Solving a System of Nonlinear Functional Equations Using Revised New Iterative Method

Transcription

1 Solving a Sysem of Nonlinear Funcional Equaions Using Revised New Ieraive Mehod Sachin Bhalekar and Varsha Dafardar-Gejji Absrac In he presen paper, we presen a modificaion of he New Ieraive Mehod (NIM proposed by Dafardar-Gejji and Jafari [J Mah Anal Appl 6;316:73 763] and use i for solving sysems of nonlinear funcional equaions This modificaion yields a series wih faser convergence Illusraive eamples are presened o demonsrae he mehod Keywords Capuo fracional derivaive, Sysem of nonlinear funcional equaions, Revised new ieraive mehod I INTRODUCTION Nonlinear differenial equaions play very imporan role in modelling numerous problems in Physics, Chemisry, Biology and Engineering Science [1], [] Many problems can be modelled as sysems of differenial equaions/inegral equaions/inegro-differenial equaions/parial differenial equaions/fracional order differenial equaions Since mos realisic funcional equaions are nonlinear and do no possess eac analyical soluions, ieraive and numerical mehods are widely used o solve hese equaions Adomian decomposiion mehod (ADM [3], variaional ieraion mehod (VIM [], homoopy perurbaion mehod (HPM [] are some of he sandard mehods Recenly Dafardar-Gejji and Jafari [6] have inroduced a new ieraive mehod (NIM o solve general funcional equaion: y = f +N(y, where f is specified funcion and N a given nonlinear funcion of y NIM is simple in is principles and easy o implemen on compuer using symbolic compuaion packages such as Mahemaica This mehod is beer han numerical mehod as i is free from rounding off errors and does no require large compuer power NIM has proven successful over oher mehods in many cases [7], [8] In he presen paper we presen a modificaion of he NIM o solve he following sysem of funcional equaions wih improved convergence: y i = f i + N i (y 1,y,,y n i =1,,,n The revised mehod has been applied o solve various eamples, some of which have already been solved by oher mehods A comparison wih oher soluions reveals he usefulness of his modificaion S Bhalekar is wih Deparmen of Mahemaics, Shivaji Universiy, Vidyanagar, Kolhapur - 16, India address: sachinmah@yahoocoin V Dafardar-Gejji is wih Deparmen of Mahemaics, Universiy of Pune, Ganeshkhind, Pune - 117, India address: vsgejji@mahunipuneacin II PRELIMINARIES AND NOTATIONS We review some basic definiions from fracional calculus [1], [9] Definiion 1: A real funcion f(, > is said o be in space C α,α Rif here eiss a real number p (> α, such ha f( = p f 1 ( where f 1 ( C[, Definiion : A real funcion f(, > is said o be in space C m α,m IN {} if f (m C α Definiion 3: Le f C α and α 1, hen he (lefsided Riemann-Liouville inegral of order μ is given by I μ f(, = 1 Γ(μ ( τ μ 1 f(, τ dτ, > (1 Definiion : The (lef sided Capuo fracional derivaive of f, f C m 1,m IN {}, is defined as: D μ f(, where m 1 <μ<m,m IN Noe ha = m f(,, μ = m, m = I m μ m f(, m, ( I μ D μ k f f(, =f(, (, k, <μ m, m IN, k k! k= (3 I μ ν Γ(ν +1 = Γ(μ + ν +1 μ+ν ( III NEW ITERATIVE METHOD FOR A SYSTEM OF NONLINEAR FUNCTIONAL EQUATIONS Consider he sysem of nonlinear funcional equaions: y i = f i + N i (y 1,y,,y n (i =1,,,n, ( where f i are known funcions and N i are nonlinear operaors Le ȳ =(y 1,,y n be a soluion of sysem ( where y i having he series form: y i = y i,j, i =1,,,n (6 j= Inernaional Scholarly and Scienific Research & Innovaion 6( scholarwaseorg/19997/19

