SOLVING NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS USING THE NDM

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1 Jornal of Applied Analyi and Comptation Volme 5, Nmber, Febrary 205, Webite: doi:0.948/ SOLVING NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS USING THE NDM Mahmod S. Rawahdeh and Sheh Maitama Abtract In thi reearch paper, we examine a novel method called the Natral Decompoition Method (NDM). We e the NDM to obtain exact oltion for three different type of nonlinear ordinary differential eqation (NLODE). The NDM i baed on the Natral tranform method (NTM) and the Adomian decompoition method (ADM). By ing the new method, we cceflly handle ome cla of nonlinear ordinary differential eqation in a imple and elegant way. The propoed method give exact oltion in the form of a rapid convergence erie. Hence, the Natral Decompoition Method (NDM) i an excellent mathematical tool for olving linear and nonlinear differential eqation. One can conclde that the NDM i efficient and eay to e. Keyword Natral tranform, Smd tranform, Laplace tranform, Adomian decompoition method, ordinary differential eqation. MSC(2000) 35Q6, 44A0, 44A5, 44A20, 44A30, 44A35, 8V0.. Introdction Nonlinear differential eqation have received a coniderable amont of interet de to it broad application. Nonlinear ordinary differential eqation play an important role in many branche of applied and pre mathematic and their application in engineering, applied mechanic, qantm phyic, analytical chemitry, atronomy and biology. From lat decade, reearcher pay attention toward analytical and nmerical oltion of nonlinear ordinary differential eqation. Therefore, it become increaingly important to be familiar with all traditional and recently developed method for olving linear and nonlinear ordinary differential eqation. We preent a new integral tranform method called the Natral Decompoition Method (NDM) 29, and apply it to find exact oltion to nonlinear ODE. There are many integral tranform method 3, 3 9 exit in the literatre to olve ODE. The mot ed one i the Laplace tranformation 30. Other method ed recently to olve PDE and ODE, ch a, the Smd tranform 6, the Redced Differential Tranform Method (RDTM) and the Elzaki tranform 4 9. Fethi Belgacem and R. Silambaraan, 2, ed the N Tranform to olve the Maxwell eqation, Beel differential eqation and linear and nonlinear Klein Gordon Eqation and more. Alo, Zafar H. Khan and Waqar A. Khan 2, ed the correponding athor. addre: malrawahdeh@jt.ed.jo(m. Rawahdeh) Department of Mathematic and Statitic, Jordan Univerity of Science and Technology, P.O.Box 3030, Irbid 220, Jordan

2 78 M. Rawahdeh & S. Maitama the N Tranform to olve linear differential eqation and they preented a table with ome propertie of the N Tranform of different fnction. We preent everal application in the field of Phyic and Engineering to how the efficiency and the accracy of the NDM. The Adomian decompoition method (ADM),2, propoed by George Adomian, ha been applied to a wide cla of linear and nonlinear PDE. For the nonlinear model, the NDM how reliable relt in pplying exact oltion and analytical approximate oltion that converge rapidly to the exact oltion. Or aim in thi paper i to develop an efficient algorithm for nmerical comptation by natral decompoition method for ch problem. The natral decompoition method provide oltion a rapidly convergent erie. In thi paper, we olve the following NLODE: Firt, conider the nonlinear econd order differential eqation of the form: d 2 v dt 2 + bject to the initial condition ( ) 2 dv + v 2 (t) = in(t), (.) dt v(0) = 0, v (0) =. (.2) Second, the firt order nonlinear ordinary differential eqation of the form: bject to the condition dv dt = v2 (t), (.3) v(0) = 0. (.4) Third, the nonlinear Riccati differential eqation of the form: bject to the condition dv dt = t2 + v 2 (t), (.5) v(0) = 0. (.6) The ret of thi paper i organized a follow: In Section 2 and 3, we give ome backgrond material abot the NDM. In ection 4, we explain the methodology of the NDM. In ection 5, we apply the NDM to three tet problem to how the effectivene of or method. Section 6 i for dicion and conclion of thi paper. 2. Baic Idea of The Natral Tranform Method In thi ection, we preent ome backgrond abot the natre of the Natral Tranform Method (NTM). Ame we have a fnction f(t), t (, ), and then the general integral tranform i defined a follow, 2: I f(t) () = K(, t) f(t) dt, (2.) where K(, t) repreent the kernel of the tranform, i the real (complex) nmber which i independent of t. Note that when K(, t) i e t, t J n (t) and t (t),

