Numerical solution of the first order linear fuzzy differential equations using He0s variational iteration method

Transcription

1 Malaya Jounal of Matematik, Vol. 6, No. 1, 80-84, 2018 htts://doi.og/16637/mjm0601/0012 Numeical solution of the fist ode linea fuzzy diffeential equations using He0s vaiational iteation method M. Ramachandan1 * and M. Shobanaiya2 Abstact In this eseach, fist ode linea fuzzy diffeential equations is consideed. This ae comaes the He s vaiational iteation method (HVIM) and Leafog method [17] fo solving these equations. He s vaiational iteation method is an analytical ocedue fo finding the solutions of oblems which is based on the constucting a vaiational iteations. The Leafog method, based uon Taylo seies, tansfoms the fuzzy diffeential equation into a matix equation. The esults of alying these methods to the fist ode linea fuzzy diffeential equations show the simlicity and efficiency of these methods. Keywods Fuzzy diffeential equations, Fuzzy initial value oblems, Leafog method, He s vaiational iteation method. AMS Subject Classification 65L80, 65L05. 1 Deatment of Mathematics, Govenment Ats and Science College, Sathyamangalam , Tamil Nadu, India. Assistant (Mathematics), Municial Gils Highe Seconday School, Gobichettialayam , Tamil Nadu, India. *Coesonding autho: 1 d.amachandan64@gmail.com; 2 mshobanaiya@gmail.com Aticle Histoy: Received 24 Setembe 2017; Acceted 09 Decembe BT Contents 1 Intoduction He s Vaiational Iteation Method Geneal fomat fo Fuzzy initial value oblems Numeical Exeiments Examle Examle Examle Conclusion Refeences Intoduction Knowledge about dynamical systems modelled by diffeential equations is often incomlete o vague. It concens, fo examle, aamete values, functional elationshis, o initial conditions. The well-known methods fo solving analytically o numeically initial value oblems can only be used fo finding a selected system behaviou, e.g., by fixing the unknown aametes to some lausible values. Howeve, in this c 2017 MJM. case, it is not ossible to descibe the whole set of system behavious comatible with ou atial knowledge. The toics of fuzzy diffeential equations, which attacted a gowing inteest fo some time, in aticula, in elation to the fuzzy contol, have been aidly develoed ecent yeas. The concet of a fuzzy deivative was fist intoduced by S. L. Chang, L. A. Zadeh [2]. It was followed u by D. Dubois, H. Pade [3], who defined and used the extension incile. Othe methods have been discussed by M. L. Pui, D. A. Ralescu [16] and R. Goetschel and W. Voxman [5]. Fuzzy diffeential equations and initial value oblems wee egulaly teated by O. Kaleva [12, 13], S. Seikkala [20]. A numeical method fo solving fist ode linea fuzzy diffeential equations has been intoduced by M. Ma, M. Fiedman and A. Kandel [15] via the standad Eule method. Recently, T. Jayakuma, D. Maheskuma and K. Kanagaajan [14] solved the fist ode linea fuzzy diffeential equations using Runge-Kutta method of ode five. S. Seka and S. Senthilkuma [18] solved the same fist ode linea fuzzy diffeential equations using single tem Haa wavelet seies method. The objective of this ae is to use the He s vaiational iteation method to solve the fist ode linea fuzzy diffeential equations (discussed by T. Jayakuma, D.

