New Modified Adomian Decomposition Recursion Schemes for Solving Certain Types of Nonlinear Fractional Two-Point Boundary Value Problems

We apply new modified recursion schemes obtained by the Adomian decomposition method (ADM) to analytically solve specific types of two-point boundary value problems for nonlinear fractional order ordinary and partial differential equations. The new modified recursion schemes, which sometimes utilize the technique of Duan’s convergence parameter, are derived using the DuanRach modified ADM. The Duan-Rach modified ADM employs all of the given boundary conditions to compute the remaining unknown constants of integration, which are then embedded in the integral solution form before constructing recursion schemes for the solution components. New modified recursion schemes obtained by the method are generated in order to analytically solve nonlinear fractional order boundary value problems with a variety of two-point boundary conditions such as Robin and separated boundary conditions. Some numerical examples of such problems are demonstrated graphically. In addition, the maximal errors (MEn) or the error remainder functions (ERn(x)) of each problem are calculated.

The ADM has been extensively utilized to solve IVPs and BVPs for nonlinear ordinary or partial differential equations, integral equations, or integrodifferential equations since it can provide approximate analytic solutions without linearization, perturbation, discretization, guessing the initial term, or using Green's functions which are quite difficult to determine in most cases.The ADM has been used as a tool to investigate analytical and numerical solutions of real-world problems by Hashim et al. [18] and Sweilam and Khader [19].Modifications of the ADM have been developed for different purposes for IVPs for both integer order and fractional order differential equations.
Several researchers proposed expressing the initial solution component  0 as a series of orthogonal polynomials, such as Chebyshev polynomials [20], or Legendre polynomials [21,22].In 2013 Duan et al. [11] combined the ADM with convergence acceleration techniques such as diagonal Padé approximants and iterated Shanks transforms to solve nonlinear fractional ordinary differential equations.It was found that the modified techniques can efficiently extend the convergence region of the decomposition series solution.In 2014 Ramana and Prasad [23] modified the ADM to 2 International Journal of Mathematics and Mathematical Sciences solve parabolic equations and the results obtained by their modified method converged very quickly and were more accurate than the standard ADM results.
There are now several different resolution techniques using the ADM for solving BVPs of nonlinear integer order differential equations.These techniques were developed by many authors as follows.Tatari and Dehghan [24] gave the solution of the general form of multipoint BVPs using the ADM.Al-Hayani [25] used the ADM with Green's function to solve sixth-order BVPs.Duan and Rach [26] proposed the Duan-Rach modified decomposition method for solving BVPs for higher order nonlinear differential equations.Duan et al. [27] developed the multistage ADM for solving BVPs for second-order nonlinear ordinary differential equations with Robin boundary conditions.In particular, modified approaches for solving nonlinear fractional order BVPs can be found in [10,14,28].Most of the resolution techniques are fundamentally based on two principles of using the ADM.The first approach is the method of undetermined coefficients (see, e.g., [10,14,29,30]) in the ADM which approximates the constants of integration embedded in the recursion scheme of the ADM by solving numerically a sequence of nonlinear algebraic equations obtained by employing boundary conditions.The Duan-Rach modified ADM (see, e.g., [26,27,30,31]), which is the second approach, determines the remaining unknown constants of integration of the solution by using the remaining boundary conditions before designing a suitable modified recursion scheme.In the Duan-Rach modified ADM, the constants of integration are calculated simultaneously along with the solution components.Details of the Duan-Rach modified ADM will be discussed in Section 2. 3.
In our work, we study the use of the Duan-Rach modified ADM to solve nonlinear high fractional order boundary value problems with a variety of boundary conditions such as Robin and separated boundary conditions.We will study these fractional BVPs for the Caputo fractional derivative which allows some boundary conditions to be included into the formulas of the solutions.To the best of the authors' knowledge, our paper will be the first to develop formulas for the recursion schemes obtained by using the Duan-Rach modified ADM for the above types of problems.This paper is organized as follows.In Section 2, we review necessary definitions and important properties of the fractional order integrals and derivatives that are needed in our work.The principal reviews of the ADM and the Duan-Rach modified ADM are briefly given in this section as well.In Section 3, we give the formulas for the recursion schemes obtained by using the Duan-Rach modified ADM for selected types of nonlinear fractional BVPs.In Section 4, we give numerical examples of solutions obtained using the proposed recursion schemes for some scientific fractional BVPs with mixed sets of Dirichlet, Neumann, Robin, and separated boundary conditions.These examples include a Bratu-type fractional BVP, an oscillating base temperature equation, and an elastic beam problem.

