This research develops a new non-standard scheme based on the Adomian decomposition method (ADM) to solve nonlinear equations. The ADM was adopted to solve the nonlinear differential equation resulting from the discretization of the differential equation. The new scheme does not need to linearize or non-locally linearize the nonlinear term of the differential equation. Two examples are given to demonstrate the efficiency of this schem

Alomari .A.K,

Hashim .I,

Moaddy .K,

Momani .S,

UKM Journal Article Repository

Sains Malaysiana 40(5)(2011): 515–519  A New Hybrid Non-standard Finite Difference-Adomian Scheme for Solution of Nonlinear Equations(Skim Hibrid Baru Beza-terhingga Tak Piawai-Adomian bagi Penyelesaian Persamaan Tak Linear)K. MoADDy, I. HASHIM*, A. K. ALoMArI & S. MoMANIABSTrACTThis research develops a new non-standard scheme based on the Adomian decomposition method (ADM) to solve nonlinear equations. The ADM was adopted to solve the nonlinear differential equation resulting from the discretization of the differential equation. The new scheme does not need to linearize or non-locally linearize the nonlinear term of the differential equation. Two examples are given to demonstrate the efficiency of this scheme.Keywords: Adomian decomposition method; Logistic equation; Lotka-Volterra system; non-standard schemesABSTrAKPenyelidikan ini membangunkan satu skim tak piawai baru berdasarkan pada kaedah penguraian Adomian (KPA) bagi menyelesaikan persamaan tak linear. KPA ini diadaptasi untuk menyelesaikan persamaan tak linear yang terhasil daripada pendiskretan persamaan terbitan. Skim baru ini tidak perlu melinearkan atau melinearkan secara tak setempat sebutan tak linear persamaan terbitan itu. Dua contoh diberi untuk medemonstrasikan keefisienan skim ini.Kata kunci: Kaedah penguraian Adomian; persamaan logistik; skim tak piawai; sistem Lotka-VolterraINTroDuCTIoNOne of the shortcomings of the standard finite difference method is that the qualitative properties of the exact solution usually are not transferred to the numerical solution. Furthermore, many problems may affect the stability properties of the standard approach. Also, in practice, using the standard method the limit of the step-size is not reached. What we obtain is the numerical solution for one or several values of the step-size (Ibijola et al. 2008). Non-standard finite difference schemes (NSFD) have emerged as an alternative method for solving a wide range of problems whose mathematical models involve algebraic, differential and biological models as well as chaotic systems (Mickens 2005). These techniques have many advantages over classical techniques and provide an efficient numerical solution. In fact, the non-standard finite difference method is an extension of the standard finite difference method. Non-standard schemes as introduced by Mickens (1989,1990,1994) are used to help resolve some of the issues related to numerical instabilities. Furthermore, Mickens (1999,2000,2005) introduced certain rules for obtaining the best difference equations, one of the most important of which is that the nonlinear terms of f (t, y(t))  are approximated in a non-local form.  If we do not linearize non-locally the system of differential equations, a somewhat better method is chosen - the Newton iteration method-to numerically solve the algebraic equation. This requires that f (t, y(t)) be smooth and that the inverse of the derivative operator fy exist. Furthermore, to solve a system of equations, Newton’s method might require a long time, so it is not economical. In this paper, a new non-standard finite difference scheme is presented in which the non-local linearization step is ignored when using the Adomian decomposition method (ADM) (Adomian 1988) to solve the nonlinear algebraic equations. The decomposition method yields rapidly convergent series solutions for both