermore, i has formally been shown ha if some erms in u 0 are cancelled by idenical erms in u 1 wih opposie signs, even hough u 1 includes addiional erms, he remaining non-cancelled erms in u 0 consiue he exac soluion of he given equaion [14, 15]. However, i is necessary and essenial o verify ha he remaining non-canceled erms saisfy he given equaion. Since in he examples given in he nex secion of his paper we will be comparing he approximae soluion from he ADM wih he exac soluion obained by d Alember s formula, he following heorem is perinen. Theorem.. (d Alember s Formula) For he Cauchy iniial value problem u = v u xx, u(x, 0) = f(x), u (x, 0) = g(x), x R, > 0, v > 0 (9) where f, g C (, ) are given by he iniial condiions, here exiss a unique C (R R)-soluion given by d Alember s formula u(x, ) = φ(x v)+ψ(x+v) = 1 [f(x v)+f(x+v)]+ 1 v φ, ψ being arbirary C -funcions. x+v x v g(s)ds, (10) Proof. Assume ha here is a soluion u(x, ) of he general linear homogeneous wave equaion (9). We sar by wriing he general soluion of he wave equaion as u(x, ) = φ(x v) + ψ(x + v). (11) Now we choose φ and ψ in such a way ha he iniial condiions are saisfied. Subsiuing φ(x v) + ψ(x + v) for u in he iniial condiions resuls in By differeniaing (1), we obain u(x, 0) = φ(x) + ψ(x) = f(x) (1) u (x, 0) = vφ (x) + vψ (x) = g(x) (13) φ (x) + ψ (x) = f (x). (14) Now (13) and (14) form an algebraic sysem of wo equaions in wo unknowns φ (x) and ψ (x) since f, g and v are all known from he given PDE and he iniial condiions. Muliplying (14) by v gives vφ (x) + vψ (x) = vf (x). (15)

7 Solving he linear homogeneous one-dimensional wave equaion 45 Adding (13) and (15) eliminaes φ (x) so ha we have vψ (x) = vf (x) + g(x). (16) Now subracing (13) from (15) eliminaes ψ (x) o give vφ (x) = vf (x) g(x). (17) Dividing (16) and (17) by v gives, respecively, which, upon inegraion, become Therefore, ψ (x) = 1 f (x) + 1 v g(x) φ (x) = 1 f (x) 1 v g(x) ψ(x) = 1 f(x) + 1 v φ(x) = 1 f(x) 1 v u(x, ) = φ(x v) + ψ(x + v) = 1 1 f(x v) v x v = 1 1 [f(x v) + f(x + v)] + v = 1 1 [f(x v) + f(x + v)] + v 0 x 0 x 0 g(s)ds g(s)ds g(s)ds f(x + v) + v 0 x v x+v x v g(s)ds + 1 v g(s)ds x+v 0 x+v 0 g(s)ds g(s)ds Thus, each C -soluion of he Cauchy IVP is given by d Alember s formula. On he oher hand, he funcion u(x, ) defined by he RHS of (10) is a soluion of he given IVP. 3 Examples In his secion we give hree numerical examples of linear homogeneous wave equaions and show heir soluion by he ADM. All he compuaions associaed wih hese examples were performed using a Samsung Series 3 PC wih an Inel Celeron CPU 847 a 1.10GHz and 6.0GB inernal memory. The figures were consruced using MATLAB R016a.

8 46 Chrisian Kasumo EXAMPLE 1. Consider he wave equaion u v u xx = 0 wih iniial condiions u(x, 0) = f(x) = sin x, u (x, 0) = g(x) = x. Applying d Alember s formula (Theorem.) o he given equaion we have he soluion u(x, ) = 1 [sin(x v) + sin(x + v)] + 1 v x+v x v s ds By he rigonomeric ideniies sin(a B) = sin A cos B cos A sin B and sin(a + B) = sin A cos B + cos A sin B, we have sin(x v) + sin(x + v) = sin x cos(v) cos x sin(v) + sin x cos(v) + cos x sin(v) i.e., = sin x cos(v) u(x, ) = 1 [ ] 1 s 3 x+v [ sin x cos(v)] + v 3 x v = sin x cos(v) + 1 [ (x + v) 3 (x v) 3] 6v = sin x cos(v) + x v 3 This is he exac soluion agains which he soluion obained by he ADM will be compared. The recursive relaionship of he ADM is given by i.e., u 0 = f(x) + g(x) [ ( )] u n+1 = v L 1 L x u n n=0 = v L 1 [( ) u n n=0 xx ] u 0 = sin x + x u 1 = v L 1 u = v L 1 [ ( [L x (u 0 )] = v L )] 1 Lx sin x + x = v sin x + v 3 )] 3 [L x (u 1 )] = v L 1 [L x ( v sin x + v 3 = v sin x ( )] v [L 4 4 x 4 sin x = v sin x u 3 = v L 1 [L x (u )] = v L 1 and so on. Adding he approximans u 0, u 1, u, u 3,... gives he approximae