Review of Fractional Order Integrals and Derivatives.
In this section, we present basic definitions and important theorems of the fractional calculus (see [5,[32][33][34][35]) required in this paper.
Definition 1 (see [5]).A function () ( > 0) is said to be in the space   ( ∈ R) if it can be expressed as () =    1 () for some  > , where  1 () is continuous in [0, ∞), and it is said to be in the space Definition 2 (see [5]).The Riemann-Liouville fractional integral operator of order  > 0 of a function  ∈   with  ≥ 0 is defined as where Γ(⋅) is the gamma function.
For 0 <  ≤ ,  ≥ 0, and  ∈ N, we have the following important properties [5]: International Journal of Mathematics and Mathematical Sciences 3 where  − 1 <  ≤ ,  ∈    ,  ≥ −1. (6) 2.2.Review of the Decomposition Methods.We first review the important concepts of the Adomian decomposition method (ADM) [9] introduced by George Adomian who is an American physicist.The ADM combined with the use of symbolic algebra packages such as MATHEMATICA or MAPLE is a powerful method for solving nonlinear operator equations including ordinary or partial differential equations [14,36].
Here we describe the ADM to solve integer order IVPs and BVPs as follows.
Consider the ordinary differential equation in the following operator form: where  is the highest order derivative; that is,  = (  /  )(⋅), where  is the integer order of the derivative, which is assumed to be easily invertible,  is a linear differential operator of order less than ,  denotes a nonlinear operator assumed to be analytic,  is a source term, and  is the solution of the equation.The ADM decomposes the solution () and the analytic nonlinear term  of the nonlinear operator equation ( 7) into a rapidly convergent series of solution components and a series of the Adomian polynomials.
Applying the inverse linear operator  −1 , which is a fold definite integration, to both sides of (7) and using the given conditions, that is, the initial conditions or boundary conditions, and the fact that  −1  =  − Φ, we obtain where Φ denotes the terms arising from using the given conditions.The ADM decomposes the solution () into an infinite series and then it decomposes the nonlinear term () into a series where   =   ( 0 (),  1 (), . . .,   ()) are the Adomian polynomials that are obtained by the following formula (see the derivation of the formula in [27,37] and the references therein): where  is a grouping parameter.The first six Adomian polynomials obtained by using (11) for the general analytic nonlinear term () = (()) are as follows: We observe that the Adomian polynomials are of the following forms: where    are the sums of all possible products of  components from  1 ,  2 , . . .,  −+1 , whose subscripts sum to , divided by the factorial of the number of repeated subscripts.
From ( 8), (9), and (10), we have and the classic (or standard) Adomian recursion scheme [36,38] is as follows: Then the -term approximation of the solution is which in the lim →∞ yields the exact solution to (7) as International Journal of Mathematics and Mathematical Sciences The choice of different initial solutions  0 () can generate different recursion schemes which can remedy problems in the classic scheme caused by the difficulty of integration for  0 () or the slowness of convergence to the series solution (see [39] for details).One criterion that can be used for choosing the initial solution is to achieve a simple integration for the initial solution component of the series solution.Then, to obtain a fast rate of convergence and an extended region of convergence, we can apply Duan's convergence parameter technique [40][41][42] to the recursion scheme for that initial solution component.Duan and coresearchers [26,27] have shown that the parametrized recursion scheme usually gives a sequence of decreasing maximal errors which approach zero when the value of  in the approximation   () increases.

The Duan-Rach Modified Decomposition Method.
In this section, we will provide the idea of the modified ADM called the Duan-Rach modified decomposition method to solve integer order BVPs.The method generates a recursion scheme for computing successive solution components without any undetermined coefficients.Unlike the method of undetermined coefficients, this new Duan-Rach modified decomposition method [26,27,30] does not require the solution of a sequence of the nonlinear algebraic equations obtained from the approximation   () for the constants of integration.In the Duan-Rach modified decomposition method, we first incorporate as many of the given boundary conditions as possible into the solution () in (8) of the BVP and then we determine the remaining unknown constants of integration before constructing the modified recursion scheme.
We will now give an example of the use of the Duan-Rach modified decomposition method to solve the following twopoint BVP: where the operator  in ( 18) is  = ( 2 / 2 )(⋅).Using the first condition of the boundary conditions in (18), we obtain the solution form as follows: where  −1 = ∫   ∫   (⋅) .We then apply the second condition of the boundary conditions in (18) to solve for   () which can be expressed as follows: Substituting ( 20) into (19) yields Applying the Adomian polynomials for the nonlinear terms in (21), we then obtain the following modified recursion scheme:

Description of the Proposed Recursion Schemes for Solving Certain Types of Fractional Order BVPs
The main advantage of using the Duan-Rach modified ADM for solving nonlinear integer order BVPs which we can see from Section 2.3 is that evaluating the inverse operator directly at the boundary points allows us to obtain the solution components without using numerical methods to calculate the values of unknown constants of integration as in the method of undetermined coefficients.In this section, we construct the recursion schemes developed by using the Duan-Rach modified ADM for solving fractional higher order two-point BVPs with their boundary conditions.We give examples for a set of Robin conditions and separated boundary conditions.We consider the following nonlinear fractional order differential equation: where  is a fractional order of the differential equation with The specific types of boundary conditions imposed on the fractional differential equation in (23) or the fractional integral equation in (24) depend upon the value of .

The Fractional Order Differential Equation (23) with a
Set of Robin Boundary Conditions.We consider a nonlinear fractional order differential equation of the form subject to a set of Robin boundary conditions where (()) is an analytic nonlinear term and , , , and  satisfy the following condition: In order to have the two boundary conditions required for the problem and to make the condition (28) hold, it is necessary to have ,  not both zero, ,  not both zero, and ,  not both zero.
From (24), we have that the solution of this BVP can be written in the form We now apply the Duan-Rach modified ADM to the problem in (25).Using (29), we evaluate () at  =  to obtain where [ RL    (⋅)] = is the Riemann-Liouville fractional integral operator of order  evaluated at  = .Differentiating (29) and then using the property that (/) RL    (⋅) = RL  −1  (⋅) and evaluating   () at  = , we obtain where ] = is the Riemann-Liouville fractional integral operator of order  − 1 evaluated at  = .
Substituting ( 30) and ( 31) into ( 27), we get After manipulating the terms in the above equation, we obtain It is possible to solve the system of two linearly independent equations ( 26) and (33) for the two remaining undetermined coefficients () and   () if the determinant of the coefficient matrix denoted by Δ is not zero, that is, if For Δ ̸ = 0, the values of () and   () can be expressed in terms of the specified values of the system parameters , , , , , , , and  as follows: International Journal of Mathematics and Mathematical Sciences Substituting ( 35) into (29), we obtain the following equivalent nonlinear Fredholm-Volterra integral equation: which is free of any undetermined coefficients.Therefore (36) represents the solution of the fractional order nonlinear differential equation ( 25) subject to the Robin boundary conditions ( 26) and ( 27).Next we apply the decomposition to the solution () and the nonlinear term (); that is, respectively, where   () are the Adomian polynomials defined in (11).
Inserting the equations in (37) into (36), the solution components are determined by the following modified recursion scheme: where the resulting integrals are assumed to exist.The -term approximation of the solution to the BVP obtained by the ADM is the following truncated decomposition series: With the above decomposition obtained by the Duan-Rach modified ADM, each approximation   (),  ≥ 1, must exactly satisfy the boundary conditions ( 26) and ( 27).In addition, other techniques such as partitioning initial terms into two appropriate terms [39,43,44] or using the Duan's convergence parameter [40][41][42] can be incorporated, if necessary, into the recursion scheme (38) for solving the BVP described in ( 25)- (27).Using the general formulas given above, we can derive the equivalent nonlinear Fredholm-Volterra integral equations and their associated recursion schemes for (25) for the special cases of the boundary conditions in (26) and (27).The results are as follows.
Case 1.The nonlinear fractional BVP consists of (25) and the following Dirichlet boundary conditions: The boundary conditions (40) correspond to the case of  =  = 1 and  =  = 0 in ( 26) and ( 27).Thus we have Δ = − ̸ = 0 and then ( 36) is reduced to where RL    (⋅) and [ RL    (⋅)] = are the Riemann-Liouville fractional integral operator of order  and the Riemann-Liouville fractional integral operator of order  evaluated at  = , respectively.Substituting equations in (37) into (41), we can determine the solution components from the following modified recursion scheme: provided that the resulting integrals exist.
Case 2. The nonlinear fractional BVP consists of (25) and the following mixed set of Neumann and Dirichlet boundary conditions: The boundary conditions (43) correspond to the case of  =  = 0 and  =  = 1 in ( 26) and ( 27).Thus we have Δ = −1 ̸ = 0 and then (36) becomes where RL    (⋅) and [ RL    (⋅)] = are the Riemann-Liouville fractional integral operator of order  and the Riemann-Liouville fractional integral operator of order  evaluated at  = , respectively.
Substituting the equations in (37) into (44), we obtain the following modified recursion scheme: where we assume the resulting integrals exist.
Case 3. The nonlinear fractional BVP consists of ( 25) and the following mixed set of Robin and Neumann boundary conditions: The boundary conditions (46) correspond to the case of  = 0 and  = 1 in ( 27).Thus we have Δ =  ̸ = 0 and then E (36) is reduced to where RL    (⋅) and [ RL  −1  (⋅)] = are the Riemann-Liouville fractional integral operator of order  and the Riemann-Liouville fractional integral operator of order  − 1 evaluated at  = , respectively.
Insertion of the equations in (37) into (47) gives the following modified recursion scheme: where we assume the resulting integrals exist.(23) with Separated Boundary Conditions.We consider a nonlinear fractional order two-point BVP consisting of the fractional order differential equation