linear and nonlinear deterministic and stochastic equations (Cherruault &  Adomian 1993). The technique has several advantages over the classical techniques, mainly it avoids discretization and provides an efficient numerical solution with high accuracy, minimal calculations and avoidance of physically unrealistic assumptions. However, the ADM can be used to solve this problem effectively (Momani et al. 2006). The rest of the paper is organized as follows. In the next section we present the non-standard finite difference method (NSFD) for solution of a system of first-order differential equations. Then we briefly describe ADM for systems of nonlinear algebraic equations. Next, we merge the NSFD and ADM to develop the non-standard scheme based on Adomian decomposition method to solve a system of nonlinear differential equations. Logistic equations and the Lotka-Volterra system are considered as test examples, and we discuss numerical approximations to the solutions. In the last section we summarize the conclusions.516 NoN-STANDArD FINITE DIFFErENCE METHoDWe seek to obtain the NSFD (Mickens 2000) solution for a system of differential equations of the form  = f (t, y1, y2, …, ym),   k = 1, 2, …, m,  (1)where )(,( tytf k  is the nonlinear term in the differential equation. Using the finite difference method we have  (2)   (3)                M  (4)where φk and ψk are functions of the step size h = Δt. The ψk and φk have the following properties:  ψk  = 1 + o(h), (5) φk (h,λ) = h + o(h2), (6)where h → 0 and λ  is fixed. The numerator functions ψk  are usually equal to one (Lubuma & Patidar 2005), unless the system has dissipation. Examples of functions φk (h, λ) that satisfy (6) are h, sin (h), sinh (h),  Non-linear terms can in general be replaced by nonlocal discrete representations, for example,   (7)  (8) Let h = T/N, tn =  nh, n = 0, 1, …, N ∈ Z+. Then (1) can be discretized as follows:  yk(tn+1) = ψk(h)yk(tn) + φk(h)f (tn+1, yk(tn+1), yk(tn)). (9)where f (tn+1, yk(tn+1), yk(tn)) is the product the non-local linearization of f(tn+1, yk(tn+1)). ADoMIAN DECoMPoSITIoN METHoD (ADM)The properties of ADM can found in (Vahidi et al. 2009). Consider the following system of nonlinear equations:  gi(u1, u2, …, un) = 0,   i = 1, 2, …, n, (10)where gi : ℜn → ℜ. Equation (10) can be written in the form:   ui = vi,0 + Ni,0(u1, …, un)   i = 1, 2, …, n, (11)where vi,0 are constants and Ni,0 are nonlinear operators. The decomposition method allows a solution to equations having the series form:    (12) The nonlinear operators Ni,0 are decomposed as an infinite series called Adomian polynomials:    (13)where Ai,j depends upon u1,0, u1,1, …, u1,j, u2,0, u2,1, …, u2,j, un,1, …, un,j. In view of (12) and (13) one has:    (14)which yields:  (15)where λ is the parameter introduced for convenience to represents. Therefore, (11) can be rewritten as:   (16) The Adomian decomposition method identifies ui,j, j ≥ 0 by the following recursive relation: vi,0 = vi,0, (17)  ui,j+1 = Ai,j,   i = 1, 2, …, n,   j = 1, 2,…. (18) The solution ui,j can be approximated by the truncated series:    (19)such that:     (20)THE NuMErICAL ALgorITHM BASED oNADoMIAN DECoMPoSITIoNApplying the first derivatives developed by Mickens (1994) to the solution of a system of nonlinear differential equations in (1) gives:   (21)  (22) M  (23)  517where the functions φk(h) satisfy the condition in (6). Solving the above system for yi,k+1, i = 1,2,…,m yields   (24)  (25) M  (26)  We now use ADM to solve this system of algebraic equations. Assume:   (27)where:  u0 = yk, (28) u1 = φ1(h) f (A0), (29) u2 = φ2(h) f (A1), (30) M un = φn(h) f (An–1). (31) For the n-term of the ADM solution we have   (32)APPLICATIoN AND rESuLTSSoLVINg A LogISTIC EquATIoN uSINg NSFD-ADMWe use the NSFD-ADM technique to solve a logistic equation (Mickens 1994) of the form:   (33)with the initial condition y(0) = 