Appl. Math. Inf. Sci. 10, No. 4, (2016) 1231

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1 Al. Math. Inf. Sci. 0, No. 4, Alied Mathematics & Information Sciences An International Journal htt://dx.doi.org/0.8576/amis/00403 Solution of Higher-Order, Multioint, Nonlinear Boundary Value Problems with High-Order Robin-Tye Boundary Conditions by the Adomian Decomosition Method Randolh Rach, Jun-Sheng Duan and Abdul-Majid Wazwaz 3, Emeritus, 36 South Male Street, Hartford, MI , U.S.A. School of Sciences, Shanghai Institute of Technology, Shanghai 048, P.R. China 3 Deartment of Mathematics, Saint Xavier University, Chicago, IL 60655, U.S.A. Received: 7 Mar. 06, Revised: 3 May 06, Acceted: 4 May 06 Published online: Jul. 06 Abstract: In this aer, we roose a new modified recursion scheme for the aroximate solution of higher-order, multioint, nonlinear boundary value roblems with higher-order Robin-tye boundary conditions by the Adomian decomosition method. Our new aroach utilizes all of the boundary conditions to derive an equivalent nonlinear Fredholm-Volterra integral equation before establishing the new modified recursion scheme for the solution. We solve several comlex numerical examles obtaining a raidly convergent sequence of analytic functions as the solution. In all cases investigated, we achieved an aroximately exonential rate of convergence, thus confirming that only a few terms of the Adomian decomosition series can rovide an accurate engineering model for arametric simulations. Keywords: Adomian decomosition method, Adomian olynomials, boundary value roblem, Robin boundary condition, nonlinear differential equation Introduction Boundary value roblems BVPs for nonlinear ordinary differential equations are extensively alied in science and engineering. For examle, the temerature distribution of convective straight fins with temerature-deendent thermal conductivity is modeled by a second-order nonlinear BVP []. The diffusion of oxygen in a sherical cell with Michaelis Menten kinetics is modeled by a second-order nonlinear BVP []. The magneto hydrodynamics Jeffery Hamel flow roblem leads to a third-order nonlinear BVP [3]. The Euler-Bernoulli beam is described by a fourth-order nonlinear BVP, e.g. the beam-tye nanoscale electromechanical system actuator [4]. In all of these alications, Robin and Robin-tye boundary conditions can often be involved [, 5 9]. The Adomian decomosition method ADM is a well-known systematic method for ractical solution of linear or nonlinear and deterministic or stochastic oerator equations, including ordinary differential equations, artial differential equations, integral equations, integro-differential equations, etc. [0 0]. The ADM is a owerful technique, which rovides efficient algorithms for analytic aroximate solutions and numeric simulations for real-world alications in the alied sciences and engineering. In the ADM, the solution ux is reresented by the Adomian decomosition series and the nonlinearity Nux is reresented by the series of the Adomian olynomials that are tailored to the articular nonlinear function as ux= u n x and Nux= A n x, resectively, the Adomian olynomials are deendent uon the solution comonents from u 0 x through u n x, inclusively, i.e. A n x = A n u 0 x,..., u n x. Adomian and Rach [] ublished the definitional formula for the Adomian olynomials in 983 for the simle nonlinearity Nux = Fx, ux and the differential multivariable nonlinearity Corresonding author wazwaz@sxu.edu c 06 NSP

2 3 R. Rach et al.: Solution of higher-order, multioint, nonlinear boundary... Nu = Fx,ux,u x,u x, or one-variable nonlinearity and differential multivariable nonlinearity, resectively, F is assumed to be analytic, as A n x = n n! λ n Fx, λ n u n x, λ=0 n A n x = n! λ n Fx, λ n u n x, λ n u nx,..., λ n u n x, 3 λ=0 λ is a grouing arameter of convenience, and similarly for more comlex nonlinearities. We observe that only the deendent variables, such as the solution ux and its derivatives, are arameterized while the indeendent variables such as x are not arameterized. The Adomian olynomials are in the form A 0 = Fx,u 0, and A n = n k= C k n Fk x,u 