2 We decompose he nonlinear operaor N i as N i (ȳ = N i j= j= = N i (y 1,,,y n, k k + N i y 1,j,, y n,j k=1 j= j= (7 j= j= By virue of equaions (6 and (7, sysem ( is equivalen o y i,j = f i + N i (y 1,,,y n, j= k k + N i y 1,j,, y n,j k=1 j= j= (8 j= j= (i =1,,,n For i =1,,,n, we define he recurrence relaion: y i, = f i, y i,1 = N i (y 1,,,y n,, m m y i,m+1 = N i j= j=, m =1,, j= j= Then y i = j= y i,j The k-h order approimaion o y i is given by y i = j= y i,j IV REVISED NIM In his secion we sugges a modificaion o NIM for solving sysem of nonlinear funcional equaions To illusrae he mehod we consider he sysem of equaions ( Iniial sep: Firs ieraion: y i, = f i, i =1,,,n y 1,1 = N 1 (y 1,,y,,,y n, y,1 = N (y 1, + y 1,1,y,,,y n, y 3,1 = N 3 (y 1, + y 1,1,y, + y,1,y 3,,,y n, y n,1 = N n (y 1, + y 1,1,y, + y,1,,y n 1, + y n 1,1,y n, k h ieraion (k =, 3, ( y 1,k = N 1 i= i= i= ( N 1 i= ( k y,k = N y 1,i, y,i,, i= ( N i= i= y 1,i, y j,k = N j ( k i= ( N j i= i= i= y,i,, k i= y n,k = N n ( k i= ( N n i= i= y j 1,i, y j,i,, i= k i= i= y j 1,i, i= y n 1,i, i= i= y n 1,i, i= ( Thus N i (ȳ =N i j= y 1,j,, j= y n,j Hence y i = j= y i,j V NUMERICAL EXAMPLES i= y j,i,, i= = j=1 y i,j E1: Consider he sysem of linear differenial equaions [1]: y 1 = y 3 cos(, y 1 ( = 1, y = y 3 e, y ( =, y 3 = y 1 y, y 3 ( = (9 Equivalen sysem of inegral equaions is y 1 = (1 sin( + y = (1 e + y 3 = + y 3 d = f 1 (+N 1 (y 1,y,y 3 ; y 3 d = f (+N (y 1,y,y 3 ; (y 1 y d = f 3 (+N 3 (y 1,y,y 3 Using revised NIM we ge an ieraive scheme: y 1, =1 sin(, y, =1 e,y 3, =; y 1,1 =, y,1 =, y 3,1 = +e + cos(; y 1, = 1+e +sin(, y, = 1+e +sin(, y 3, =; y 1,3 =,y,3 =,y 3,3 = Thus, he soluion of sysem (9 is y 1 = e,y =sin(, y 3 = e + cos( Inernaional Scholarly and Scienific Research & Innovaion 6( scholarwaseorg/19997/19

3 E: Consider he sysem of nonlinear differenial equaions: Inegraing we ge y 1 = 1+ y = 1+ y 3 = y 1 = y, y 1 ( = 1, y = e y 1, y ( = 1, y 3 = y + y 3, y 3 ( = (1 y d = f 1 (+N 1 (y 1,y,y 3, e y 1 d = f (+N (y 1,y,y 3, (y y 3 d = f 3 (+N 3 (y 1,y,y 3 The revised NIM leads o E3: Consider sysem of nonlinear parial differenial equaions [11]: y 1, =1,y, =1,y 3, =; y 1,1 =, y,1 =3 3e e, u y 3,1 = + +e +e ; + v u + u =1, u(, = e, v y 1, = e e 16e 17e, y, = e 3 8e +33e + 6 u v v =1, v(, = e (11 9 e 3 The sysem (11 is equivalen o 16e 3e + 3 e 3, ( y 3, = e 3 +8e 1e + 61 u =(e + u + v u d, 7 ( 6 7 e 3 +8e +8e + v =(e + + v + u v d (1 9 e 3, and so on In Fig1, Fig and Fig3 we compare he soluions The eac soluion of (1 is u(, =e,v(, =e of (1 wih he soluions by sandard NIM and by revised Employing revised NIM o (1, we ge ADM [1] Solid line shows eac soluion, doed line shows soluion by revised NIM, dashed line shows sandard NIM u = e +, v = e + ; soluion and long dashed line shows revised ADM soluion u 1 = ( + (1 + e, v 1 = ( 6 e 6+ + e ( ; Fig 1: (E, y Fig : (E,y y Solid 1 1 Fig 3: (E, y 3 line = eac soluion, dashed line = sandard NIM, long dashed line = revised ADM, doed line = revised NIM and so on In Fig, Fig, Fig6 and Fig7 we draw 3-erm soluions and eac soluions of (11, i is clear from figures ha he 3-erm soluions are in agreemen wih he eac soluions E: Consider sysem represening nonlinear chemical reacion [1] y 1 = y 1, y 1 ( = 1, y = y 1, y ( =, y 3 =, y 3 ( = (13 The sysem (13 is equivalen o y 1 = 1 y = y 3 = y 1 d; y 1 d y d y d; Inernaional Scholarly and Scienific Research & Innovaion 6( scholarwaseorg/19997/19

