An Efficient Higher-Order Quasilinearization Method for Solving Nonlinear BVPs

In this research paper, we present higher-order quasilinearization methods for the boundary value problems as well as coupled boundary value problems. The construction of higher-order convergent methods depends on a decomposition method which is different from Adomain decomposition method (Motsa and Sibanda, 2013). The reported method is very general and can be extended to desired order of convergence for highly nonlinear differential equations and also computationally superior to proposed iterative method based on Adomain decomposition because our proposed iterative scheme avoids the calculations of Adomain polynomials and achieves the same computational order of convergence as authors have claimed in Motsa and Sibanda, 2013. In order to check the validity and computational performance, the constructed iterative schemes are also successfully applied to bifurcation problems to calculate the values of critical parameters. The numerical performance is also tested for one-dimension Bratu and Frank-Kamenetzkii equations.


Introduction
In the very beginning, the Newton-Raphson method constructed for nonlinear single-variable algebraic equation whose efficiency index is optimal according to Kung-Traub conjecture [1], actually Newton-Raphson uses two function evaluations, and its computational order of convergence is two.Many authors [2][3][4][5][6][7] have made good effort to construct the iterative methods for algebraic equations.Similarly, for the system of algebraic equation, the version of Newton-Raphson gives us second-order convergence.In the literature, the efficiency index is only defined for single-variable algebraic equation on the basis of function evaluations.For multivariable case, if we consider system of  algebraic equations, then we require the computation of Jacobian inverse  −1 of dimension  ×  and  function evaluations to perform one newton iteration and it is clearly evident the computational cost of matrix inversion is dominant over other binary operations to complete one newton iteration.The iterative methods which require only once the Jacobian inverse for the whole cycle of iterations are clearly efficient.
When we talk about nonlinear boundary value problems, the quasilinear method (QLM) [8][9][10][11] is an iterative method which starts from initial guess for a boundary value problem (BVP) which is quadratically convergent.Initially, Bellman and Kalaba [12] proposed QLM and later Mandelzweig and coauthors [8][9][10][11] provide the second-order convergence proof for the BVPs.In [13], recently authors proposed higher-order quasilinearization method for single BVP as well as coupled BVPs.The original idea in [13] is to decompose the nonlinear operator as an infinite sum of Adomain [14] polynomials.The reported algorithm is efficient in the case of couple BVPs.The computation of Jacobian inverse is performed at the initial guess, but the calculation of Adomain polynomials is somehow difficult and also increases the computation cost of iterative scheme.To avoid the computational burden of Adomain polynomials, we use a different decomposition method for nonlinear operator which was actually introduced in [15].Our proposed scheme uses only one calculation of Jacobian inverse  −1 and does not require any calculation of Adomain polynomials, and this fact increases its computational efficiency in comparison with [13].The sequences of and Frank-Kamenetzkii [24] boundary value problem ( The Frank-Kamenetzkii BVP (2) has no solution if  > 2, unique solution if  = 2 and two solutions if  < 2. The closed form solution of ( 2) is reported in [24][25][26].The solutions of (2) in [25] are given as The closed form solution for Bratu equation [27] can be written as The critical parameter   for Bratu problem satisfies 4 = √2  sinh(  /4) and if  <   ,  =   , and  >   , then there are two solutions, unique solution and no solution for (1).The numerical reported value of critical parameter is   = 3.51383071912516 [18].

Construction of Iterative Methods
Suppose, that we can decompose the solution  into infinite series sum Further, we obtain the decomposition of nonlinear operator () as follows: By substituting (11) in (10), we get By using (13), we obtain By comparing left and right sides in (15), we have If we approximate solution By adding (16) to (18), we get Equations ( 16) and (21) give After renaming the variables, we obtain the following iterative schemes.
Note that  0 corresponds to the QLM scheme which is quadratically convergent.
By calculating computational order of convergence, we show that the order of convergence of   is  + 2.
Consider the following nonlinear coupled boundary value problem: where  = [ (1) ,  (2) ,  (3) , . . .,  () ]  and  1 ,  2 ,  3 , . . .,   are linear derivative operators.Equation ( 27) can be rewritten as where  = [ 1 , where  is a Jacobian  =         = , Equation ( 28) can be written as where  can be decomposed into infinite series sum Substituting ( 33) in (32), we get By using the same decomposition for nonlinear operator () for multivariable case which is given in (12), we obtain If we approximate the solution by we get by adding (35) From (32), we have After renaming the variables, we get the following iterative schemes.