3 Solving NLODE ing the NDM 79 then Eq. (2.) give, repectively, Laplace tranform, Hankel tranform and Mellin tranform. Now, for f(t), t (, ) conider the integral tranform defined by: and I f(t) () = I f(t) (, ) = K(t) f(t) dt, (2.2) K(, t) f(t) dt. (2.3) It i worth mentioning here when K(t) = e t, Eq. (2.2) give the integral Smd tranform, where the parameter replaced by. Moreover, for any vale of n the generalized Laplace and Smd tranform are repectively defined by, 2: and l f(t) = F () = n e n+t f( n t) dt, (2.4) 0 S f(t) = G() = n e nt f(t n+ ) dt. (2.5) Note that when n = 0, Eq. (2.4) and Eq. (2.5) are the Laplace and Smd tranform, repectively. 3. Definition and Propertie of the N Tranform The natral tranform of the fnction f(t) for t (, ) i defined by, 2: N f(t) = R(, ) = 0 e t f(t) dt;, (, ), (3.) where N f(t) i the natral tranformation of the time fnction f(t) and the variable and are the natral tranform variable. Note that Eq. (3.) can be written in the form 4, 5: N f(t) = = 0 e t f(t) dt;, (, ) e t f(t) dt;, (, 0) + e t f(t) dt;, (0, ) = N f(t) + N + f(t) = N f(t)h( t) + N f(t)h(t) = R (, ) + R + (, ), where H(.) i the Heaviide fnction. It hold be mentioned here, if the fnction f(t)h(t) i defined on the poitive real axi, with t R, then we define the Natral tranform (N Tranform) on the et { f(t) : M, τ, τ A = 2 > 0, ch that f(t) < Me t } τ j, if t ( ) j 0, ), j Z + 0

4 80 M. Rawahdeh & S. Maitama a: N f(t)h(t) = N + f(t) = R + (, ) = 0 e t f(t) dt;, (0, ), (3.2) where H(.) i the Heaviide fnction. Note if =, then Eq. (3.2) can be redced to the Laplace tranform and if =, then Eq. (3.2) can be redced to the Smd tranform. Now we give ome of the N Tranform and the converion to Smd and Laplace, 2. Table. Special N Tranform and the converion to Smd and Laplace f(t) N f(t) S f(t) l f(t) t 2 2 e at a a a t n (n )!, n=, 2,... n n n n in(t) Remark 3.. The reader can read more abot the Natral tranform in, 2. Now we give ome important propertie of the N Tranform are given a follow, 2, 20, 2: Table 2. Propertie of N Tranform Fnctional Form Natral Tranform y(t) Y (, ) y(at) ay (, ) y (t) y(0) Y (, ) y (t) 2 Y (, ) 2 y(0) y (0) 2 γ y(t) ± β v(t) γ Y (, ) ± β V (, ) 4. The Natral Decompoition Method In thi ection, we illtrate the applicability of the Natral Decompoition Method to ome nonlinear ordinary differential eqation. Methodology of the NDM:

5 Solving NLODE ing the NDM 8 Conider the general nonlinear ordinary differential eqation of the form: bject to the initial condition Lv + R(v) + F (v) = g(t), (4.) v(0) = h(t), (4.2) where L i an operator of the highet derivative, R i the remainder of the differential operator, g(t) i the nonhomogeneo term and F (v) i the nonlinear term. Sppoe L i a differential operator of the firt order, then by taking the N Tranform of Eq. (4.), we have: V (, ) V (0) + N+ R(v) + N + F (v) = N + g(t). (4.3) By btitting Eq. (4.2) into Eq. (4.3), we obtain: V (, ) = h(t) + N+ g(t) N+ R(v) + F (v). (4.4) Taking the invere of the N Tranform of Eq. (4.4), we have: v(t) = G(t) N N+ R(v) + F (v), (4.5) where G(t) i the orce term. We now ame an infinite erie oltion of the nknown fnction v(t) of the form: v(t) = v n (t). (4.6) Then by ing Eq. (4.6), we can re-write Eq. (4.5) in the form: v n (t) = G(t) N N+ R v n (t) + A n (t), (4.7) where A n (t) i an Adomian polynomial which repreent the nonlinear term. Comparing both ide of Eq. (4.7), we can eaily bild the recrive relation a follow: v 0 (t) = G(t), v (t) = N N+ Rv 0 (t) + A 0 (t), v 2 (t) = N N+ Rv (t) + A (t), v 3 (t) = N N+ Rv 2 (t) + A 2 (t). Eventally, we have the general recrive relation a follow: v n+ (t) = N N+ Rv n (t) + A n (t), n 0. (4.8) Hence, the exact or approximate oltion i given by: v(t) = v n (t). (4.9)