2 Numeical solution of the fist ode linea fuzzy diffeential equations using He0 s vaiational iteation method 81/84 deivative of y and y(t0 = y0 is a aallelogam o a aallelogam shaed fuzzy numbe. We denote the fuzzy function y by y = [y, y. It means that the -level set of y(t) fo t [t0, T ] is Maheskuma and K. Kanagaajan [14] and S. Seka and S. Senthilkuma [18]). In this aticle we develoed numeical methods fo fist ode linea fuzzy diffeential equations to get discete solutions via He s vaiational iteation method which was studied by Seka et al. [19]. The subject of this ae is to ty to find numeical solutions of fist ode linea fuzzy diffeential equations using He s vaiational iteation method and comae the discete esults with the Leafog method which is esented eviously by Seka et al. [17]. Finally, we show the method to achieve the desied accuacy. Details of the stuctue of the esent method ae exlained in sections. We aly He s vaiational iteation method and Leafog method fo fist ode linea fuzzy diffeential equations. In Section 4, it s oved the efficiency of the He s vaiational iteation method. Finally, Section 5 contains some conclusions and diections fo futue exectations and eseaches. [y(t)] = [y(t; ), y(t; )], [y(t0 )] = [y(t0 ; ), y(t0 ; )], (0, 1] we wite f (t; y) = [ f (t; y), f (t; y)] and f (t; y) = F[t, y, y], f (t; y) = G[t, y, y]. Because of y0 (t) = f (t, y) we have f (t; y(t; )) = F[t; y(t; ), y(t; )] f (t; y(t; )) = F[t; y(t; ), y(t; )] By using the extension incile, we have the membeshi function f (t; y(t))(s) = Su{y(t)(τ)/s = f (t, τ)}, s R 2. He s Vaiational Iteation Method In this section, we biefly eview the main oints of the oweful method, known as the He s vaiational iteation method [6]-[11]. This method is a modification of a geneal Lagange multilie method oosed by [6]-[11]. In the vaiational iteation method, the diffeential equation L[u(t)] + N[u(t)] = g(t) so fuzzy numbe f (t; y(t)). Fom this it follows that [ f (t; y(t))] = [ f (t, y(t; )), f (t, y(t; ))], [0; 1] whee f (t, y(t; )) = min{ f (t, u)/u [y(t)] } (2.1) is consideed, whee L and N ae linea and nonlinea oeatos, esectively and g(t) is an inhomogeneous tem. Using the method, the coection functional Z un+1 (t) = un (t) + f (t, y(t; )) = max{ f (t, u)/u [y(t)] } Definition 3.1. A function f : R RF is said to be fuzzy continuous function, if fo an abitay fixed t0 R and ε > 0, δ > 0 such that t t0 < δ D[ f (t), f (t0 )] < ε exists. λ [L[un (s)] + N[u n (s)] g(s)]ds (2.2) is consideed, whee λ is a geneal Lagange multilie, un is the nth aoximate solution and u n is a esticted vaiation which means δ u n = 0. In this method, fist we detemine the Lagange multilie λ that can be identified via vaiational theoy, i.e. the multilie should be chosen such that the coection functional is stationay, i.e. δ u n+1 (un (t),t) = 0. Then the successive aoximation un, n 0 of the solution u will be obtained by using any selective initial function u0 and the calculated Lagange multilie λ. Consequently u = limn un. It means that, by the coection functional (2.2) seveal aoximations will be obtained and theefoe, the exact solution emeges at the limit of the esulting successive aoximations. In this ae we also conside fuzzy functions which ae continuous in metic D. Then the continuity of f (t, y(t); ) guaantees the existence of the definition of f (t, y(t); ) fo t [t0, T ] and [0, 1] [M. Ma, M. Fiedman and A. Kandel [15]]. Theefoe, the functions