0, n, 4 etc. Several convenient algorithms to readily generate the Adomian olynomials have been develoed by Adomian and Rach [, ], Rach [3, 4], Wazwaz [6, 5], Abdelwahid [6], and several others [7 30]. Recently, Duan [3 34] has develoed several new algorithms and subroutines for fast generation of the one-variable and the multi-variable Adomian olynomials. For fast comuter generation, we esecially recommend Duan s new Corollary 3 algorithm [33] to generate the coefficients Cn k and hence the Adomian olynomials quickly and to high orders as Cn = u n, for n, and Cn k = n n k j=0 j+ u j+cn k j, for k n. 5 It does not involve the differentiation oerator, but only requires the analytic oerations of addition and multilication, which is eminently convenient for symbolic imlementation by MATHEMATICA, MAPLE or MATLAB as well as for debugging. Furthermore, it has been timed to be one of the fastest subroutines on record using a commercially available lato comuter [33]. We note that the solution comonents u n x may be determined by one of several advantageous recursion schemes, which differ from one another by the choice of the initial solution comonent u 0 x, beginning with the classic Adomian recursion scheme [35 38]. In next section, we first consider the second-order and third-order BVPs with Robin and Robin-tye boundary conditions, resectively, as u x+ f x,u 0 x,u x = 0, α, u +α, u 0 =β, α, u ξ +α, u 0 ξ =β, < ξ, and u 3 x+ f x,u 0 x,u x,u x = 0, α, u +α, u +α,3 u 0 =β, α, u ξ +α, u ξ +α,3 u 0 ξ =β, α 3, u ξ 3 +α 3, u ξ 3 +α 3,3 u 0 ξ 3 =β 3, ξ ξ 3 and < ξ 3. We emhasize that our formulation ermits the number of distinct boundary oints to be less than or equal to the order of the differential equation. Then we consider the general case: u x+ f x,u 0 x,...,u x = 0, i= α j,i u i ξ j =β j, j =,,...,,, ξ j ξ and < ξ. Derivation of the equivalent integral equations and decomosition of the solutions Case : the second-order differential equation We begin by considering the second-order nonlinear differential equation u x+ f x,u 0 x,u x = 0, 6 subject to the classic Robin boundary conditions α, u +α, u 0 =β, 7 α, u ξ +α, u 0 ξ =β, 8 we have assumed that < ξ, i.e. there are two distinct boundary oints. We assume that Eqs. 7 and 8 are two linearly indeendent equations in four unknowns. Also the nonlinear function f is assumed to be analytic in all of its arguments. We rewrite Eq. 6 in Adomian s oerator-theoretic form as L ux+nux=0, 9 L = d dx, Nux= f x,u 0 x,u x. Alying the inverse oerator L vx= vxdxdx to both sides of Eq. 9 yields L L ux= L Nx, c 06 NSP

3 Al. Math. Inf. Sci. 0, No. 4, / 33 L L ux=ux Φx, Φx=u 0 + u, = x. Thus ux=φx L Nux, 0 or, uon substitution, we obtain ux=u 0 +u L Nux, which is the equivalent nonlinear Volterra integral equation for the solution ux with two undetermined constants of integration, u 0 and u. Calculating the first-order derivative yields u x=u L Nux, L = dx. Next, we evaluate the formulas for the solution and its firstorder derivative at x=ξ as uξ = u 0 +u, L,Nux, u ξ = u L,Nux, 3, = ξ, L, = ξ L ξ, = dx. dxdx, Substituting Eqs. and 3 into the remaining boundary condition 8 yields α, + α,, u +α, u 0 = β + α, L, Nux+ α,l, Nux. By denoting α, = α, + α,,, β = β + α, L, Nux+α,L, Nux, 4 we obtain two linearly indeendent equations solely in terms of the two unknowns u 0 and u in a similar attern as the boundary conditions 7 and 8 as α, u +α, u 0 =β, 5 α, u +α, u 0 = β. 6 Or equivalently α, α, u β α, α, u 0 =. β Introducing the artitioned matrix system coefficient α and the artitioned vector inutβ, we have αu=β, α= α, α,, u= α, α, u β u 0, β=. β From our assumtion that the boundary conditions 7 and 8 are linearly indeendent, it follows that detα 0, thus u=α β, α = adjα detα = α, α, α, α, α, α, α, α, ᾱ, ᾱ =,, ᾱ, ᾱ, and the inverse matrix elements ᾱ i, j are comuted by an aroriate algorithm, such as by the Lalace exansion, Gauss-Jordan elimination, Gauss-Seidel iteration, and LU or Cholesky decomosition alicable, or by a native routine imlemented within an available comuter algebra system such as MATHEMATICA, MAPLE or MATLAB. For our matrix, we readily obtain i.e. u ᾱ, ᾱ u 0 =, β, ᾱ, ᾱ, β u =ᾱ, β + ᾱ, β, u 0 =ᾱ, β + ᾱ, β, 7 which therefore determines the values of the constants of integration u and u 0 by formula. Substituting Eq. 7 into Eq., we have } } ux = {ᾱ, β + ᾱ, β + {ᾱ, β + ᾱ, β L Nux. Inserting β in Eq. 4 leads to ux=ᾱ, β + ᾱ, β +ᾱ, β + ᾱ, β +ᾱ, α, L, Nux+ᾱ,α, L, Nux L Nux + ᾱ, α, L, Nux+ᾱ,α, L, Nux. Uon aroriate algebraic maniulations, we deduce that ux=ᾱ, β + ᾱ, β +ᾱ, β + ᾱ, β +ᾱ, + ᾱ, α, L, Nux+ α,l, Nux L Nux, 8 which is the equivalent nonlinear Fredholm-Volterra integral equation for the solution without any undetermined constants of integration. Decomosing the solution and nonlinearity as ux= u n x, Nux= A n x, 9 c 06 NSP

4 34 R. Rach et al.: Solution of higher-order, multioint, nonlinear boundary... A n x = A n u 0 x,...,u n x are the Adomian olynomials, and substituting the decomosition series 9 into Eq. 8 yields L u n x=ᾱ, β + ᾱ, β +ᾱ, β + ᾱ, β α, L A n x+ᾱ, + ᾱ, +α, L A n x.,, A n x For simle nonlinearities such as Nu = u,u 3,uu, e.g., ositive integer owers, olynomials and roduct nonlinearities, etc., we can use the classic Adomian recursion scheme as u 0 x=ᾱ, β + ᾱ, β +ᾱ, β + ᾱ, β, u n+ x= L A nx 0 +ᾱ, + ᾱ, α, L, A nx+α, L, A nx, n 0, to calculate the solution comonents. We note that each such aroximate solution φ n x= n m=0 u m x satisfies all of the boundary conditions, i.e. when using Adomian s choice for the initial solution comonent u 0 x. For more comlicated nonlinearities such as Nu = e u,sinu, +u, e.g., exonential, sinusoidal, radical, negative-ower, and even decimal-ower nonlinearities, etc., we instead use one of the arameterized recursion schemes, e.g., u 0 x=c, u x= c q+ᾱ, β + ᾱ, β +ᾱ, + ᾱ, α, L, A 0x+α, L, A 0x +ᾱ, β + ᾱ, β L A 0x, u n+ x= c qq n L A nx +ᾱ, + ᾱ, α, L, A nx+α, L, A nx, n, c and q are two redetermined constants and 0 q <. For examle, by the mean value theorem of integral calculus we can take c as the average value of the original solution comonent u 0 in the classic Adomian recursion scheme over the domain, i.e. c = ξ ξ ᾱ, β + ᾱ, β +ᾱ, β + ᾱ, β dx = ᾱ, β + ᾱ, β +ᾱ, β + ᾱ, β ξ. 3 For a roblem without an a riori exact closed-form analytic solution, the error analysis can be considered by calculating the sequence of error remainder functions ER n x=φ n x+ f x,φ n 0 x,φ n x, and the maximal error remainder arameters MER n = max x ξ ER n x, for n 0. We remark that the logarithmic lot of the MER n versus the index n rovides a reliable measure of the rate of convergence, e.g., a nearly linear relation with a negative sloe demonstrates an aroximate exonential rate of convergence. Case : the third-order differential equation Consider the third-order nonlinear differential equation u 3 x+ f x,u 0 x,u x,u x = 0, 4 subject to the set of three Robin-tye boundary conditions α, u +α, u +α,3 u 0 =β, 5 α, u ξ +α, u ξ +α,3 u 0 ξ =β, 6 α 3, u ξ 3 +α 3, u ξ 3 +α 3,3 u 0 ξ 3 =β 3, 7 ξ ξ 3 and < ξ 3. We assume that Eqs. 5 7 are three linearly indeendent equations in nine unknowns. Using Adomian s oerator-theoretic form, we have L 3 ux= Nux, 8 L 3 = d3 dx 3, Nux= f x,u 0 x,u x,u x. Alying the inverse oerator L 3 vx= to both sides of Eq. 8 yields and vxdxdxdx L 3 L3 ux= L 3 Nux, L 3 L3 ux=ux Φx, Φx=u 0 +u + u!. Thus we have ux=φx L 3 Nux, 9 or, uon substitution, we obtain ux=u 0 +u + u! L 3 Nux, 30 c 06 NSP

5 Al. Math. Inf. Sci. 0, No. 4, / 35 which is the equivalent nonlinear Volterra integral equation for the solution ux with three undetermined constants of integration, u 0, u and u. From Eq. 30, we calculate the first- and second-order derivatives as u x = u +u L Nux, 3 u x = u L Nux, 3 L vx= vxdxdx, L vx= vxdx. Next, we evaluate the formulas for the solution and its first- and second-order derivatives at x = ξ j, for j =, 3, as uξ j = u 0 +u j, + u j,! Nux, 33 L 3, j u ξ j = u +u j, L, jnux, 34 u ξ j = u L, jnux, 35 j, = ξ j and L 3, j = ξ j L, j = ξ j dxdxdx, 36 dxdx, 37 L, j = ξ j dx. 38 Substituting Eqs into the boundary conditions 6 and 7, for j =, 3, yields u α j, + α j, j, + α j, j,3! +u α j, + α j,3 j, + u 0 α j,3 = β j + α j, L, j Nux+ α j,l, j Nux+α j,3l 3, j Nux. By denoting α j, = α j, + α j, j, + α j, j,3!, α j, = α j, + α j,3 j,, β j = β j + α j, L, j Nux+α j,l, j Nux+α j,3l 3, j Nux, 39 we obtain three linearly indeendent equations solely in terms of the three unknowns u 0, u and u in a similar attern as the boundary conditions 5, 6 and 7 as α, u +α, u +α,3 u 0 =β, 40 α, u + α, u +α,3 u 0 = β, 4 α 3, u + α 3, u +α 3,3 u 0 = β 3. 