Approximate Analytical Solution for Quadratic Riccati Differential Equation

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1 Iranian J. of Numerical Analyi and Optimization Vol 3, No. 2, 2013), pp Approximate Analytical Solution for Quadratic Riccati Differential Equation H. Aminikhah Abtract In thi paper, we introduce an efficient method for olving the quadratic Riccati differential equation. In thi technique, combination of Laplace tranform and new homotopy perturbation method LTNHPM) are conidered a an algorithm to the exact olution of the nonlinear Riccati equation. Unlike the previou approach for thi problem, o-called NHPM, the preent method, doe not need the initial approximation to be defined a a power erie. Four example in different cae are given to demontrate implicity and efficiency of the propoed method. Keyword: Riccati differential equation; Laplace tranform method; NHPM; LTNHPM. 1 Introduction The quadraic Riccati differential equation, deriving it name from Jacopo Franceco, Count Riccati ). Thee kind of differential equation are a cla of nonlinear differential equation of much importance, and play a ignificant role in many field of applied cience [1]. At an early tage, the occurrence of uch differential equation in the tudy of Beel function led to it appearance in many related application and, to the preent time, the literature on the Riccati equation ha been extenive. For everal reaon, a Riccati equation comprie of a highly ignificant cla of nonlinear ordinary differential equation. Firtly, thi equation i cloely related to ordinary linear homogeneou differential equation of the econd order. Secondly, the olution of Riccati equation poee a very particular tructure in that the general olution i a fractional linear function of the contant of integration. Thirdly, the olution of an Ricaati equation i involved in the reduction of nth-order linear homogeneou ordinary differential equation. Fourthly, a one-dimenional Schrodinger equation i cloely related to Riccati differential Recieved 18 February 2013; accepted 12 June 2013 H. Aminikhah Department of Applied Mathematic, School of Mathematical Science, Univerity of Guilan, P.O. Box 1914, Raht, Iran. aminikhah@guilan.ac.ir 21

2 22 H. Aminikhah equation. Solitary wave olution of a nonlinear partial differential equation can be expreed a a polynomial in two elementary function atifying a projective Riccati equation [3]. Moreover, uch type of problem alo arie in the optimal control literature, and can be derived in olving a econd-order linear ordinary differential equation with contant coefficient [2]. In conformity with the general tudy of differential equation, much of the early work were concerned with the tudy of particular clae of Riccati differential equation with the aim of determining the olution in finite form. Many mathematician uch a Jame Bernouli, Jahn Bernouli ), Leonhard Euler ), Jean-le-Rondd Alembert ), and Adrian Marine Legendre ) contributed to the tudy of uch differential equation [1]. Deriving analytical olution for Riccati equation in an explicit form eem to be unlikely except for certain pecial ituation. Of coure, having known it one particular olution, it general olution can be eaily derived. Therefore, one ha to go for numerical technique or approximate approache for getting it olution. Recently, Adomian decompoition method ha been propoed for olving Riccati differential equation [7, 8]. HPM wa introduced by He [4] and ha been already ued by many mathematician and engineer to olve variou functional equation [9, 10, 5, 6, 12, 11]. Abbabandy olved a Riccati differential equation uing He VIM, homotopy perturbation method HPM) and iterated He HPM and compared the accuracy of the obtained olution to that derived by Adomian decompoition method [13, 14, 16]. Moreover, Homotopy analyi method HAM) and a piecewie variational iteration method VIM) are propoed for olving Riccati differential equation [16]. In [15], Liao ha hown that HPM equation are equivalent to HAM equation when ħ = 1. Aminikhah and Hemmatnezhad [6] propoed a new homotopy perturbation method NHPM) to obtain the approximate olution of ordinary Riccati differential equation. In thi work, we preent the olution of Riccati equation by combination of placelaplace tranform and new homotopy perturbation method. An important property of the propoed method, which i clearly demontrated in example, i that pectral accuracy i acceible in olving pecific nonlinear Riccati differential equation which have analytic olution function. 2 LTNHPM for quadratic Ricatti equation Conider the nonlinear Riccati differential equation a the following form u t) = At) + Bt)ut) + Ct)u 2 t), 0 t T, u0) = α. where At), Bt) and Ct) are continuou and α i an arbitrary contant. By the new homotopy technique, we contruct a homotopy U : Ω [0, 1] R, 1)