The Fractional Order Differential Equation
and the following separated boundary conditions: where (()) is an analytic nonlinear term, , , ,  ∈ R, and , , ,  ≥ 0 satisfy the following condition In order to satisfy four necessary boundary conditions required for the problem and to have condition (52), it is necessary to have ,  not both zero, ,  not both zero, and ,  not both zero.From (24) and the first condition in (50), we have that the solution of this BVP can be written in the form We now apply the Duan-Rach modified ADM to the problem in (49).Using (53) and the first condition in (51), we evaluate () at  =  to obtain International Journal of Mathematics and Mathematical Sciences Differentiating (53) three times and then using the properties that (  /  ) RL    (⋅) = RL  −  (⋅),  = 1, 2, 3, to the resulting equations, we obtain Evaluating (55) at  = , we have Insertion of (56) into (51) gives the following relation: It is possible to solve the system consisting of the second condition of (50) and (57) for the two remaining undetermined coefficients   () and   () if the determinant of the coefficient matrix denoted by Δ is not zero, that is, if For Δ ̸ = 0, the values of   () and   () can be expressed in terms of the specified values of the system parameters , , , , , , , and  as follows: Substituting (59) for   () and   () into (54) and then solving the resulting equation for   (), we obtain the value of   () as follows: For the computational convenience, we set (61) Then we substitute (60) and ( 59) into (53) to obtain the following equivalent nonlinear Fredholm-Volterra integral equation: where RL    (⋅) is the Riemann-Liouville fractional integral operator of order  and where [ RL    (⋅)] = , [ RL  −2  (⋅)] = , and [ RL  −3 a (⋅)] = are the operators of orders ,  − 2, and  − 3 evaluated at  = .
Substituting the equations in ( 37) into (62), we can determine the solution components from the following modified recursion scheme: where the resulting integrals are assumed to exist.The -term approximation of the solution to the BVP can be obtained using (16).In addition, other techniques such as partitioning initial terms into two appropriate terms [39,43,44] or using Duan's convergence parameter [40][41][42] can be incorporated, if necessary, into the recursion scheme (63) for solving the BVP described in (49)-(51).
Using the general formulas derived in (62) and (63), we can derive the equivalent nonlinear Fredholm-Volterra integral equations and their associated recursion schemes for (49) for the special cases of the boundary conditions in (50) and (51).The results are as follows.
Case 1.The nonlinear fractional BVP consists of (49) and the following two-point boundary conditions: The derivative boundary conditions (64) correspond to the case of  =  = 1 and  =  = 0 in (50) and (51).Thus (62) is then reduced to Insertion of ( 37) into (65) gives the following modified recursion scheme for solution components: where we assume the resulting integrals exist.
where we assume the resulting integrals exist.