4 The matrix form is α α α 3 α α α 3 u β u = β, α 3 α 3 α 33 u 0 β 3 Introducing the artitioned matrix system coefficient α and the artitioned vector inut β, we have αu=β, α= α α α 3 u β α α α 3, u= u, β= β. α 3 α 3 α 33 u 0 β 3 If detα 0, then u=α β, we define α = adjα detα = ᾱ, ᾱ, ᾱ,3 ᾱ, ᾱ, ᾱ,3, ᾱ 3, ᾱ 3, ᾱ 3,3 the inverse matrix elements ᾱ i, j are rather comuted by an aroriate algorithm. For our 3 3 matrix, we readily obtain u u = ᾱ, ᾱ, ᾱ,3 β ᾱ, ᾱ, ᾱ,3 β, u 0 ᾱ 3, ᾱ 3, ᾱ 3,3 β 3 or u =ᾱ, β + ᾱ, β + ᾱ,3 β3, 43 u =ᾱ, β + ᾱ, β + ᾱ,3 β3, 44 u 0 =ᾱ 3, β + ᾱ 3, β + ᾱ 3,3 β3, 45 which therefore determines the values of the constants of integration u, u and u 0 by formula. Substituting Eqs into Eq. 30, we have ux = ᾱ 3, β + ᾱ 3, β + ᾱ 3,3 β3 +ᾱ, β + ᾱ, β + ᾱ,3 β3 +ᾱ, β + ᾱ, β + ᾱ,3 β3! L 3 Nux, 46 β and β 3 are given in Eq. 39. Uon aroriate algebraic maniulations, we deduce that ux=ᾱ 3, β + ᾱ 3, β + ᾱ 3,3 β 3 +ᾱ, β + ᾱ, β + ᾱ,3 β 3 +ᾱ, β + ᾱ, β + ᾱ,3 β 3 L 3 Nux + ᾱ 3, + ᾱ, + ᾱ, α, L, Nux+α,L, Nux+ α,3l 3, Nux + ᾱ 3,3 + ᾱ,3 + ᾱ,3 α 3, L,3 Nux+α 3,L,3 Nux+ α 3,3L 3,3, Nux 47 c 06 NSP

6 36 R. Rach et al.: Solution of higher-order, multioint, nonlinear boundary... which is the equivalent nonlinear Fredholm-Volterra integral equation for the solution ux without any undetermined constants of integration. Next the decomosition series of the solution and the nonlinearity and the recursion scheme are similarly obtained as in the last case. Here we omit the details to avoid self-evident reetitions. Case 3: the general th-order differential equation for Consider the th-order nonlinear differential equation u x+ f x,ux,...,u x = 0, 48 subject to the set of Robin-tye boundary conditions i= α j,i u i ξ j =β j, j =,,...,, 49 ξ j ξ and < ξ. Eq. 49 denotes linearly indeendent equations in unknowns. Using Adomian s oerator-theoretic notation, we write Eq. 48 as L ux= Nux, 50 L = d dx, Nux = f x,ux,...,u x. Alying the -fold integral oerator L vx= }{{} f old to both sides of Eq. 50, we have L vx dx dx }{{} f old L L ux= L Nux, L ux=ux Φx, 5 Φx= So we obtain ux= u ν ν ν!. 5 u ν ν ν! L Nux, 53 which is the equivalent nonlinear Volterra integral equation for the solution ux with undetermined constants of integration, u 0,..., u. Calculating the kth-order derivatives of ux, for 0 k, yields u k x = dk dx k = ν=k u ν ν ν! dk dx k L Nux ν k u ν ν k! L k Nux, 54 L k vx= vx ξ }{{} } dx dx {{}. k f old k f old Next we calculate the derivatives u k ξ j, for k=0,,..., and j =, 3,...,, as u k ξ j = ν=k ν k j, u ν ν k! L k, j Nux, the oerator L k, j is defined as ξ L k j, j vx= ξ }{{} k f old vx } dx dx {{} dx. k f old For the substitution k := i, then we have ν k := ν+ i and k := i, and u i ξ j = ν= i ν+i j, u ν ν+ i! L, j i Nux, i=,...,, j =,...,. 55 Substituting the last equation into the boundary conditions 49 at ξ j, for j =,...,, we have [ α j,i u ν ν+i ] j, i= ν= i ν+ i! L, j i Nux = β j, j =,...,. 56 Using the following summation formula i= ν= i = C,0 + C i,ν = l= ν=l i i= l= C i, l = C ν, l = C,0 + C i, l l= i=l i= ν=i C ν, i, we obtain linearly indeendent equations solely in terms of the unknowns u 0, u,..., u as for j =, i= α,i u i =β, 57 for j =,...,, { } α j,i u i + α j, u 0 = β j, 58 i= β j = β j + ν= α j,ν L, ν j Nux, α j,i = α j,ν ν=i Or equivalently, we have the matrix form ν i j, ν i!. 59 αu=β, 60 c 06 NSP