0.5. The exact solution of this equation is:   (34) The new scheme (NSFD-ADM) for solution of the logistic equation is:    (35)when φ(h) = 1 – e–h (Mickens, 2006). Solving Equation (35) for yk+1 gives:  yk+1 = yk + φ(h)yk+1 – φ(h)(yk+1)2. (36)using ADM to solve Equation (36) yields:    (37)where:  u0 = yk, (38)  u1 = φ(h)(u0 – u02), (39) u2 = φ(h)(u1 – 2u0u1), (40)  M  (41)SoLVINg THE LoTKA-VoLTErrA SySTEM uSINg NSFD-ADMThe prey-predator differential equation system takes the form:  x´ = ax – bxy, (42) y´ = –cy + dxy,  (43)where a,b,c,d are positive parameters. For the initial conditions x(0) = x0 > 0 and y(0) = y0 > 0, all solutions except the fixed point at  are periodic (Mickens, 2003). In the Lotka-Volterra system, the values for the parameters (a,b,c,d)  in the prey-predator system (42) and (43) are all taken to be one, yielding:  x´ = x – xy,  (44) y´ = –y + xy.  (45) We now illustrate Mickens’s scheme (Mickens 2003) and our scheme to solve the Lotka-Volterra system. 1. The finite difference scheme proposed by Mickens (NSFD) to solve this system is to take:   (46)  (47) where φ(h) = sin h. 2. The new non-standard scheme incorporating Adomian decomposition (NSFD-ADM) is to take:    (48)  (49) 518  where φ(h) = sin h. Solving (48) and (49) for xk+1 and yk+1, respectively, giving:  xk+1 = xk + φ(h)(xk+1 – xk+1yk+1), (50) yk+1 = yk + φ(h)(–yk+1 + xk+1yk+1). (51)  using ADM to solve (50) and (51) yields   (52)  where:  u0 = xk,   v0 = yk, (53) u1 = φ(h)(u0 – u0v0),  v1 = φ(h)(–v0 + u0v0), (54)  M  (55)  (56)rESuLTS AND DISCuSSIoNThe Lotka-Volterra system in (44) and (45) was numerically integrated using the NSFD-ADM scheme as coded in the computer algebra package Maple. In Maple, the number of variable digits controlling the number of significant digits is set to 35. In all the calculations done in this paper, we set the parameters (a,b,c,d) of the Lotka-Volterra system equal to one with initial conditions x(0) = 20, y(0) = 1.  Table 1 shows the accuracy of the NSFD-ADM for solution of the logistic equation. Comparing NSFD-ADM rersults with the exact solution, we see that the maximum difference between the NSFD-ADM solution and the exact solution at time steps Δt = 0.05 and Δt = 0.01 is of the order of magnitude of 10–6. Thus we can conclude that the NSFD-ADM solutions for the time step Δt = 0.01 is sufficiently accurate for our comparison purposes.  In Table 2 we present the absolute errors between NSFD and NSFD-ADM solutions and the fourth-order runge-Kutta method (rK4) solutions at time step Δt = 0.01 for the Lotka-Volterra system. It was found that the NSFD-ADM solutions agree very well with the rK4 solutions for t up to t = 20. We note that increasing the number of terms improves the accuracy of the NSFD-ADM solutions. TABLE 2. Differences between the rK4 solution for the Lotka-Volterra system and solutions obtained using 6-term NSFD-ADM and NSFDt  Δ=|rK40.01–NSFD0.01| Δ=|rK40.01–NSFD–ADM0|  Δx   Δy   Δx   Δy 2  1.510E-06  1.635E-02  4.285E-07  4.201E-02 4  4.001E-07  7.449E-03  1.229E-07  5.291E-04 6  1.813E-06  1.724E-03  5.848E-07  6.333E-04 8  1.217E-05  3.312E-04  4.117E-06  1.821E-04 10  8.623E-05  5.831E-05  3.058e-05  3.785E-05 12  6.153E-04  9.821E-06  2.289E-04  6.959E-06 14  4.395E-03  1.679E-06  1.714E-03  1.225E-06 16  3.139E-02  3.610E-07  1.284E-02  2.380E-07 18  0.224  1.662E-07  9.62E-02  7.972E-08 20  1.6  3.496E-07  0.720  9.166E-08TABLE 1. Absolute errors between the exact solution and the 6-term NSFD-ADM solution for the logistic equationt |Exact–NSED–ADM0.05| |Exact–NSFD–ADM0.01|  1 2 3 4 5 6 7 8 9 10  5.975E-03  7.450E-03 5.372E-03 3.035E-03 1.519E-03  7.110 E-04  3.195E-04  1.398 E-04  5.998E-05  2.536E-05 1.214E-03  1.502E-03  1.066E-03  5.913E-04  2.902E-04  1.332E-04  5.868E-05 2.516E-05 1.058E-05 4.383E-06 The x -y phase portraits obtained using the NSFD-ADM, NSFD and rK4 solutions at Δt = 0.01 for the Lotka-Volterra system are shown in Figure 1. In this case, we see that the NSFD-ADM has the advantage over the NSFD on achieving a good accuracy with different time steps. xyFIgurE 1. Numerical solution for the Lotka-Volterra system using rK4, NSFD and 6-term NSFD-ADM where h = 0.01, x(0) = 20 and y(0) = 1 In Figure 2, xk and yk are shown to be periodic, and the corresponding xk and yk phase-space curves were closed for the Lotka-Volterra system with initial conditions x(0) = 0.1, y(0) =1  for time step Δt = 0.001.   