9 Solving he linear homogeneous one-dimensional wave equaion 47 soluion as u = u0 + u1 + u + u3 + v v 3 v 4 4 v 6 6 = sin x + x sin x + + sin x sin x v 3 (v) (v)4 (v) x + = sin x 1! 4! 6! 3 1 = sin x cos(v) + x + v 3 3 This is, in fac, he exac soluion obained earlier by d Alember s formula. The resuls for v = 3, 0 < x < 1 and 0 < < 3 are given in Figure 1. (a) Surface u(x, ) = sin x cos(v) + x + 31 v 3 (b) Conour diagram Figure 1: Approximae and exac soluion and conour diagram for given IVP EXAMPLE. Consider he following one-dimensional wave equaion: u = 4uxx, u(0, ) = u(1, ) = 0, u(x, 0) = f (x) = 0, u (x, 0) = g(x) = π sin(πx). Here v = and by d Alember s formula we have he exac soluion 1 u(x, ) = 4 Z x+ x 1 π sin(πs)ds = [cos(πs)]x+ x 1 = [cos(π(x + )) cos(π(x ))] 1 = [cos(πx) cos(π) sin(πx) sin(π) cos(πx) cos(π) sin(πx) sin(π)] = sin(πx) sin(π) Now applying he concep of ADM o he given wave equaion, we have

10 48 Chrisian Kasumo he recursive relaionship u0 = g(x) = π sin(πx) u1 = 4L 1 [Lx (u0 )] = 4L [Lx (π sin(πx))] = (π) sin(πx) u = 4L 1 Lx (π)3 sin(πx) = (π)5 sin(πx) [Lx (u1 )] = 4L u3 = 4L [Lx (u )] = 4L Lx (π) sin(πx) = (π)7 sin(πx) and so on. Adding he approximans gives he soluion u = u0 + u1 + u + u3 + 8 = π sin(πx) (π)3 sin(πx) = π sin(πx) (π)3 sin(πx) + 6 = sin(πx) sin(π) 3 (π)5 sin(πx) 10 3 (π)5 sin(πx) (π)7 sin(πx) (π)7 sin(πx) which is he exac soluion obained from d Alember s formula. Figure gives he resuls for 0 < x < 1 and 0 < < 3. (a) Surface u(x, ) = sin(πx) sin(π) (b) Conour diagram Figure : Approximae and exac soluion and conour diagram or given IVP EXAMPLE 3. Consider he Dirichle problem u = uxx for 0 < x < π, u(x, 0) = sin 3x, u (x, 0) = 0, u(0, ) = u(π, ) = 0. To find he soluion,

11 Solving he linear homogeneous one-dimensional wave equaion 49 we firs use d Alember s formula which gives u(x, ) = 1 [f(x ) + f(x + )] = 1 [sin 3(x ) + sin 3(x + )] = 1 [sin 3x cos 3 cos 3x sin 3 + sin 3x cos 3 + cos 3x sin 3] = sin 3x cos 3 By he ADM, we have, recursively, he approximans u 0 = sin 3x u 1 = L 1 u = L 1 [L x (u 0 )] = L 1 [L x (u 1 )] = L 1 u 3 = L 1 [L x (u )] = L 1 [L x (sin 3x)] = 9 sin 3x ( [L x 9 )] sin 3x [ ( )] 81 L x 4 4 sin 3x = sin 3x = sin 3x and so on. The sum of he approximans u 0, u 1, u,... is he desired approximae soluion, i.e., u = u 0 + u 1 + u + u 3 + = sin 3x 9 sin 3x sin 3x sin 3x + = sin 3x (3) sin 3x + (3)4 sin 3x (3)6 sin 3x +! 4! 6! ] = sin 3x [1 (3) + (3)4 (3)6 +! 4! 6! = sin 3x cos 3 which is simply he exac soluion. The resuls for 0 < x < π and 0 < < are shown in Figure 3. 4 Conclusion In his paper we proposed he applicaion of he Adomian decomposiion mehod for solving he linear homogeneous one-dimensional wave equaion. This is a semi-analyical mehod ha lends iself well o he soluion of ordinary and parial differenial equaions wih boh linear and non-linear powers. Illusraive numerical examples were given and from he resuls i is concluded ha he ADM leads o boh exac and approximae numerical soluions for

12 50 Chrisian Kasumo u(x,) (a) Surface u(x, ) = sin 3x cos 3 (b) Conour diagram Figure 3: Approximae and exac soluion and conour diagram for given IVP he wave equaion. The work of his paper can be improved by (1) using one of he many modificaions of he ADM, e.g., [15, 16, 17], insead of he sandard ADM. () applying he ADM o he soluion of Volerra inegro-differenial and inegral equaions ha frequenly occur in insurance mahemaics; (3) applying he mehod o wave equaions wih a source erm; (4) applying he ADM o parial differenial equaions wih non-linear erms. Acknowledgemens. The auhor wishes o hank Mulungushi Universiy for funding his research and he anonymous referees for heir valuable commens which significanly improved he paper. References [1] P.M. Shearer, Inroducion o seismology: he wave equaion and body waves, Cambridge Universiy Press, Cambridge, 009. [] G. Adomian, Solving fronier problems of physics: he decomposiion mehod, Springer, New York, hps://doi.org/ / [3] J.-S. Duan, R. Rach, D. Băleanu and A.-M. Wazwaz, A review of he Adomian decomposiion mehod and is applicaions o fracional differenial equaions, Commun. Frac. Calc., 3() (006), [4] H. Bulu, M. Ergü, V. Asil and R.H. Bokor, Numerical soluion of a viscous incompressible flow problem hrough an orifice by Adomian decomposiion mehod, Appl. Mah. Compu., 153 (004), hps://doi.org/ /s (03)00667-

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14 5 Chrisian Kasumo [16] J. Biazar and K. Hosseini, A modified Adomian decomposiion mehod for singular iniial value Emden-Fowler ype equaions, In. J. Appl. Mah. Research, 5(1) (016), hps://doi.org/ /ijamr.v5i [17] R. Rach, G. Adomian and R.E. Meyers, A modified decomposiion, Compu. Mah. Appl., 3 (199), hps://doi.org/ / (9) Received: February 7, 019; Published: March 7, 019