Analytical Method for Solution Two-Dimensional Partial Differential Equations with Boundary Integral Conditions
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1 Ieraioal Joural of Egieerig Sciece Iveio ISSN Olie: ISSN Pri: Volume 6 Issue 8 Augus 7 PP Aalical Mehod for Soluio Two-Dimesioal Parial Differeial Equaios wih Boudar Iegral Codiios Ima Isho Maerials Egieerig Deparme Uiversi of Techolog Iraq Correspodig Auhor: Ima Isho Absrac:This paper deals wih he boudar ad iiial value problems for wo-dimesioal parial differeial equaio model b usig he aalical mehod. Tesed eamples ad he obaied resuls demosrae efficiec of he proposed mehod. The resuls are preseed usig he Malab sofware package. Ke words: Modified decomposiio mehod wo-dimesioal parial differeial equaio boudar iegral codiio problem Dae Of Submissio: -8-7 Dae Of Accepace: I. Iroducio Ma classes of liear ad oliear differeial equaios ca solve usig Adomia decomposiio mehod ad i compuaio ad faser i covergece i is much simpler ha a oher mehod available i he ope lieraure. There are ma lieraures developed cocerig Adomia decomposiio mehod [-4] ad he relaed modificaio o ivesigae various scieceific model [5-8]. E. Babolia e al. iroduced he resar mehod o solve he equaio f = [9] ad he iegral equaios []. H. Jafari e.al used a correcio of decomposiio mehod for ordiar ad oliear ssems of equaios ad show ha he correcio acceleraes he covergece [ ]. I his paper we prese compuaioall efficie umerical mehod for solvig he parial differeial equaio wih boudar iegral codiios: D D D h wih he iiial codiio ad he iegral codiios f T T dd g T dd g T Where f g g ad h are kow fucios. T is give cosa. I he prese work we appl he modified Adomia s decomposiio mehod for solvig eq. ad compare he resuls wih eac soluio. The paper is orgaized as follows: I secio he wo-dimesioal parial differeial equaios wih boudar iegral codiios ad is soluio b modified decomposiio mehod is preseed. I secio 3 a eample is solved umericall usig he modified decomposiio mehod. Fiall we prese coclusio abou soluio of he wo-dimesioal parial differeial equaio. Modified Adomia s Decomposiio Mehod of Soluio he Two-Dimesioal Parial Differeial Equaios I his secio we prese modified decomposiio mehod for solvig wo-dimesioal parial differeial equaios wih boudar iegral give i eq.. I his mehod we assume ha: 39 Page
2 Aalical Mehod for Soluio Two-Dimesioal Parial Differeial 4 Page eq. ca be rewrie: h Where ad The iverse is give b d 3 Take o boh sides of eq we have h The we ca wrie h 4 The modified decomposiio mehod was iroduced b Wazwaz [6]. This mehod is based o he assumpio ha he fucio ca be divided io wo pars amel ad. Uder his assumpio we se The he modificaio u < Numerical Illusraio: Eample : Cosider wo-dimesioal parial differeial equaio wih boudar iegral codiio for he equaio : D D D T
3 dd T dd T 3 4 We appl he above proposed mehod; we obai: 3 The he series form is give b: 3 Aalical Mehod for Soluio Two-Dimesioal Parial Differeial This is he eac soluio. Table shows he aalical soluios for parial differeial equaio wih boudar iegral codiio obaied for differe values ad compariso bewee eac soluio ad aalical soluio. Table. Compariso bewee eac soluio ad aalical soluio for eample = Eac Modified e - MADM Soluio Adomia Decomposiio Mehod Eample : Cosider he problem wih he followig codiios: D D D e 5 T 4 Page
4 Aalical Mehod for Soluio Two-Dimesioal Parial Differeial dd 4e T dd e T Now afer modified decomposiio mehod we obai: 5 e 3 The he series form is give b: 3 5 e Which gives he eac soluio 5 e. Table shows he aalical soluios for parial differeial equaio wih boudar iegral codiio obaied for differe values ad compariso bewee eac soluio ad aalical soluio. Table. Compariso bewee eac soluio ad aalical soluio for eample whe =.4 = Eac Modified e - MADM Soluio Adomia Decomposiio Mehod Eample 3: Cosider he problem wih he followig boudar iegral ad iiial codiios: D D D T 4 Page
5 Aalical Mehod for Soluio Two-Dimesioal Parial Differeial dd 3 3 T 7 7 dd T Now we appl he above modified decomposiio mehod we obai: The he series form is give b: This is he eac soluio Table 3 shows he aalical soluios for parial differeial equaio wih boudar iegral codiio obaied for differe values ad compariso bewee eac soluio ad aalical soluio. Table3. Compariso bewee eac soluio ad aalical soluio for eample 3 whe = = Eac Modified e - MADM Soluio Adomia Decomposiio Mehod II. Coclusio I his paper we have applied he modified decomposiio mehod for he soluio of he wodimesioal he parial differeial equaio wih boudar iegral codiio.. This algorihm is simple ad eas o impleme. The obaied resuls cofirmed a good accurac of he mehod. O he oher had he calculaios are simpler ad faser ha i radiioal echiques. Refereces [] Adomia G. ad RachR. Noise erms i decomposiio soluio series Compuers Mah. Appl [] AdomiaG. Solvig Froier Problems of Phsics: The Decomposiio Mehod Kluver Academic Publishers Boso 994. [3] Cao J.R ad Va der HoekJ. Diffusio equaio subjec o he specificaio of mass J.Mah. Aal. Appl [4] Wazwaz A. M J. Necessar codiios for appearece of oise erms i decomposiio soluio series Mah.Aal.Appl Page
6 Aalical Mehod for Soluio Two-Dimesioal Parial Differeial [5] A.M. Wazwaz A ew algorihm for calculaig Adomia polomials for oliear operaors Appl Mah Compu [6] A.M. Wazwaz A reliable modificaio of Adomia decomposiio mehod Appl Mah Compu 999 pp [7] M.M. Hosseii H Nasabzadeh Modified Adomia decomposiio mehod for specific secod order ordiar differeial equaios Appl. Mah. Compu [8] Y.Q. Hasa.M. Zhu Solvig sigular boudar valued problems of higher-order ordiar differeial equaios b modified Adomia decomposiio mehod Commu Noliear Sci Numer Simula [9]. E. Babolia ad S. H. Javadi Resared Adomia Mehod for Algebraic Equaios App. Mah. Ad Compuaio []. E. Babolia S. H. Javadi H. Sadeghi Resared Adomia Mehod for Iegral Equaios App. Mah. Ad Compuaio []. Hossei Jafari Varsha Dafardar-Gejji Revised Adomia decomposiio mehod for Solvig a ssem of oliear equaios App.Mah. Ad Compuaio []. Hossei Jafari Varsha Dafardar-Gejji Revised Adomia decomposiio mehod for Solvig ssems of Ordiar ad racioal Differeial Equaios App. Mah. Ad Compuaio Ima Isho. Aalical Mehod for Soluio Two-Dimesioal Parial Differeial Equaios wih Boudar Iegral Codiios. Ieraioal Joural of Egieerig Sciece Iveio IJESI vol. 6 o. 8 7 pp Page
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