519CoNCLuSIoNSIn this paper, we derived a hybrid NSFD-ADM algorithm for nonlinear differential equations. Non-local linearization procedure of the nonlinear terms in NSFD was replaced with the Adomian polynomials. Solutions to the logistic equation and Lotka-Volterra system were presented to demonstrate the efficiency of the new scheme.ACKNoWLEDgMENTSAuthors gratefully acknowledge the financial support received from the ZAMALAH scheme of universiti Kebangsaan Malaysia and the research grant uKM-ST-07-FrgS-0028-2009.rEFErENCESAdomian, g. 1988. A review of the decomposition method in applied mathematics. J. Math. Analy. Appl. 135: 501-544.Cherruault, y. & Adomian, g. 1993. Decomposition method: A new proof ofconvergence. Numer. Mathl. Comput. Model., 18: 103-106.Lubuma, J.M.S. & Patidar, K.C. 2005. Numerical Contributions to the theory of non-standard finite difference method and applications to singular perturbation problems. In Advances In the Applications of Nonstandard Finite Difference Schemes, edited by R.E. Mickens. Singapore: World Scientific 513-560.Ibijola, E.A., Lubuma, J.M.S. & Ade-Ibijola, o.A. 2008. on nonstandard finite difference schemes for initial value problems in ordinary differential equations, Int. J. Phys. Sci. 3(2): 59-64.Mickens, R.E. 1989. Exact solutions to a finite difference model of anonlinear reaction-advection equation: Implications for numerical analysis, Numer. Meth. Partial Diff. Eq. 5: 313-325.Mickens, r.E. & Smith, A. 1990. Finite difference models of ordinary differential equations: Influence of denominator models, J. Franklin Institute 327: 143-145.Mickens, r.E. 1994. Nonstandard Finite Difference Models of Differential Equations. Singapore: World Scientific.Mickens, R.E. 1999. Nonstandard finite difference schemes for reaction-diffusion equations, Numer. Meth. Partial Diff. Eq. 15: 201-214.Mickens, r.E. 2000. Applications of Nonstandard Finite Difference Schemes. Singapore: World Scientific.Mickens, R.E. 2003. Non-standard finite-difference schemes for the Lokta-Volterra system. Applied Numerical Mathematics 45: 309-314.Mickens, r.E. 2005. Advances in the Applications of Nonstandard Finite Difference Schemes. Singapore: World Scientific.Mickens, r.E. 2006. Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition. Numer. Meth. Partial Diff. Eq. 23(3): 672-691.Momani, S., Moadi, K. & Noor, M.A. 2006. Decomposition method for solving a system of fourth-order obstacle boundary value problems. Appl. Math. Comput. 175: 923-931.Vahidi, A.r., Babobian, E., Asadi, Cordshooli, g.H. & Mirzaie, M. 2009. Adomian decomposition method to systems of nonlinear algebraic equations. Appl. Math. Sci. 3: 883-889.K. Moaddy & I. Hashim*School of Mathematical Sciences universiti Kebangsaan Malaysia 43600 uKM Bangi Selangor D.E.MalaysiaA.K. AlomariDepartment of SciencesFaculty of Nursing and ScienceJerash Private university26150 JerashJordan S. MomaniDepartment of Mathematics Faculty of ScienceThe university of JordanAmman 11942 Jordan*Corresponding author; email: ishak_h@ukm.myreceived: 20 May 2010Accepted: 7 July 2010xyFIgurE 2. Numerical solution for the Lotka-Volterra system using rK4, NSFD and 6-term NSFD-ADM where h = 0.001, x(0) = 0.1 and y(0) = 1

A new hybrid non-standard finite difference-adomian scheme for solution of nonlinear equations

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