G and F can be definite too. 4. Numeical Exeiments In this section, the exact solutions and aoximated solutions obtained by He s vaiational iteation method and Leafog method. To show the efficiency of the He s vaiational iteation method, we have consideed the following oblem taken fom C. Duaisamy and B. Usha [4] and T.Jayakuma, D.Maheskuma and K.Kanagaajan [14], with ste size = along with the exact solutions. The discete solutions obtained by the two methods, He s vaiational iteation method and Leafog method. The absolute eos between them ae tabulated and ae esented in Tables 1 4. To distinguish the effect of the eos in accodance with the exact solutions, gahical eesentations ae given fo selected values of 0 0 and ae esented in Figues 1 6 fo the following oblem, using thee dimensional effects. 3. Geneal fomat fo Fuzzy initial value oblems Conside a fist-ode fuzzy initial value diffeential equation is given by y0 (t) = f (t, y(t)),t [t0, T ], y(t0 ) = y0 whee y is a fuzzy function of t, f (t, y) is a fuzzy function of the cis vaiable t and the fuzzy vaiable y, y0 is the fuzzy 81

3 Numeical solution of the fist ode linea fuzzy diffeential equations using He0 s vaiational iteation method 82/ Examle Conside the initial value oblem [C. Duaisamy and B. Usha [4]] y0 (t) = t f (t),t [0, 1] with initial condition y(0) = (1 + e, e) The exact solution at t = is given by Y (, ) = [(1+ e)e0.0005, (1.5+ e)e ], Examle Conside the fuzzy initial value oblem [M. Ma, M. Fiedman and A. Kandel [15]] Figue 1. Eo estimation of Examle 4.1 at 0 y (t) = y(t),t I = [0, 1] with initial condition y(0) = (5 + 5, ), 0< 1 The exact solution is given by Y1 (t, ) = (0; )et, Y2 (t, ) = (0; )et which at t = Examle Conside the fuzzy initial value oblem [James J. Buckley and Thomas Feuihg [1]] y0 (t) = c1 (t) + c2 with initial condition y(0) = 0 whee ci > 0, fo i = 1, 2 ae tiangula fuzzy numbes. The exact solution is given by Y1 (t; ) = l1 ()tan(w1 ()t), Y2 (t; ) = l2 ()tan(w2 ()t), with l1 () = c2,1 ()/c1,1 (), l2 () = c2,2 ()/c1,2 () w1 () = c1,1 ()/c2,1 (), w2 () = c1,2 ()/c2,2 () whee [c1 ] = [c1,1 (), c1,2 ()] and [c2 ] = [c2,1 (), c2,2 ()] c1,1 () = +, c1,2 () = 1.5 c2,1 () = 5 + 5, c2,2 () = The -level sets of y0 (t) ae Y10 (t; ) = c2,1 ()sec2 (w1 ()t), Y20 (t; ) = c2,2 ()sec2 (w2 ()t), which defines a fuzzy numbe. We have f1 (t, y, ) = min{ c1 u2 + c2 u [ (t; ), (t; )], c1 [c1,1 (), c1,2 ()], c2 [c2,1 (), c2,2 ()]} f2 (t, y, ) = max{ c1 u2 + c2 u [ (t; ), (t; )], c1 [c1,1 (), c1,2 ()], c2 [c2,1 (), c2,2 ()]} Figue 2. Eo estimation of Examle 4.1 at Figue 3. Eo estimation of Examle 4.2 at 5. Conclusion He s vaiational iteation method is a oweful, accuate, and flexible tool fo solving many tyes of fuzzy diffeential equations (oblems) in scientific comutation. The obtained aoximate solutions of the fist ode linea fuzzy diffeential equations ae comaed with exact solutions and it eveals that the He s vaiational iteation method woks well fo finding the aoximate solutions. Fom the Tables 1-4, one can obseve that fo most of the time intevals, the absolute eo Figue 4. Eo estimation of Examle 4.2 at 82

