FUZZY ALTERNATING DIRECTION IMPLICIT METHOD FOR SOLVING PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS IN THREE DIMENSIONS

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1 FUZZY ALTERNATING DIRECTION IMPLICIT METHOD FOR SOLVING PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS IN THREE DIMENSIONS N.Mugutha *1, B.Jessaili Jeba #2 *1 Assistat Professor, Departmet of Mathematics, M.V.Muthiah Govt. Arts College for Wome, Didigul. #2 M.Phil Research Scholar, Departmet of Mathematics, M.V.Muthiah Govt. Arts College for Wome, Didigul. ABSTRACT I this paper, we solved parabolic partial differetial equatios i three-dimesios usig the fiite differece method such as Alteratig Directio Implicit method. Here we solved heat equatio usig Alteratig Directio Implicit method ad also we applied fuzzy cocepts to solve heat equatio usig this Alteratig Directio Implicit method. Keywords: parabolic equatio, alteratig directio implicit method, heat trasform, fiite differece I.INTRODUCTION Partial differetial equatios form the basis of very may mathematical models of physical, chemical ad biological pheomea, ad more recetly they spread ito ecoomics, fiacial forecastig, image processig ad other fields [9]. Parabolic partial differetial equatios i three space dimesios with over-specified boudary data feature i the mathematical modelig of may importat pheomea [3]. Calculatio of the solutio of fuzzy partial differetial equatios is i geeral very difficult. We ca fid the exact solutio oly i some special cases. The theory of fuzzy differetial equatios has attracted much attetio i recet times because this theory represets a atural way to model dyamical systems ucertaity. The cocept of the fuzzy derivative was first itroduced by Chag ad Zadeh [1]; it was followed up by Dubois ad prade [4], who used the extesio priciple i their approach. The study of fuzzy differetial equatios has bee iitiated as a idepedet subject i cojuctio with fuzzy valued aalysis[2] ad [10] ad set-valued differetial equatios. I this paper, we apply fuzzy to solve Alteratig directio Implicit method with the parabolic partial differetial equatios i three dimesios. JETIRAW06023 Joural of Emergig Techologies ad Iovative Research (JETIR) 174

2 PRELIMINARIES The basic defiitio of fuzzy umbers is give i [5] We deote the set of all real umbers by R ad the set of all fuzzy umbers o R is idicated by RF. A fuzzy umber is mappig u : R [0,1] with the followig properties: (a) u is upper-semicotiuous, (b) u is fuzzy covex, i.e., u(λ x+(1-λ)y) mi {u(x), u(y)} for all x, y R, λ [0,1], (c) u is ormal, i.e., x0 R for which u(x0) = 1, (d) Supp u = { x R u(x) > 0} is the support of the u, ad its closure cl(supp u) is compact. Defiitio 2.1[5] A arbitrary fuzzy umbers represeted by a ordered pair of fuctios (u(α), u(α)), 0 α 1 that, satisfies the followig requiremets: u(α) is a bouded left cotiuous o decreasig fuctio over [0,1], with respect to ay α. u(α) is a bouded left cotiuous o icreasig fuctio over [0,1], with respect to ay α. u(α) u(α), 0 α 1 The the α-level set [u] α of a fuzzy set u o R is defied as: [u] α = {x R; u(x) α}, for each α (0,1], Ad for α = 0 [u] 0 = α (0,1] [u] α Where A deotes the closure of A. For u, v RF ad λ R the sum u+v ad the product λu are defied by [u + v] α = [u] α +[v] α, [λu] α = λ[u] α, α [0,1] where [u] α +[v] α meas the usual additio of two itervals (subsets) of R ad λ[u] α meas the usual product betwee a scalar ad a subset of R. The metric structure is give by the Hausdroff distace D: RF RF R+ {0}, D(u,v) = supα (0,1] max{ u(α) v(α), u(α) v(α) }. Where [u] α = [u(α), u(α)] ad [v] α = [v(α), v(α)]. (RF, D) is a complete metric space ad the followig properties are well kow: D(u+w, v+w) = D(u, v) D(ku,kv) = k D(u, v) u, v, w RF, k R, u, v RF, D(u+v, w+e) D(u, w) + D(v,e) u, v, w, e RF. JETIRAW06023 Joural of Emergig Techologies ad Iovative Research (JETIR) 175