3 Approximate Analytical Solution for which atifie HUt), p) = U t) u 0 t)+pu 0 t) p[at)+bt)ut)+ct)u 2 t)] = 0, 2) where p [0, 1] i an embedding parameter, u 0 t) i an initial approximation of olution of equation 1). Clearly, we have from equation 2) HUt), 0) = U t) u 0 t) = 0, 3) HUt), 1) = U t) At) Bt)Ut) Ct)U 2 t) = 0 4) By applying Laplace tranform on both ide of 2), we have L U t) u 0 t) + pu 0 t) p[at) + Bt)Ut) + Ct)U 2 t)] } = 0 5) or Uing the differential property of Laplace tranform we have LUt)} U0) = L u 0 t) pu 0 t) + p[at) + Bt)Ut) + Ct)U 2 t)] } 6) LUt)} = 1 U0) + L u0 t) pu 0 t) + p[at) + Bt)Ut) + Ct)U 2 t)] }} By applying invere Laplace tranform on both ide of 7), we have Ut) = L 1 1 U0) + Lu 0t) pu 0 t) + p[at) + Bt)Ut) + Ct)U 2 t)]}}} 8) According to the HPM, we ue the embedding parameter p a a mall parameter, and aume that the olution of equation 8) can be repreented a a power erie in p a Ut) = pn U n. Now let u write the Eq. 8) in the following form p n U n t) =L 1 1 U0) + L u 0 t) pu 0 t) [ + p At) + Bt) 7) p n U n t) + Ct) p n U n t)) 2]})} 9) Comparing coefficient of term with identical power of p lead to p 0 : U 0 t) = L 1 1 U0) + L u 0t)}) } p 1 : U 1 x) = L 1 1 L u0 t) At) Bt)U 0 t) Ct)U0 2 t) })} p j : U j x) = L 1 1 L Bt)U j 1 t) + Ct) })} j 1 k=0 U kt)u j k 1 t), j = 2, 3,... 10)

4 24 H. Aminikhah Suppoe that the initial approximation ha the form U0) = u 0 t) = α, therefore the exact olution may be obtained a following ut) = lim p 1 Ut) = U 0 t) + U 1 t) + 11) 3 Example Example 1. Conider the following quadratic Riccati differential equation taken from [1] u t) = 16t tut) + u 2 t), 12) u0) = 1 The exact olution of above equation wa found to be of the form ut) = 1 4t. 13) To olve equation 12) by the LTNHPM, we contruct the following homotopy U t) u 0 t) + p [ u 0 t) t 2 8tUt) U 2 t) ] = 0 14) Applying Laplace tranform on both ide of 14), we have L U t) u 0 t) + p [ u 0 t) t 2 8tUt) U 2 t) ]} = 0 15) Uing the differential property of Laplace tranform we have or LUt)} U0) = L u 0 t) p [ u 0 t) t 2 8tUt) U 2 t) ]} 16) LUt)} = 1 U0) + L u0 t) p [ u 0 t) t 2 8tUt) U 2 t) ]}} By applying invere Laplace tranform on both ide of 17), we have Ut) = L ) U0) + L u0 t) p [ u 0 t) t 2 8tUt) U 2 t) ]})} Suppoe the olution of equation 18) to have the following form 18) Ut) = U 0 t) + pu 1 t) + p 2 U 2 t) +, 19) where U i t) are unknown function which hould be determined. Subtituting equation 19) into equation 18), collecting the ame power of p and equating each coefficient of p to zero, reult in

5 Approximate Analytical Solution for p 0 : U 0 t) = L 1 1 U0) + L u 0t)}) } p 1 : U 1 x) = L 1 1 L u0 t) t 2 8tU 0 t) U0 2 t) })} p j : U j x) = L 1 1 L 8tU j 1 t) + })} j 1 k=0 U kt)u j k 1 t) j = 2, 3,... 20) Auming u 0 t) = U0) = 1, and olving the above equation for U j t), j = 0, 1, lead to the reult U 0 t) = 1 + t, U 1 t) = 5t 3 5t 2 + 3t 3 ), U 2 t) = 5t2 3 10t t 2 8t 3 ), U 3 t) = 5t t t 3 399t 2 399t + 63 ), U 4 t) = t t t t t t ),. Therefore we gain the olution of Eq. 12) a which i exact olution. ut) = U 0 t) + U 1 t) + U 3 t) + = 1 4t Example 2. Conider the following quadratic Riccati differential equation taken from [13, 7, 14, 6, 8] u t) = 1 + 2ut) u 2 t), 21) u0) = 0. The exact olution of above equation wa found to be of the form ut) = 1 + [ )] 2t tanh + 2 log ) The Taylor expanion of ut) about t = 0 give ut) = t + t t3 1 3 t t t t t ) To olve equation 21), by the LTNHPM, we contruct the following homotopy U t) = u 0 t) p [ u 0 t) 1 2Ut) + U 2 t) ] 24) Applying Laplace tranform, we have L U t) u 0 t) + pu 0 t) p[1 + 2Ut) U 2 t)] } = 0 25) Uing the differential property of Laplace tranform we have