7 Al. Math. Inf. Sci. 0, No. 4, / 37 α, α, α,3 α, α, α, α, α,3 α, α, α= α 3, α 3, α 3,3 α 3, α 3, α, α, α,3 α, α, u β u β u= u 3, β= β 3... u 0. β If detα 0, then u = α β, the inverse matrix α = ᾱ i, j is comuted by an aroriate algorithm or native routine imlemented within an available comuter algebra system such as MATHEMATICA, MAPLE or MATLAB. We calculate for ν = 0,,...,, u ν =ᾱ ν, β + j= ᾱ ν, j β j, 6 which therefore determines the values of the constants of integration u, u,..., u 0 by formula. Substituting these values into Eq. 53, we have ux= { ᾱ ν, β + j= ᾱ ν, j β j } ν ν! L Nux, Inserting the β j, for j=,...,, from Eq. 59 into the last revious equation leads to { ux= [ β j + k= ᾱ ν, β + α j, k L k, j Nux j= ]} ᾱ ν, j ν ν! L Nux. Uon aroriate algebraic maniulations, we deduce that ux= β j ν ᾱ ν, j L j= ν! Nux + j= ν ᾱ ν, j α j,k L k, j ν! Nux k=, 6 which is the equivalent nonlinear Fredholm-Volterra integral equation for the solution without any undetermined constants of integration. Next we decomose the solution and the nonlinearity as ux= u n x, Nux= A n x, A n x=a n u 0 x,...,u n x, 63 and substitute them into Eq. 6 to obtain u n x= β j ᾱ ν ν, j ν! + j= j= ᾱ ν ν, j ν! k= α j,k L k L, j A n x A n x. For simle nonlinearities such as the quadratic nonlinearity, roduct nonlinearity, etc., we can use the classic Adomian recursion scheme as u 0 x = j= u n+ x = j= { β j ᾱ ν, j ν ν! }, { ᾱ ν ν, j ν! α j,k L k, j A nx k= L A nx, n 0, } 64 to calculate the solution comonents. We note that each aroximate solution φ n x= n m=0 u m x, n=,,..., 65 satisfies all of the boundary conditions. For comlicated nonlinearities such as Nu = e u,sinu, +u, etc., we instead use one of the arameterized recursion schemes, e.g., u 0 x=c, u x= c q+ j= β j + j= u n+ x= c qq n+ L + j= ᾱ ν ν, j ν! ᾱ ν ν, j ν! α j,k L k, j A 0x k= A n+x, ᾱ ν ν, j ν! α j,k L k, j A n+x k= L A 0x, n 0, 66 c and q are two redetermined constants and 0 q<. For examle, we can take c as the average value of the initial solution comonent u 0 in the classic Adomian recursion scheme over the domain, i.e. c = = ξ { j= ξ j= { β j ν ᾱ ν, j ν! } ξ β j ν ᾱ ν, j ν+! } dx, 67 For the error analysis for a roblem without an a riori exact closed-form analytic solution, we can consider the sequence of error remainder functions ER n x=φ n x+ f x,φ 0 n x,...,φ n x, 68 and the sequence of maximal error remainder arameters MER n = max x ξ ER n x. 69 c 06 NSP

8 38 R. Rach et al.: Solution of higher-order, multioint, nonlinear boundary... 3 Numeric examles Examle. Consider the third-order linear differential equation E n x x u x+u x u x ux=0, 70 subject to the set of three Robin-tye boundary conditions u 0 u 0 u0=, u 0 u0=, u + u=0. The exact solution is x u x= 0e 0e 4e e x 0e 4e e x. 7 The equivalent Fredholm-Volterra integral equation is ux = 3 3x 5 x x 5 L,3 Nu+L 3,3 Nu L 3 Nu, Nu=u x u x ux and L 3,3 = 0 0 The recursion scheme is u 0 x= 3 5 u n+ x= 5 + x 5 n 0 and Φ n x dxdxdx, L,3 = 3x x+ 5, 0 L,3 A n+ L 3,3 A n L 3 A n, A n = u n x u n x u nx, n x 0 dxdx. Fig. : Curves of the exact solution u x solid line, and the aroximate solutions φ x dash line, φ 3 x dot line and φ 4 x dot-dash line. In Fig., we lot the curves of the exact solution u x and the aroximate solutions φ x, φ 3 x and φ 4 x, φ 3 x, φ 4 x and the exact solution overla Fig. : Curves of the error functions E x solid line, E 3 x dash line, E 4 x dot line and E 5 x dot-dash line. MEn Fig. 3: The logarithmic lots of the maximal error arameters ME n, n= through 0. We consider the error function E n x=φ n x u x and the maximal error arameters ME n = max 0 x E nx. In Fig., we lot the curves of the error functions E n x for n=,3,4 and 5. In Fig. 3, we dislay the logarithmic lots of the maximal error arameters ME n, n= through 0, the oints are distributed almost on a straight line thus indicating an aroximate exonential rate of convergence. Examle. Consider the third-order nonlinear differential equation n u x u x+xsinhux=0, 7 subject to the set of three Robin-tye boundary conditions u 0 u 0+ u0=, u 0.5+ u 0.5+ u0.5=, u 3u + u=. For this BVP, we calculate that α= 9/4 3, α = , 3/ c 06 NSP

