Application of the Adomian Decomposition Method (ADM) and the SOME BLAISE ABBO (SBA) method to solving the diffusion-reaction equations

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1 Advaces i Theoreical ad Alied Mahemaics ISSN Volume 9, Number (4), Research Idia Publicaios h:// Alicaio of he Adomia Decomosiio Mehod (ADM) ad he SOME BLAISE ABBO (SBA) mehod o solvig he diffusio-reacio equaios Joseh BONAZEBI YINDOULA*, Gabriel BISSANGA*, PARE Youssouf**, Fracis BASSONO** ad Blaise SOME**. *Uiversiy Marie NGOUABI Brazzaville Cogo **Uiversiy of Ouagadougou Buria Faso Absrac I his aer, he SBA mehod is used o cosruc he soluio of he diffusio-reacio equaio. AMS Subjec Classificaio: 47H4; 34G; 47J5; 65J5. Keywords: Adomia decomosiio mehod (ADM), SBA mehod, diffusioreacio. Iroducio I [], we cosruced he soluio of he covecio-diffusio-dissiaio equaio, here we examie he diffusio-reacio equaio. We use he SOME BLAISE ABBO (SBA) mehod o ge he exac soluio. The diffusio-reacio equaio The geeral heory of SBA mehod ca be foud i [7], [8]. The liear diffusio-reacio equaio Le s cosider he followig mixed roblem: u x, u x, u x, x x u x, x : u, x u, x ()

2 98 Joseh BONAZEBI YINDOULA e al wih x T, ; ;. We are goig o solve he roblem usig he SBA mehod. From () we have, o oe had: u x, s u x, u x, u x, s xs ds x () u x, s x x u x, s ds x ad, o he oher had: u, u, u z, u z, z dz x x (3) (3) u z, u z, dz (4) (4) u z, u z, dz (5) From () ad (5), we obai: u x, s u x, x x u x, s ds x u z, u z, dz x x u x, s ds N u x, Where u x, s u z, N u x, ds u z, dz x (7) Accordig o he SBA mehod, we suose ha he soluio of () has he followig form: u x, lim u x, (8) where u x, u x, ; (9) Ad, for every algorihm:, we ge, (6) u x for, hrough he followig SBA

3 Alicaio of he Adomia Decomosiio Mehod (ADM) 99 u x, x x N u x, ; u x, u x, s ds ; () For, we have he followig SBA algorihm: u x, x x N u x, u x, u x, s ds ; (), N u x,, we obai he Le s suose ha oe ca fid u x as followig SBA algorihm: u x, x x () u x, u x, s ds ; From (), we have: u x, x x 3 u x, x x 3! 4 u x, x x (3)! 4!... u x, x x!! Le s u i i x, u x, we obai: i i x, x x x x (4) i i! i i! Thus, for, we have: u x, lim x, x (5) From (5), we have:

4 Joseh BONAZEBI YINDOULA e al,, u x s u z N u x, ds dz u z, dz x x xsds z z dz z dz x = For, we obai he followig SBA algorihm: u x, x x (7) u x, u x, s ds ; From () ad (5), we have: i i x, x x x x (8) i i! i i! Thus for, we have: u x, lim x, x (9) Usig he same rocedure for 3, we have: i i x, x x x x i i! i i! () ad u x, lim x, x () From which we obai: u x, lim u x, x. (6) The oliear diffusio-reacio equaio Le s cosider he followig mixed roblem: u x, u x, u x, x e x u x, x q : u, e x u, e x x,,, T. 3 wih We are goig o solve roblem q usig he SBA mehod. From (), we have: (3)

5 Alicaio of he Adomia Decomosiio Mehod (ADM) where, u x, u x, 3, u x u x x e x From (3), we obai: u x, s 3s u x, u x, ds u x, sds u x, sds x e ds x (3) (4) (4) u x, s u x, x x e x ds u x, sds u x, sds x (5) From (5), we have also: u, u x,, 3 u z dz u z, dz u z, dz e z dz x x (6) (6) u z, e e dz u z, dz u z, dz e 4 (7) (7) u z, 3 dz 3 u z, dz u z, dz e 4 From (5) ad (8), we obai: (8) u x, s 3 u x, x x e x ds x (9) 3 u z, u x, s ds u x, s ds dz u z, dz u z, dz e 4,,, (9) u x x u x s ds N u x (3),,,, u z, u x s 3 3 N u x, e x e x 4 ds dz x u z dz u z dz u x s ds Accordig o he SBA mehod, we suose ha he soluio of () has he followig form: u x, lim u x, (3) (3)