6 82 M. Rawahdeh & S. Maitama 5. Worked Example In thi ection, we employ the NDM to three phyical application and then compare or oltion to exiting exact oltion. Example 5.. Conider the firt order nonlinear differential eqation of the form: d 2 ( ) 2 v dv dt v 2 (t) = in(t), (5.) dt bject to the initial condition v(0) = 0, v (0) =. (5.2) We begin by taking the N tranform to both ide of Eq. (5.), we obtain: (dv 2 ) 2 V (, ) V (0) 2 2 v (0) + N+ + N + v 2 (t) = dt (5.3) By btitting Eq. (5.2) into Eq. (5.3) we obtain: (dv ) 2 V (, ) = N+ + v (t) 2. (5.4) dt Then by taking the invere N Tranform of Eq. (5.4), we have: (dv v(t) = t2 2 ) 2 + in(t) N 2! 2 N+ + v (t) 2. (5.5) dt We now ame an infinite erie oltion of the nknown fnction v(t) of the form: v(t) = v n (t). (5.6) By ing Eq. (5.6), we can re-write Eq. (5.5) a follow: v n (t) = t2 2 + in(t) N 2! 2 N+ A n + B n, (5.7) where A n and B n are the Adomian polynomial of the nonlinear term ( dv dt ) 2 and v 2 (t) repectively. Then by comparing both ide of Eq. (5.7), we can drive the general recrive relation a follow: v 0 (t) = t2 2! + in(t), v (t) = N 2 2 N+ A 0 + B 0, v 2 (t) = N 2 2 N+ A + B, v 3 (t) = N 2 2 N+ A 2 + B 2.

7 Solving NLODE ing the NDM 83 Therefore, the general recrive relation i given by: v n+ (t) = N 2 2 N+ A n + B n, n 0. (5.8) Then by ing the recrive relation derived in Eq. (5.8), we can eaily compte the remaining component of the nknown fnction v(t) a follow: v (t) = N 2 2 N+ A 0 + B 0 = N 2 2 N+ (v 0) 2 + v0 2 = N 2 2 N+ (v 0) 2 + v0 2 = N 2 2 N+ + = N = t2 2! +. Hence, by canceling the noie term that appear between v 0 (t) and v (t), one can ee that the non-canceled term of v 0 (t) till atifie the given differential eqation which lead to an exact oltion of the form: v(t) = in(t). The exact oltion i in cloed agreement with the relt obtained by (ADM) 3. Example 5.2. Conider the firt order nonlinear ordinary differential eqation of the form 3: bject to the initial condition dv dt = v2 (t), (5.9) v(0) = 0. (5.0) Taking the Natral tranform to both ide of Eq. (5.9), we obtain: Sbtitting Eq. (5.0), we obtain: V (, ) V (, ) = N+ v 2 (t). (5.) V (, ) = 2 + Taking the invere Natral tranform of Eq. (5.2), we obtain: N + v 2 (t). (5.2) v(t) = t + N N + v 2 (t). (5.3)