3 Model of equatio [8]: I the case of three dimesios, the mathematical model is such a iitial ad boudary value problem is give by [8] as follows: u = 2 u + 2 u + 2 u, (0 x, y, z 1, t 0) t x 2 y 2 z 2...(1) u (x, y, z, 0) = g(x, y, z), (0 x, y, z 1).... (2) u (0, y, z, t) = f1(y, z, t), u (1, y, z, t) = f2(y, z, t), (0 y, z 1, t 0)... (3) u (x, 0, z, t) = f3(x, z, t), u (x, 1, z, t) = f4(x, z, t), ( 0 x, z 1, t 0)... (4) u (x, y, 0, t) = f5(x, y, t), u (x, y, 1, t) = f6(x, y, t), ( 0 x, y 1, t (5) Where u (x, y, z, t) deotig temperature or cocetratio of chemical, while g, f1, f2, f3, f4, f5 ad f6 are kow fuctios ad heat trasferred i three dimesio system of Legth L, Width W ad Depth D with grid poits i cubic. III. FUZZY ALTERNATING DIRECTION IMPLICIT METHOD The alteratig directio implicit method was first suggested by Douglas, Peacema ad Richard for solvig the heat equatio i two spatial variables ad alteratig directio implicit methods have proved valuable i the approximatio of the solutios of parabolic ad elliptic differetial equatios i two ad three variables [6,7] I the ADI approach, the fiite differece equatios are writte i terms of quatities at three x levels. However three differet fiite differece approximatios are used alterately, oe JETIRAW06023 Joural of Emergig Techologies ad Iovative Research (JETIR) 176

4 to advace the calculatios from (+1) plae to the (+2) plae ad the third to advace the calculatios from (+2) plae to the (+3) plae [8] Assume Ũ is a fuzzy fuctio of the idepedet crisp variables x ad t. subdivided the x-t plae ito sets of equal rectagles of sides x, y, z, t by equally spaced grid lies parallel to Ox [12] Deote the parametric for of fuzzy umber, Ũi,j as follows Ũ i,j = ( U i,j, U i,j ) The by Taylor s theorem ad defiitio of stadard differece (Dx Dx ) Ũi,j = ((Dx Dx )Ũi,j (Dy Dy ) Ũi,j (Dz Dz ) Ũi,j) The, we advace th3e solutio of the parabolic partial differetial equatio i three dimesios from the th plae to (+1)th plae by replacig 2 u by implicit fiite differece approximatio x 2 at the (+1)th plae. Similarly, 2 u ad 2 u are replaced by a explicit fiite differece approximatio at the th plae. y 2 z 2 With these approximatio eq(1) i pa rabolic model ca be writte as u +1 u u +1 2u +1 + u +1 u 2u +u u 2u + u i,j,k i,j,k = i 1,j,k i,j,k i+1,j,k + i,j 1,k i,j,k i,j+1,k + i,j,k 1 i,j,k i,j,k+1..(6) t ( x) 2 ( y) 2 ( z) 2 We set ( x= y= z)= h ad t=k the we have a square ad multiply equatio (6) by t the we get u +1 u = k (u +1 2u +1 + u +1 + u 2u + u +u 2u + i,j,k i,j,k h 2 i 1,j,k i,j,k i+1,j,k i,j 1,k i,j,k i,j+1,k i,j,k 1 i,j,k+1 Let r = k h, we get 2 u ) u +1 u = r(u +1 2u +1 + u +1 ) + r(u 2u + u ) i,j,k i,j,k i 1,j,k i,j,k i+1,j,k i,j 1,k i,j,k i,j+1,k + r(u 2u + u ) Ad i,j,k 1 i,j,k i,j,k+1 u +1 r(u +1 2u +1 + u +1 ) = u + r(u 2u + u ) i,j,k i 1,j,k i,j,k i+1,j,k i,j,k i,j 1,k i,j,k i,j+1,k + r(u 2u + u ) i,j,k 1 i,j,k i,j,k+1 i,j,k JETIRAW06023 Joural of Emergig Techologies ad Iovative Research (JETIR) 177