4 Applying revised NIM, we ge y 1, =1,y, =,y 3, =; y 1,1 =, y,1 = (, y 3,1 = 3 6 ( 1 +3 ; y 1, =,y, = 3 6 (1 1 +3, y 3, = 8316 ( , Fig8, Fig9 and Fig1 represens he 3-erm soluions of (13 Noe ha hese graphs are in agreemen wih he graphs given in [1] 1 u - 1 u Fig : (E3, 3-erm soluion u Fig : (E3, Eac soluion u = e 1 v - 1 v Fig 6: (E3, 3-erm soluion v Fig 7: (E3, Eac soluion v = e Fig 8: (E, 3-erm soluion y Fig 9: (E, 3-erm soluion y Inernaional Scholarly and Scienific Research & Innovaion 6( scholarwaseorg/19997/19

5 1 y Fig 1: (E, 3-erm soluion y Fig 11: (E, -erm soluions E: Consider he sysem of nonlinear fracional differenial equaions D 13 y 1 = y 1 +, y 1 ( =, y 1( = 1, D y = y 1 +y, y ( =, y ( = y ( = 1 (1 In view of (3, his sysem is equivalen o he following sysem of equaions y 1 = + I 13 ( + ; y = + + I (y 1 +y (1 Applying revised NIM o (1 we ge y 1, =, y, = + ; y 1,1 = , y,1 = ; y 1, = , y, = , and so on Fig11 represens he -erm approimae soluions of (1 E6: Consider he sysem of nonlinear fracional differenial equaions D α y 1 = y 1 + y y 3, y 1 ( = 1, D α y = y y 3, y ( =, D α y 3 =, y 3 ( =, <α 1 (16 Applying (3, we ge equivalen sysem of inegral equaions In view of revised NIM, y 1 = 1+I α ( y 1 + y y 3 ; y = +I α ( y 3 ; y 3 = I α ( y y 1, =1,y, =,y 3, =; α y 1,1 = Γ(α +1, y,1 = 8α Γ(α +1, α y 3,1 = (1 3 α π α Γ(α +1 Γ(α + + +α α Γ(α +, πγ(1 + 3α and so on VI CONCLUSIONS In his aricle a modificaion of NIM, ermed as revised NIM has been presened I has been applied successfully o solve a variey of problems formulaed in erms of sysems of funcional equaions Revised NIM gives series soluion which converges faser relaive o he series obained by NIM The soluions obained are highly in agreemen wih he eac soluions ACKNOWLEDGMENT V Dafardar-Gejji acknowledges he Deparmen of Science and Technology, N Delhi, India for he Research Grans [Projec No SR/S/HEP-/9] REFERENCES [1] I Podlubny, Fracional Differenial Equaions, Academic Press, San Diego, 1999 [] L Debnah, In J Mah and Mah Sci, 3, 1(3 [3] G Adomian, Solving Fronier Problems of Physics: The Decomposiion Mehod, Kluwer, 199 [] J H He, Compu Meh Appl Mech Eng, 167, 7(1998 [] J H He, Compu Meh Appl Mech Eng, 178, 7(1999 [6] V Dafardar-Gejji, H Jafari, J Mah Anal Appl, 316, 73(6 [7] S Bhalekar, V Dafardar-Gejji, Solving Riccai differenial equaions of fracional order using he new ieraive mehod, (submied for publicaion [8] S Bhalekar, V Dafardar-Gejji, New Ieraive Mehod: Applicaion o Parial Differenial Equaions, (submied for publicaion [9] S G Samko, A A Kilbas, O I Marichev, Fracional Inegrals and Derivaives: Theory and Applicaions, Gordon and Breach, Yverdon, 1993 [1] H Jafari, V Dafardar-Gejji, Appl Mah Compu, 181, 98(6 [11] A M Wazwaz, Compu Mah Appl,, 89(7 [1] DD Ganji, M Nourollahi, E Mohseni, Compu and Mah wih Appl, (In press, doi:1116/jcamwa6178 Inernaional Scholarly and Scienific Research & Innovaion 6( scholarwaseorg/19997/19