4 Numeical solution of the fist ode linea fuzzy diffeential equations using He0 s vaiational iteation method 83/84 Table 1. He s vaiational iteation method Eo Table 4. Leafog Method Eo Calculations Calculations Examle 4.1 0E-11 0E E E E E E E E E-11 0E-10 0E-10 Examle 4.2 0E-10 0E E E E E E E E E E E-10 Examle 4.3 0E-09 0E E E E E E E E E-09 0E E-09 Table 2. Leafog Method Eo Calculations Examle 4.1 0E-09 0E E E E E E E E E-09 0E-08 0E-08 Examle 4.2 0E-08 0E E E E E E E E E E E-08 Figue 5. Eo estimation of Examle 4.3 at vaiational iteation method is moe suitable fo studying fist ode linea fuzzy diffeential equations. Table 3. He s vaiational iteation method Eo Calculations Refeences [1] Examle 4.3 0E-11 0E E E E E E E E E-11 0E E-11 [2] [3] [4] [5] [6] is less in He s vaiational iteation method when comaed to the Leafog method [18], which yields a little eo, along with the exact solutions. Fom the Figues 1-6, it can be edicted that the eo is vey less in He s vaiational iteation method when comaed to the Leafog method. Hence, He s [7] 83 J. J. Buckley and T. Feuihg, Intoduction to Fuzzy Patial Diffeential Equations, Fuzzy Sets and Systems, 110(1) (2000), S. L. Chang and L. A. Zadeh, On fuzzy maing and contol, IEEE Tansactions on Systems, Man, and Cybenetics, 2 (1972), D. Dubois and H. Pade, Towads fuzzy diffeential calculus.iii. Diffeentiation, Fuzzy Sets and Systems, 8(3) (1982), C. Duaisamy and B. Usha, Anothe Aoach to Solution of Fuzzy Diffeential Equations, Alied Mathematical Sciences, 4(16) (2010), R. Goetschel and W. Voxman, Elementay calculus, Fuzzy Sets and Systems, 18 (1986), J. H. He, A new aoach to nonlinea atial diffeential equations, Commun Nonlinea Sci Nume Simulat, 2(4) (1997), J. H. He, Aoximate analytical solution fo seeage flow with factional deivatives in oous media, Comut Methods Al Mech Eng., 167(1998),

5 Numeical solution of the fist ode linea fuzzy diffeential equations using He0 s vaiational iteation method 84/84????????? ISSN(P): Malaya Jounal of Matematik ISSN(O): ????????? Figue 6. Eo estimation of Examle 4.3 at [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] J. H. He, Vaiational iteation method a kind of nonlinea analytical technique: some examles, Int. J. Nonlinea Mech., 34(1999), J. H. He, Vaiational iteation method fo autonomous odinay diffeential systems, Al. Math. Comut., 114(2 3) (20000, J. H. He, Vaiational iteation method Some ecent esults and new inteetations, J. Comut. Al. Math., 207 (2007), J. H. He, Vaiational incile fo some nonlinea atial diffeential equations with vaiable coefficients, Chaos Solitons Factals, 19 (2004), O. Kaleva, Fuzzy diffeential equations, Fuzzy Sets and Systems, 24(3) (1987), O. Kaleva, The Cauchy oblem fo fuzzy diffeential equations, Fuzzy Sets and Systems, vol. 35, no. 3, , T. Jayakuma, D. Maheskuma and K. Kanagaajan, Numeical Solution of Fuzzy Diffeential Equations by Runge Kutta Method of Ode Five, Alied Mathematical Sciences, 6(60) (2012), M. Ma, M. Fiedman and A. kandel, Numeical Solutions of Fuzzy Diffeential Equations, Fuzzy Sets and Systems, 105 (1999), M. L. Pui and D. A. Ralescu, Diffeentials of fuzzy functions, Jounal of Mathematical Analysis and Alications, 91(2) (1983), S. Seka and K. Pabhavathi, Numeical solution of fist ode linea fuzzy diffeential equations using Leafog method, IOSR Jounal of Mathematics, 10(5) (2014), S. Seka and S. Senthilkuma, Single Tem Haa Wavelet Seies fo Fuzzy Diffeential Equations, Intenational Jounal of Mathematics Tends and Technology, 4(9) (2013), S. Seka and B. Venkatachalam alias Ravikuma, Numeical Solution of linea fist ode Singula Systems Using He s Vaiational Iteation Method, IOSR Jounal of Mathematics, 11(5) (2015), S. Seikkala, On the fuzzy initial value oblem, Fuzzy Sets and Systems, 24(3) (1987),