Numerical Examples
In this section, we demonstrate a use of the proposed recursion schemes in Section 3 derived from the Duan-Rach modified decomposition method to analytically and numerically solve nonlinear fractional BVPs.Several nonlinear fractional BVPs presented in this section correspond to the problems and their formulas described in Section 3 and some of these problems include physical and engineering problems such as problems of the Bratu type, a problem of the periodic base temperature in convective longitudinal fins, and an elastic beam problem.Numerical results obtained by the method are demonstrated graphically.Moreover, if the presented nonlinear fractional BVPs have their exact solutions then we will compute their corresponding maximal errors.Otherwise, we will investigate the error remainder function for the remaining problems.
In general, we consider a nonlinear fractional BVP: () +  = (),  ≤  ≤ , where  is a fractional order of the equation imposed by some boundary conditions.If the exact solution  * () of the problem is known then we can examine the convergence of the -term approximation   () = ∑ −1 =0   () from the error functions expressed as and the maximal errors defined as For each value of , we can calculate ME  using the MATHEMATICA command "NMaximize" over an interval of interest.Then the logarithmic plot of these values of ME  can be made using the MATHEMATICA command "ListLogPlot"; however, if the exact solution  * () of such a problem is unknown, we compute the error remainder functions defined as We observe that ER  is the indicator for measuring how well the approximation   () satisfies the original nonlinear fractional differential equation.
For each specified value of  ̸ = 2, since the exact solution to this BVP is not known, we calculate the corresponding error remainder function ER  () for  = 10. Figure 2 displays the graphs of ER 10 () for the specified values of .We can deduce from the graphs in Figure 2 that the approximations  10 () obtained by this method give remarkable accuracy, as expected, since when  is sufficiently large the magnitude of each function ER  () approaches zero.
For  = 2, it is possible to compute both the error function   () and the maximal errors ME  ; however, we only compute the values of ME  for  = 2, 3, . . ., 9 listed in Table 1.We show the logarithmic plots of ME  versus  for  = 2, 3, . . ., 9 in Figure 3.We can observe in Figure 3 that all of the data points after  = 2 lie almost on a straight line which demonstrates that the maximal errors ME  are reduced approximately at an exponential rate.homogeneous partial differential equation of the engineering model in [47] as follows: where the notation   /  in (79) represents a fractional partial derivative with respect to the space  in the Caputo sense with the fractional order  ∈ (1, 2] and where (, ) has the domain of definition  ∈ [0, 1] and  ∈ [0, ∞).The physical variables , ,  are the dimensionless temperature, distance, and time, respectively.For (79), the following mixed set of homogeneous Neumann and inhomogeneous Dirichlet boundary conditions is given as The above conditions consist of a sinusoidally varying boundary value.The parameters , , , and  in (79) and (80) represent thermal conductivity parameter, fin parameter, amplitude of oscillation, and frequency of oscillation, respectively.The BVP in (79) and (80) describes physically the periodic base temperature in convective longitudinal fins.
For each specified value of  ̸ = 2, since the exact solution to this BVP is not known, we calculate the corresponding error remainder functions ER  () for  = 15.Figure 7 displays the graphs of ER 15 () for the specified values of .We can deduce from the graphs in Figure 7 that the approximations  15 () obtained by this method give the remarkable accuracy as expected that when  is sufficiently large then the magnitude of each function ER  () approaches zero.
For  = 3.  (71) as ME  = max 0≤≤1 |  ()|.However, we only compute the values of ME  for  = 2, 3, 4, . . ., 9 listed in Table 3.In Figure 11 we show the logarithmic plots of ME  versus  for  = 2, 3, 4, . . ., 9 obtained by the method.All of the data points lie almost on a straight line so the maximal errors are decreasing approximately at an exponential rate.

Conclusions
We have established new recursion schemes using the Duan-Rach modified decomposition method to solve a variety of nonlinear fractional BVPs.The obtained recursion schemes have been derived for solving the nonlinear fractional BVPs with a set of Robin boundary conditions (order 1 <  ≤ 2) and with separated boundary conditions (order 3 <  ≤ 4).We have applied the new recursion schemes to four      which is imposed with a set of Robin boundary conditions.The beam-type fractional BVP of order 3 <  ≤ 4 with separated boundary conditions and the product nonlinearity is provided in Example 4.Besides the obtained approximate solutions, we also provided the maximal errors (ME  ) and the error remainder functions (ER  ()) for each problem if possible.The results for all examples shown confirm that increasing the number of solution components (i.e., the value of ) reduces the errors in the numerical solutions.Furthermore, unlike the method of undetermined coefficients in the ADM, the Duan-Rach modified decomposition method does not require solving a system of nonlinear algebraic equations obtained from using the -term approximation   () for the remaining unknown constants of integration, which are sometimes multiple roots or nonphysical roots.Hence, the method is very efficient and has provided very accurate approximate solutions when compared with the exact solutions (if any).

Figure 8 :
Figure 8: Logarithmic plots of the maximal errors ME  versus  for  = 2 through 15 obtained by the Duan-Rach modified ADM.

Figure 11 :
Figure 11: Logarithmic plots of the maximal errors ME  versus  for  = 2 through 9 obtained by the Duan-Rach modified ADM.
Since the expressions of the solution   (),  ≥ 2, are quite long, we show only the first two solution components  1 () and  2 () computed using the Adomian polynomials  0 and  1 in (91) as follows:
with the sum of an exponential nonlinearity in the solution and a quadratic nonlinearity in the derivative of the solution International Journal of Mathematics and Mathematical Sciences