5 We simply ad rearrage the above equatio ad we get -r u +1 + (1 + 2r)u +1 ru +1 = (1 4r)u + r(u + u ) i 1,j,k i,j,k i+1,j,k i,j,k i,j 1,k i,j+1,k + r(u 2u i,j,k 1 i,j,k i,j,k+1 + u )..... (7) Also, we advace the solutio from the( +1)th plae to (+2)th plae by replacig 2 u implicit fiite differece approximatio at (+2)th plae. by y 2 Similarly, 2 u ad 2 u are replaced by a explicit differece approximatio at the (+1)th plae. x 2 z 2 With these approximatio equatio (1) i parabolic model becomes u +2 u +1 u +1 2u +1 +u +1 u +2 2u +2 +u +2 u +1 2u +1 + u +1 i,j,k i,j,k = i 1,j,k i,j,k i+1,j,k + i,j 1,k i,j,k i,j+1,k + i,j,k 1 i,j,k i,j,k+1...(8) t ( x) 2 ( y) 2 ( z) 2 We set ( x= y= z) = h ad t=k the we have a square regio ad multiply equatio (8) by t the we get, u +2 u +1 = k (u +1 2u +1 + u +1 + u +2 2u +2 + u +2 i,j,k i,j,k h 2 i 1,j,k i,j,k i+1,j,k i,j 1,k i,j, i,j+1,k +u +1 2u +1 + u +1 ) Let r = k h we get, 2 i,j,k 1 i,j,k i,j,k+1 u +2 u +1 = r(u +1 2u +1 + u +1 ) + r(u +2 2 u +2 + u +2 ) i,j,k Ad i,j,k i 1,j,k i,j,k i+1,j,k i,j 1,k i,j,k i,j+1,k u +2 r(u +2 2u +2 + u +2 ) = u +1 + r(u +1 2u +1 + u +1 ) i,j,k i,j 1,k i,j,k i,j+1,k i,j,k i 1,j,k i,j,k i+1,j,k + r(u +1 2u +1 + u +1 ) We simplify ad rearrage the above equatio ad we get, i,j,k 1 i,j,k i,j,k+1 -ru +2 + (1 + 2r)u +2 ru +2 = (1 4r)u +1 + r( u +1 + u +1 ) i,j 1,k i,j,k i,j+1,k i,j,k i 1,j,k i+1,j,k + r(u +1 + u +1 )... (9) i,j,k 1 i,j,k+1 Now, we advace the solutio from the( +2)th plae to (+3)th plae by replacig 2 u ad 2 u by explicit fiite differece approximatio at (+2)th plae. y 2 x 2 JETIRAW06023 Joural of Emergig Techologies ad Iovative Research (JETIR) 178

6 2 u The by a implicit fiite differece approximatio at the (+3)th plae. With these z2 approximatio equatio (1) i parabolic model becomes, u +3 u +2 u +2 2u +2 + u +2 u +2 2u +2 +u +2 u+3 2u +3 + u +3 i,j,k i,j,k = i 1,j,k i,j,k i+1,j,k + i,j 1,k i,j,k i,j+1,k + i,j,k 1 i,j,k i,j,k+1.(10) t ( x) 2 ( y) 2 ( z) 2 Put t = k ad ( x= y= z) = h u +3 u +2 u +2 2u +2 + u +2 u +2 2u +2 +u +2 u+3 2u +3 + u +3 i,j,k i,j,k i 1,j,k i,j,k i+1,j,k = + i,j 1,k i,j,k i,j+1,k i,j,k 1 i,j,k i,j,k+1 + k (h) 2 (h) 2 (h) 2 Multiply equatio (10) by k whe ( x= y= z) = h the, we have a square regio ad we get u +3 u +2 = k (u +2 2u +2 + u +2 +u +2 2u +2 + u +2 i,j,k i,j,k + u +3 h2 i 1,j,k i,j,k i+1,j,k i,j 1,k i,j,k i,j+1,k i,j,k 1 2u +3 + u +3 Let r = k h we get, 2 i,j,k i,j,k+1 u +3 u +2 = r(u +2 2u +2 + u +2 ) + r(u +2 2u +2 + u +2 ) i,j,k i,j,k i 1,j,k i,j,k i+1,j,k i,j 1,k i,j,k i,j+1,k + r(u +3 2u +3 + u +3 ) Ad i,j,k 1 i,j,k i,j,k+1 u +3 r(u +3 2u +3 + u +3 ) = u +2 + r(u +2 2u +2 + u +2 ) + i,j,k i,j,k 1 i,j,k i,j,k+1 i,j,k i 1,j,k i,j,k i+1,j,k r(u +2 2u +2 + u +2 ) We simplify ad rearrage the above equatio we get, i,j 1,k i,j,k i,j+1,k -ru +3 + (1 + 2r)u +3 ru +3 = (1 4r)u +2 + r(u +2 + u +2 ) i,j,k 1 i,j,k i,j,k+1 i,j,k i 1,j,k i+1,j,k + r(u +2 + u +2 ).(11) i,j 1,k i,j+1,k Expressed from the above equatios (7), (9) ad (11) by the system AX=B ) JETIRAW06023 Joural of Emergig Techologies ad Iovative Research (JETIR) 179