9 Al. Math. Inf. Sci. 0, No. 4, / 39 and the equivalent nonlinear Fredholm-Volterra integral equation as ux= x x x x 0x 49 x L 3 Nu L, Nu+L, Nu+L 3, Nu L,3 Nu 3L,3 Nu+L 3,3, Nu Nu= u x+xsinhux. We decomose the solution and the nonlinearity as ux= u n x, Nux= B n x, B 0 = u 0 x+xa 0, B n = u n x+xa n, and the A n are the Adomian olynomials for the analytic function sinhux, i.e. A 0 = sinhu 0, A = u coshu 0, A = u sinhu 0 +u coshu 0, A 3 = u u sinhu u3 coshu 0+u 3 coshu 0,..., and design the arameterized recursion scheme accordingly as u 0 x=c, u x= c q x x x 0x x 49 x L 3 B 0, B 0+ L 3 L, B 0+ L, B 0 L,3 B 0 3L,3 B 0+ L 3,3 B 0, u n+ x= c qq n L 3 B n x x L, B n+ L, B n+ L 3, B n x 0x L,3 B n 3L,3 B n+ L 3,3 B n, n, we take c= x 49 x dx= Φ n x x Fig. 4: Curves of φ x solid line, φ 3 x dash line, φ 4 x dot line and φ 5 x dot-dash line. ER n x x Fig. 5: Curves of ER x solid line, ER 3 x dash line, ER 4 x dot line and ER 5 x dot-dash line. MERn Fig. 6: Logarithmic lots of MER n for n= through 0. n Table : The maximal error remainder arameters MER n in Examle. n MER n n MER n We take q = 0. to comute the solution aroximants. The solution aroximants φ n x, n =,3,4,5, are lotted in Fig. 4, the last three curves overla. In Fig. 5, we lot the error remainder functions ER n x for n =,3,4,5. The maximal error remainder arameters MER n, n= through 0, are listed in Table. The logarithmic lots of these values are dislayed in Fig. 6, the last 7 oints are distributed almost on a straight line thus indicating an aroximate exonential rate of convergence. Examle 3. The squeezing flow and heat transfer between two arallel disks with velocity sli and temerature jum [39] lead to the following nonlinear BVP in dimensionless form f 4 η Sη f η 3S+M f η+s fη f η=0, 73 f0=0, β f 0+ f 0=0, f=/, β f + f =0. c 06 NSP

10 40 R. Rach et al.: Solution of higher-order, multioint, nonlinear boundary... η is a similarity variable, fη characterizes the axial velocity, S is the squeeze number, M is the Hartman number, and β is the dimensionless sli arameter. For this fourth-order nonlinear BVP, 0 = = ξ < ξ 3 = ξ 4 =, α,4 = α,3 = α 3,4 = α 4,3 =, α, = β, α 4, = β, α i, j = 0 for other i, j, β = β = β 4 = 0,β 3 = /, and N fη= Sη f η 3S+M f η+s fη f η. We calculate that α= 0 β 0 6, β + β + 0 +β + β β + β α = D β 3 β + β 6 β β + β β 3 + β β, 6 D D= + β 3 + β. The equivalent nonlinear Fredholm-Volterra integral equation is fη= β+β D + D β+β η+ +β + D +β η3 β 6 η+ +β 4 η +β 6 η 3 4 η +β 6 η 3 η η L 4,3 N fη L 4 N fη β L,4 N fη+l 3,4 N fη. We decomose the solution and the nonlinearity as fη= f n η, N fη= and design the recursion scheme f 0 η= D f n+ η= D + D +β η3 β 6 L 4 A n, n 0, β+β η+ +β β+β η+ +β η η A n η, 4 η +β 6 η, 3 4 η +β 6 η 3 β L,4 A n+ L 3,4 A n L 4,3 A n A 0 = Sη f 0 η 3S+M f 0 η+s f 0η f 0 η, A n = Sη f n η 3S+M f nη +S n k=0 f kη f n k η, n. From the recursion scheme, we obtain the solution comonents and solution aroximants. We take S = M = and β = 0. to comute the solution aroximants. The solution aroximants φ n η, n =,3,4, are lotted in Fig. 7, the three curves overla. In Fig. 8, we lot the error remainder functions Φ n Η Fig. 7: Curves of φ η solid line, φ 3 η dash line and φ 4 η dot line for S=M = and β = 0.. ER n Η Fig. 8: Curves of ER η solid line, ER 3 η dash line and ER 4 η dot line for S=M = and β = 0.. Table : The maximal error remainder arameters MER n in Examle 3. n 3 4 MER n n MER n MERn Fig. 9: Logarithmic lots of MER n for n= through 7. ER n η for n =,3,4. The maximal error remainder arameters MER n, n = through 7, are listed in Table. The logarithmic lots of these values are dislayed in Fig. 9, the oints are distributed almost on a straight line thus indicating an aroximate exonential rate of convergence. n Η Η c 06 NSP

11 Al. Math. Inf. Sci. 0, No. 4, / Conclusions We have resented a new modification of the Adomian decomosition method to conveniently solve a wide class of higher-order, multioint, nonlinear boundary value roblems with higher-order Robin-tye boundary conditions. Our new aroach yields a raidly convergent sequence of analytic functions for the solution without any incororating undetermined coefficients within the modified recursion scheme for comuting successive solution. Thus we avoid the comlications resulting from the necessity of evaluating such undetermined coefficients at each stage of aroximation. As this resulting sequence of nonlinear algebraic or transcendental equations roduces more than one root for the ossible value of each undetermined coefficient at each stage, we are then confronted with the ambiguity of multile roots. Usually this difficulty is handled