8 84 M. Rawahdeh & S. Maitama We now ame an infinite oltion of the nknown fnction v(t) of the form: v(t) = v n (t). (5.4) Uing Eq. (5.4), we can re-write Eq. (5.3) in the form: v n (t) = t + N N + A n (t), (5.5) where A n (t) i the Adomian polynomial repreenting the nonlinear term v 2 (t). Then from Eq. (5.5), we can generate the recrive relation a follow: v 0 (t) = t, v (t) = N v 2 (t) = N v 3 (t) = N N + A 0 (t), N + A (t), N + A 2 (t). Th, the general recrive relation i given by: v n+ (t) = N N + A n (t), n 0. (5.6) Uing Eq. (5.6), we can eaily compte the remaining component of the nknown fnction v(t) a follow: v (t) = N N + A 0 (t) = N N + v 2 0(t) N + t 2 2 = N 3 = N 4 = 3 t3, v 2 (t) = N N + A (t) = N N + 2v 0 (t)v (t) 2t = N N = N = 2t5 5, v 3 (t) = N N + A 2 (t) = N N + 2v 0 (t)v 2 (t) + v 2 (t) 7t = N N = N = 7t7 35. Then the approximate oltion of the nknown fnction v(t) i given by: v(t) = v n (t) = v 0 (t) + v (t) + v 2 (t) + v 3 (t) + = t + 3 t3 + 2t t Hence, the exact oltion of Eq. (5.9) i given by: v(t) = tan(t). The exact oltion i in cloed agreement with the relt obtained by (ADM) 3.

9 Solving NLODE ing the NDM 85 Example 5.3. Conider the Riccati differential eqation of the form 3: dv dt = t2 + v 2 (t), (5.7) bject to the initial condition v(0) = 0. (5.8) Taking the N Tranform to both ide of Eq. (5.7), we obtain: V (, ) v(0) By btitting Eq. (5.8) into Eq. (5.9), we obtain: = N+ v 2 (t). (5.9) v(, ) = N+ v 2 (t). (5.20) Taking the invere N Tranform of Eq. (5.20), we have: v(t) = t t3 3 + N N+ v 2 (t). (5.2) We now ame an infinite erie oltion of the nknown fnction v(t) of the form: v(t) = v n (t). (5.22) Then by ing Eq. (5.22), we can re-write Eq. (5.2) in the form: v n (t) = t t3 3 + N N+ A n (t), (5.23) where A n i the Adomian polynomial which repreent the nonlinear term v 2 (t). By comparing both ide of Eq. (5.23), we can eaily bild the general recrive relation a follow: v 0 (t) = t t3 3, v (t) = N N+ A 0 (t), v 2 (t) = N N+ A (t), v 3 (t) = N N+ A 2 (t). Then the general recrive relation i given by: v n+ (t) = N N+ A n (t). (5.24)

10 86 M. Rawahdeh & S. Maitama By ing Eq. (5.24), we can eaily compte the remaining component of the nknown fnction v(t) a follow: v (t) = N N+ A 0 (t) = N N+ v0(t) 2 ( ) 2 = N N+ t t3 3 = N N+ t N N+ t N N+ t 6 2 = N ! 5 N + 9 6! 7 N 4 = t3 3 2t5 5 + t From v (t) it i obvio that one noie term appear in the component v 0 (t). Then by canceling the noie term from v 0 (t), the remaining non-canceled term of v 0 (t) provide with the exact oltion. Thi can eaily be verified by btittion. Therefore, the exact oltion of the given problem i given by: 8 v(t) = t. (5.25) The exact oltion i in cloed agreement with the relt obtained by (ADM) Conclion In thi paper, the Natral Decompoition Method (NDM) wa propoed for olving the Riccati differential eqation and two nonlinear ordinary differential eqation. We cceflly fond exact oltion to all three application. The NDM introdce a ignificant improvement in the field over exiting techniqe. Or goal in the ftre i to apply the NDM to other linear nonlinear differential eqation (PDE, ODE) that arie in other area of cience and engineering. Acknowledgement The athor wold like to thank the Editor and the anonymo referee for their comment and ggetion on thi paper. Reference G. Adomian, Solving frontier problem of phyic: the decompoition method, Klwer Acad. Pbl, G. Adomian, A new approach to nonlinear partial differential eqation, J. Math. Anal. Appl., 02(984), Sh. Sadigh Behzadi and A. Yildirim, Nmerical oltion of LR fzzy Hnter- Saxeton eqation by ing homotopy analyi method, Jornal of Applied Analyi and Comptation, 2()(202), 0.

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