7 1 2r r u r 1 2r r , j,k u 1 0 r 1 2r r 0 0 r 1 2r r r 1 2r r u 1 r 1 2r r m12, j,k u r 1 2r m11, j,k (1 4r)u (1 4r)u = 2, 2, j,k j 1,k 3, j 1,k 3, j,k r[u r[u 2, j 1,k u 3, j 1,k u 2, j,k u 1 3, j,k u 1 3, j,k u1 4, j,k 2, j,k u ] 1 3, ] j,k u 1 (1 4r)u r[u u u u ] m11, j,k m11, j 1,k m11, j 1,k m11, j,k 1 m11, j,k 1 JETIRAW06023 Joural of Emergig Techologies ad Iovative Research (JETIR) 180

8 Similarly, applyig the above procedure with remaider of equatios These systems are of a tridiagoal liear system of equatios ad ca be solved by the Gauss elimiatio. REFERENCE [1] S.L. Chag, L.A. Zadeh, O fuzzy mappig ad cotrol, IEEE Trasactios o Systems Ma Cyberetics 2 (1972) [2] P. Diamod, P. Kloede, Metric Space of Fuzzy Sets: Theory ad Applicatio, World Scietific, Sigapore, [3] Douglas, J.Jr. ad Peacema, D., "Numerical Solutio of Two-Dimesioal Heat Flow Problems", America Istitute of Chemical Egieerig Joural, 1, pp (1955). [4] D. Dubois, H. prade, Towards fuzzy differetial calculus: part 3, differetiatio, Fuzzy Sets ad Systems 8 (1982) [5] L. Jamshidi1, L. Avazpour., Solutio of the Fuzzy Boudary Value Differetial Equatios Uder Geeralized Differetiability By Shootig Method Islamic Azad Uiversity, Volume 2012 [6] Johso, S., Saad, Y. ad Schultz, M., "Alteratig Directio Methods o Multiprocessors", SIAM, J.sci. Statist. Compute., 8,pp (1987). [7] Lapidus, L. ad Pider, G.F., "Numerical Solutio of Partial Differetial Equatios i Sciece ad Egieerig", Joh Wiley & Sos Ic (1982). [8] Mahmood H.Yahya, Abdulghafor M. AI-Rozbayai, Alteratig Directio Methods for Solvig Parabolic Partial Differetial Equatios i three dimesios [9] Mig-shu, M. ad Tog-ke, W., "A family of High-order Accuracy Explicit Differece Schemes with Brachig Stability for Solvig 3-D Parabolic Partial Differetial Equatio", Uiversity Shaghai, Chia vol. 21, pp (2000). [10] Morto, K.W. ad Mayers, D.F., "Numerical Solutio of Partial Differetial Equatios", Cambridge Uiversity Press, UK. Secod ed (2005). [11] C.V. Negoita, D.A. Ralescu, Applicatios of Fuzzy Sets to System Aalysis, Birkhauser, Basel, [12] K. Nemati Ad M.Matifar., A Implicit Method For Fuzzy Parabolic Partials Differetial Equatios, The Joural of Noliear Scieces ad Applicatios. JETIRAW06023 Joural of Emergig Techologies ad Iovative Research (JETIR) 181