by discarding unhysical roots, which does not lend itself to automation. Furthermore we can also arameterize the new modified recursion scheme by inserting a redetermined arameter in order to achieve simle-to-integrate series without regard to the ossible comlexity of the original initial solution comonent, which is comrised of the boundary values and the inut function. The value of the redetermined arameter can favorably increase the rate of convergence and extend the interval of convergence of the resulting arameterized decomosition series solution. We have demonstrated the racticality and efficiency of our new modification of the Adomian decomosition method by several numerical examles. In oint of fact, the convergence was already sufficiently raid, we demonstrated an aroximately exonential rate of convergence of our new modified decomosition series. The overall efficiency of our new modification of the ADM is further enhanced by the new algorithms and subroutines crafted by Duan for generating the Adomian olynomials quickly and to high orders at will. Furthermore the rigorous mathematical convergence of the Adomian decomosition series to the exact solution has already been established by many researchers. In many hysical models, an exact closed-form solution is not always ossible; however our exository examles have once again demonstrated that only a few terms of the truncated decomosition series rovides an excellent aroximation even in the comlex mathematical models. Acknowledgment This work was suorted by the Innovation Program of the Shanghai Municial Education Commission No. 4ZZ6 and the Natural Science Foundation of Shanghai No. 4ZR References [] J.S. Duan, Z. Wang, S.Z. Fu, T. Chaolu, Parametrized temerature distribution and efficiency of convective straight fins with temerature-deendent thermal conductivity by a new modified decomosition method, Int. J. Heat Mass Transfer 59, [] R. Rach, A.M. Wazwaz, J.S. Duan, A reliable analysis of oxygen diffusion in a sherical cell with nonlinear oxygen utake kinetics, Int. J. Biomath. 7, ID [3] L. Lu, J.S. Duan, L.Z. Fan, Solution of the magnetohydrodynamics Jeffery-Hamel flow equations by the modified Adomian decomosition method, Adv. Al. Math. Mech. 05, doi: 0.408/aamm.04.m543. [4] J.S. Duan, R. Rach, A.M. Wazwaz, Solution of the model of beam-tye micro- and nano-scale electrostatic actuators by a new modified Adomian decomosition method for nonlinear boundary value roblems, Int. J. Nonlinear Mech. 49, [5] A.M. Wazwaz, R. Rach, J.S. Duan, Adomian decomosition method for solving the Volterra integral form of the Lane-Emden equations with initial values and boundary conditions, Al. Math. Comut. 9, [6] J.S. Duan, R. Rach, A.M. Wazwaz, T. Chaolu, Z. Wang, A new modified Adomian decomosition method and its multistage form for solving nonlinear boundary value roblems with Robin boundary conditions, Al. Math. Model. 37, [7] Q. Mao, Alication of Adomian modified decomosition method to free vibration analysis of rotating beams, Math. Probl. Eng. 03, ID [8] R. Rach, J.S. Duan, A.M. Wazwaz, Solving couled Lane- Emden boundary value roblems in catalytic diffusion reactions by the Adomian decomosition method, J. Math. Chem. 5, [9] J.S. Duan, R. Rach, A.M. Wazwaz, A reliable algorithm for ositive solutions of nonlinear boundary value roblems by the multistage Adomian decomosition method, Oen Engineering 5, [0] G. Adomian, Stochastic Systems, Academic, New York, 983. [] G. Adomian, Nonlinear Stochastic Oerator Equations, Academic, Orlando, FL, 986. [] G. Adomian, Nonlinear Stochastic Systems Theory and Alications to Physics, Kluwer Academic, Dordrecht, 989. [3] G. Adomian, Solving Frontier Problems of Physics: The Decomosition Method, Kluwer Academic, Dordrecht, 994. [4] E. Momoniat, C. Harley, An imlicit series solution for a boundary value roblem modelling a thermal exlosion, Math. Comut. Modelling 53, [5] X. Shang, P. Wu, X. Shao, An efficient method for solving Emden Fowler equations, J. Franklin Institute, 346, [6] A.M. 