Numerical Results and Rate of Convergence
In all numerical experimentation, we use Chebyshev pseudospectral methods (for more details, see [13]).In order to show the rate of convergence, we require the definition of computational order of convergence, The computational order of convergence can be approximated by [28] COC ≈ ln      ( +1 − ) (  − ) and  0 () = (1 − ) is an initial guess for (62).Similarly, we can obtain the scheme  1 as follows: The  2 and  3 schemes are respectively.Tables 1 and 2 show the infinity norms of error and rates of convergence for (62), ( 63 (66) The initial guess  0 () = 1 −  2 for Frank-Kamenetzkii problem is used to start the proposed schemes.The infinity norms of error and rates of convergence for Frank-Kamenetzkii problem ( = 1,  = 1) are depicted in Table 4.In order to plot bifurcation diagram for the one-dimensional Bratu problem and the Frank-Kamenetzkii problem, we rewrite the both problems as follows: () +  () = 0,  (0) =  (1) = 0, If  <   (= 3.51383071912516),  =   , and  >   , then the Bratu problem has two solutions, unique solution and no solution, respectively, and similarly, for the Frank-Kamenetzkii problem statement is valid if  = 1 and   = 2.
We define B1-problem and FK1-problem respectively.The iterative forms of (68) and (69) are The bifurcation diagrams are shown in Figures 1 and 2. The calculation of critical parameters for the B1-problem and FK1problem is performed by using proposed system of equation in [13] and Tables 2, 5, and 6 show numerical results of different iterative schemes for B1-problem and FK1-problem.The authors of [13] are pioneer to talk about higherorder iterative quasilinearization method (QLM).Tables 1, 2, and 4 confirm the orders of convergence of their respective iterative schemes for the different values of parameters under a different range of grid points for Chebyshev pseudospectral method.We make all the calculation for Tables 1, 2, and 4 in Mathematica (MinPrecision = 200).The scheme-0 in [13] and  0 in this paper are the same because both are QLM and the calculated results should be same but unfortunately this is not the case.For the QLM, the infinity norm of error for Bratu problem is (Scheme-0,  = 50, 6.66e−49), (Scheme-2,  = 100, 6.34e−49), (Scheme-2,  = 150, 6.34e−49), and (Scheme-3,  = 200, 6.34e−49) in Table 1 [13] and in this article, in Table 1 ( 0 ,  = 50, 7.7e−49), ( 1 ,  = 100, 1.67e−69), ( 2 ,  = 150, 1.67e−69), ( 3 ,  = 200, 1.67e−69) and in other cases, our results show better reduction in error as compared to [13] for Table 1 as well as for Table 2 for Bratu problem.For the case of Frank-Kamenetzkii problem, Table 4 results are comparable with the reported results in Table 4 [13] and especially for QLM case almost results are the same.The Table 3 shows the computation of critical parameter   for Bratu problem, and again the results produced in this article and in [13] for QLM are not the same which in fact should be the same.The results presented by other authors are surprisingly superior for QLM and for other schemes.Notice that we use Matlab to compute critical parameters.Tables 5 and 6 of [13] show better performance and our results are comparable with them.It is noticed that in [13], the constructed matrix systems (33) and (34) for the Bratu and the Frank-Kamenetzkii problems have not properly implemented for all boundary conditions.It is also noted that in some of the cases, if we increase the grid points by keeping the same scheme, there is an improvement in the accuracy of the calculated results, but in some cases this is not valid rule.It is also very clear from Tables 1, 2, and 4, that we ensure the convergence order which was the claim.

Conclusions
An efficient method is presented in this article to get arbitrary higher-order iterative schemes.The construction of iterative schemes is very simple and straightforward.In [13], authors used the idea to decompose the nonlinear operator by using Adomain decomposition method (ADM) which requires the calculation of Adomain polynomials and in a result, the computational cost will be high.In our case, there is no need to calculate any extra polynomial in order to enhance the order of convergence of an iterative method and both methods require only one inversion of Jacobian.Our presented method is computationally efficient because it does not require to construct any Adomain polynomial and hence implementation is also simple to achieve the same order of convergence.We think that the results presented in [13] and in this article should be the same for the case of QLM, but it is not the case and we do not know the implementation of the iterative schemes in [13].

Table 1 :
Convergence order and absolute error for Bratu problem for  = 1 and different values of .

Table 2 :
Convergence order for Bratu problem for  = 3 and different values of .

Table 3 :
Iterative convergence for different schemes for the critical value of   for Bratu problem,  = 200.

Table 4 :
Convergence order for Frank-Kamenetzkii problem for  = 1,  = 1 and different values of .

Table 5 :
Iterative convergence for different schemes for the critical value of   for Frank-Kamenetzkii problem,  = 200.

Table 6 :
Iterative convergence for different schemes for the critical value of   for Frank problem,  = 200.