6 26 H. Aminikhah LUt)} U0) = L u 0 t) p [ u 0 t) 1 2Ut) + U 2 t) ]} 26) or LUt)} = 1 U0) + L u0 t) p [ u 0 t) 1 2Ut) + U 2 t) ]}} 27) By applying invere Laplace tranform on both ide of 27), we have 1 Ut) = L 1 U0) + L u0 t) p [ u 0 t) 1 2Ut) + U 2 t) ]})} 28) Suppoe the olution of equation 28) to have the following form Ut) = U 0 t) + pu 1 t) + p 2 U 2 t) +, 29) where U i t) are unknown function which hould be determined. Subtituting equation 29) into equation 28), collecting the ame power of p, and equating each coefficient of p to zero, reult in p 0 : U 0 t) = L 1 1 U0) + L u 0t)}) } p 1 : U 1 x) = L 1 1 p j : U j x) = L 1 1 L u0 t) 1 2U 0 t) + U0 2 t) })} 2U j 1 t) + j 1 k=0 U kt)u j k 1 t) L })}, j = 2, 3,... 30) Auming u 0 t) = U0) = 0, and olving the above equation for U j t), j = 0, 1, lead to the reult U 0 t) = 0, U 1 t) = t, U 2 t) = t 2, U 3 t) = t3 3, U 4 t) = t4 3, U 5 t) = 7t5 15,. Therefore we gain the olution of equation 21) a ut) = U 0 t) + U 1 t) + U 3 t) + = t + t t3 1 3 t t t t7 +, 31) 32) and thi in the limit of infinitely many term, yield the exact olution of 21). Example 3. Conider the following quadratic Riccati differential equation taken from [1] u t) = e t e 3t + 2e 2t ut) e t u 2 t), 33) u0) = 1.

7 Approximate Analytical Solution for The exact olution of thi equation i ut) = e t. We contruct the following homotopy [ ] U t) u 0 t) + p u 0 t) e pt + e 3pt 2e 2pt Ut) + e pt U 2 t) = 0 34) Applying Laplace tranform on both ide of 34), we have [ ]} L U t) u 0 t) + p u 0 t) e pt + e 3pt 2e 2pt Ut) + e pt U 2 t) = 0 Uing the differential property of Laplace tranform we have [ LUt)} U0) =L u 0 t) p u 0 t) e pt 35) or LUt)} = 1 U0) + L u 0 t) + e 3pt 2e 2pt Ut) + e pt U t)] } 2 36) [ p u 0 t) e pt + e 3pt 2e 2pt Ut) + e pt U t)]} } 2 37) By applying invere Laplace tranform on both ide of 37) and uing the Taylor erie of e αt = αt) n n!, we have Ut) =L 1 1 [ U0) + L u 0 t) p u 0 t) p n tn n! + p n 3t)n n! 2 p n 2t)n n! Ut) + ]} )} p n tn n! U 2 t) 38) Suppoe the olution of U can be expanded into infinite erie a Ut) = pn U n t). Subtituting Ut) into equation 38), collecting the ame power of p, and equating each coefficient of p to zero, reult in

8 28 H. Aminikhah p 0 : U 0 t) = L 1 1 U0) + L u 0t)}) } p 1 : U 1 x) = L })} 1 1 L u0 t) 2U 0 + U0 2 p 2 : U 2 x) = L })} 1 1 L 2t 4tU0 2U 1 + tu U 0 U 1 p 3 : U 3 x) = L 1 1 L4t2 4tU 1 2U 2 4t 2 U 0 + 2tU 0 U 1 39) +U U 0 U t2 U0 2 })}. Auming u 0 t) = U0) = 1, and olving the above equation for U j t), j = 0, 1, lead to the reult U 0 t) = 1 + t, U 1 t) = t3 3, U 2 t) = ) t t + t 2, U 3 t) = t t + 504t 2 840t 3 240t t 5). 40) Therefore we gain the olution of equation 33) a ut) = U 0 t) + U 1 t) + U 3 t) + = 1 + t + 1 2! t ! t3 + = tn n! = et, 41) which i exact olution. Example 4. Conider the following quadratic Riccati differential equation taken from [13, 7, 14, 6, 8] u t) = ut) + u 2 t), u0) = 1 42) 2 The exact olution of above equation wa found to be of the form ut) = The Taylor expanion of ut) about t = 0 give ut) = t t t t t9 + e t 1 + e t 43) t11. 44) To olve equation 43) by the LTNHPM, we contruct the following homotopy: U t) = u 0 t) p [ u 0 t) + Ut) U 2 t) ] 45) By applying Laplace tranform we get LUt)} = 1 U0) + L u0 t) p [ u 0 t) + Ut) U 2 t) ]}} 46)