6 Joseh BONAZEBI YINDOULA e al where u x, u x, ; (33) ad for every, we ge u x, for, hrough he followig SBA algorihm: u x, x N u x, ; (34) u x, u x, s ds ; For, we have he followig SBA algorihm: u x, x N u x, (35) u x, u x, s ds ;, N u x,, we obai he Le s suose ha oe ca fid u x as followig SBA algorihm: u x, x (36) u x, u x, s ds ; From (36), we have: u x, x u x, x u x, x! (37) 3 u3 x, x 3!... u x, x! Thus,, u x u x u x, u x,... u x,... x! 3!! (38) xe (38) u x, xe (39)

7 Alicaio of he Adomia Decomosiio Mehod (ADM) 3 For, we have he followig SBA algorihm: u x, x N u x, u x, u x, sds ; From (39), we have: 3 3 u x, s u z, N u x, e x e x ds dz 4 x,,, u z dz u z dz u x s ds = e x e x e e e x e x 4 4 = From (4) ad (4), we obai: u x, x u x, x u x, x! 3 u3 x, x 3!... u x, x! ad,, u x u x = u x, u x,... u x,...! 3!! 3 = x (4) (4) (4) (43) = xe Thus u x, xe (44) Usig he same rocedure for 3, we have: u x, xe (45) From which, we obai: u x, lim u x, xe. (46)

8 4 Joseh BONAZEBI YINDOULA e al Coclusio Through hese examles, we showed agai he usefuless of he SBA mehod, i he search of a aroximae soluio of a equaio ad his mehod gave us he exac soluio. We showed also ha his mehod is owerful for liear equaios, because he SBA mehod has bee coceived hrough he ricile of Picard, o avoid he calculaio of he Adomiaolyomials geeraed by he oliear oeraor coaied i aoliar equaio. Refereces [] Joseh YINDOULA BONAZEBI, PARE Youssouf, Gabriel BISSANGA, Fracis BASSONO ad Blaise SOME, Alicaio of he Adomia decomosiio mehod ad he Lalace rasform mehod o solvig he covecio-diffusio-dissiaio equaio, Ieraioal Joural of Mahemaical Research, 3(), 3-35, 4. [] K. ABBAOUI, les fodemes de la méhodedécomosiioelle d Adomiae alicaio a la résoluio de roblèmes issus de la biologie e de la médecie, Thèse de docora de l Uiversié Paris VI, Ocobre 995. [3] K. ABBAOUI ad Y.CHERRUAULT, covergece of Adomia mehod alied o differeial equaios, Mah. Comu.Modellig (8, 5), 3-9, 994. [4] K. ABBAOUI ad Y. CHERRUAULT, covergece of Adomia mehod alied o oliear equaios, Mah. Comu.Modellig (, 9), 6-73, 994. [5] K. ABBAOUI ad Y. CHERRUAULT, The decomosiio mehod alied o he Cauchy roblem, Kyberees (8, ), 68-74, 999. [6] N. N GARASTA, B.SOME, K.ABBAOUI ad Y. CHERRUAULT, New umerical sudy of Adomia mehod alied o a diffusio model, Kyberees, Vol. 3, o, 6-75,. [7] B. ABBO, Nouvel algorihme umérique de résoluio des EquaiosDifféreielles Ordiaires(EDO) e Equaios aux Dérivées Parielles (EDP) o liéaires, Thèse de Docora Uique, Uiversié de Ouagadougou, javier 7, UFR/SEA; Déareme de Mahémaiques e Iformaique (Buria Faso). [8] B. ABBO, N. N GARASTA, B. SOME ad L. SOME, A ew Aroach of he Adomia Algorihm for solvig Noliear Parial or Ordiary Differeial Equaios, Far Eas J. Al. Mahs. 3(3) (6),.99-3.