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12 4 R. Rach et al.: Solution of higher-order, multioint, nonlinear boundary... [8] S.E. Serrano, Engineering Uncertainty and Risk Analysis: A Balanced Aroach to Probability, Statistics, Stochastic Modeling, and Stochastic Differential Equations, Second Revised Edition, HydroScience, Ambler, PA, 0. [9] J.S. Duan, R. Rach, D. Baleanu, A.M. Wazwaz, A review of the Adomian decomosition method and its alications to fractional differential equations, Commun. Frac. Calc. 3, [0] R. Rach, A bibliograhy of the theory and alications of the Adomian decomosition method, 96-0, Kybernetes 4, [] G. Adomian, R. Rach, Inversion of nonlinear stochastic oerators, J. Math. Anal. Al. 9, [] G. Adomian, R. Rach, On comosite nonlinearities and the decomosition method, J. Math. Anal. Al. 3, [3] R. Rach, A convenient comutational form for the Adomian olynomials, J. Math. Anal. Al. 0, [4] R. Rach, A new definition of the Adomian olynomials, Kybernetes 37, [5] A.M. Wazwaz, A new algorithm for calculating Adomian olynomials for nonlinear oerators, Al. Math. Comut., [6] F. Abdelwahid, A mathematical model of Adomian olynomials, Al. Math. Comut. 4, [7] K. Abbaoui, Y. Cherruault, V. Seng, Practical formulae for the calculus of multivariable Adomian olynomials, Math. Comut. Modelling, [8] Y. Zhu, Q. Chang, S. Wu, A new algorithm for calculating Adomian olynomials, Al. Math. Comut. 69, [9] J. Biazar, M. Ilie, A. Khoshkenar, An imrovement to an alternate algorithm for comuting Adomian olynomials in secial cases, Al. Math. Comut. 73, [30] M. Azreg-Aïnou, A develoed new algorithm for evaluating Adomian olynomials, CMES-Comut. Model. Eng. Sci. 4, [3] J.S. Duan, Recurrence triangle for Adomian olynomials, Al. Math. Comut. 6, [3] J.S. Duan, An efficient algorithm for the multivariable Adomian olynomials, Al. Math. Comut. 7, [33] J.S. Duan, Convenient analytic recurrence algorithms for the Adomian olynomials, Al. Math. Comut. 7, [34] J.S. Duan, New recurrence algorithms for the nonclassic Adomian olynomials, Comut. Math. Al. 6, [35] J.S. Duan, R. Rach, A new modification of the Adomian decomosition method for solving boundary value roblems for higher order nonlinear differential equations, Al. Math. Comut. 8, [36] J.S. Duan, R. Rach, Z. Wang, On the effective region of convergence of the decomosition series solution, J. Algorithms Comut. Tech. 7, [37] L. Lu, J.S. Duan, How to select the value of the convergence arameter in the Adomian decomosition method, CMES- Comut. Model. Eng. Sci. 97, [38] J.S. Duan, R. Rach, A.M. Wazwaz, A new modified Adomian decomosition method for higher-order nonlinear dynamical systems, CMES-Comut. Model. Eng. Sci. 94, [39] M. Usman, A. Nazir, Z. Naheed, S.T. Mohyud-Din, Adomian s decomosition method to squeezing flow and heat transfer between two arallel disks with velocity sli and temerature jum, Adv. Diab. Metab., Randolh Rach is retired and formerly a Senior Engineer at Microwave Laboratories Inc. His exerience includes research and develoment in microwave electronics and travelling-wave tube technology, and his research interests san nonlinear system analysis, nonlinear ordinary and artial differential equations, nonlinear integral equations and nonlinear boundary value roblems. He has both authored and co-authored more than 00 aers in Alied Mathematics and is an early contributor to the Adomian decomosition method. Jun-Sheng Duan is a Professor of Mathematics at Shanghai Institute of Technology in Shanghai, PR China. He has both authored and co-authored more than 80 aers in alied mathematics and mathematical hysics. He has contributed notably to theoretical advances in the Adomian decomosition method, including new efficient algorithms leading to fast comuter subroutines. Abdul-Majid Wazwaz is a Professor of Mathematics at Saint Xavier University in Chicago, Illinois, USA. He has both authored and co-authored more than 400 aers in alied mathematics and mathematical hysics. He is the author of five books on the subjects of discrete mathematics, integral equations and artial differential equations. Furthermore, he has contributed extensively to theoretical advances in solitary waves theory, the Adomian decomosition method and the variational iteration method. c 06 NSP