9 Approximate Analytical Solution for Uing invere Laplace tranform on both ide of 46), we have 1 Ut) = L 1 U0) + L u0 t) p [ u 0 t) + Ut) U 2 t) ]})} 47) Suppoe the olution of equation 47) to have the following form Ut) = U 0 t) + pu 1 t) + p 2 U 2 t) +, 48) Subtituting equation 48) into equation 47), collecting the ame power of p, and equating each coefficient of p to zero, reult in p 0 : U 0 t) = L 1 1 U0) + L u 0t)}) } p 1 : U 1 x) = L 1 1 p j : U j x) = L 1 1 L u0 t) + U 0 t) U0 2 t) })} U j 1 t) + j 1 k=0 U kt)u j k 1 t) L })}, j = 2, 3,... 49) Auming u 0 t) = U0) = 1 2, and olving the above equation for U jt), j = 0, 1, lead to the reult U 0 t) = t), U 1 t) = ) t t 2, U 2 t) = ) t t 2, U 3 t) = t t t 4), U 4 t) = t t t 4), U 5 t) = t t t 6),. t ) Therefore we gain the olution of equation 42) a ut) = U 0 t) + U 1 t) + U 3 t) + = t t t t t t11, 51) thi in the limit of infinitely many term, yield the exact olution of 42). 4 Concluion In the preent work, we propoed a combination of Laplace tranform method and homotopy perturbation method to olve nonlinear Riccati differential equation. The new method developed in the current paper wa teted on everal example. The obtained reult how that thee approache can olve the problem effectively. Unlike the previou approach implemented by the

10 30 H. Aminikhah preent author, o-called NHPM, the preent technique, doe not need the initial approximation to be defined a a power erie. In the NHPM we reach to a et of recurrent differential equation which mut be olved conecutively to give the approximate olution of the problem. Sometime we have to do many computation in order to reach to the higher order of approximation with acceptable accuracy. Alo a an advantage of the LTNHPM over decompoition procedure of Adomian, the former method provide the olution of the problem without calculating Adomian polynomial. The advantage of the LTNHPM over numerical method finite difference, finite element,... ) and ADM i that it olve the problem without any need to dicretization of the variable. The Computation finally lead to a et of nonlinear equation with one unpecified value in each equation. The reult how that the LTNHPM i an effective mathematical tool which can play a very important role in nonlinear cience. Reference 1. Abbabandy, S. A new application of He variational iteration method for quadratic Riccati differential equation by uing Adomian polynomial, J. Comput. Appl. Math., ), Abbabandy, S. Homotopy perturbation method for quadratic Riccati differential equation and comparion with Adomian decompoition method, Appl. Math. Comput., ), Abbabandy, S. Iterated He homotopy perturbation method for quadratic Riccati differential equation, Appl. Math. Comput., ), Aminikhah, H. and Biazar, J. A new HPM for ordinary differential equation, Numerical Method for Partial Differential Equation, ), Aminikhah, H. and Biazar, J. A new HPM for ordinary differential equation, Numer. Method Partial Differ. Equat., ), Aminikhah, H. and Hemmatnezhad, M. An efficient method for quadratic Riccati differential equation, Commun. Nonlinear Sci. Numer. Simul., ) Biazar, J. and Aminikhah, H. Study of convergence of homotopy perturbation method for ytem of partial differential, Computer & Mathematic with Application, ), Biazar, J. and Ghazvini, H. Exact olution for nonlinear Schrodinger equation by He homotopy perturbation method, Phyic Letter A, ),

11 Approximate Analytical Solution for Carinena, J. F., Marmo, G., Perelomov, A. M. and Ranada, M. F. Related operator and exact olution of Schrodinger equation, Int. J. Mod. Phy. A, ), El-Tawil, M. A., Bahnaawi, A. A. and Abdel-Naby, A. Solving Riccati differential equation uing Adomian decompoition method, Appl. Math. Comput., ), Ganji, D. D. The application of He homotopy perturbation method to nonlinear equation ariing in heat tranfer, Phyic Letter A, ), Ganji, D. D. and Sadighi, A. Application of homotopy perturbation and variational iteration method to nonlinear heat tranfer and porou media equation, Journal of Computational and Applied Mathematic, ), Geng, F., Lin, Y. and Cui, M. A piecewie variational iteration method for Riccati differential equation, Comput. Math. Appl., ), He, J. H. Homotopy perturbation technique, Computer Method in Applied Mechanic and Engineering, ), Liao, S. Comparion between the homotopy analyi method and homotopy perturbation method, Applied Mathematic and Computation, ), Reid, W. T. Riccati Differential Equation, Academic Pre, New York, Scott, M. R. Invariant Imbedding and it Application to Ordinary Differential Equation, Addion-Weley, Tan, Y. and Abbabandy, S. Homotopy analyi method for quadratic Riccati differential equation, Commun. Non-linear Sci. Numer. Simulation, ),

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