Research Article  Open Access
An Efficient HigherOrder Quasilinearization Method for Solving Nonlinear BVPs
Abstract
In this research paper, we present higherorder quasilinearization methods for the boundary value problems as well as coupled boundary value problems. The construction of higherorder 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 onedimension Bratu and FrankKamenetzkii equations.
1. Introduction
In the very beginning, the NewtonRaphson method constructed for nonlinear singlevariable algebraic equation whose efficiency index is optimal according to KungTraub conjecture [1], actually NewtonRaphson uses two function evaluations, and its computational order of convergence is two. Many authors [2–7] have made good effort to construct the iterative methods for algebraic equations. Similarly, for the system of algebraic equation, the version of NewtonRaphson gives us secondorder convergence. In the literature, the efficiency index is only defined for singlevariable 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 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) [8–11] 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 [8–11] provide the secondorder convergence proof for the BVPs. In [13], recently authors proposed higherorder 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 and does not require any calculation of Adomain polynomials, and this fact increases its computational efficiency in comparison with [13]. The sequences of iteration schemes have convergence orders two, three, four, five, and so forth. The numerical stability and efficiency are tested over two problems, namely, onedimensional Bratu problem [16–23] and FrankKamenetzkii [24] boundary value problem The FrankKamenetzkii BVP (2) has no solution if , unique solution if and two solutions if . The closed form solution of (2) is reported in [24–26]. 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 and if , , and , then there are two solutions, unique solution and no solution for (1). The numerical reported value of critical parameter is [18].
2. Construction of Iterative Methods
2.1. Single Nonlinear Boundary Value Problem
Consider a nonlinear ordinary differential equation where is a linear derivative operator; for Bratu and FrankKamenetzkii problems linear operators are and , and is any nonlinear function of . Let be an initial guess (satisfying the boundary conditions) for the solution of (5). By expanding around , we obtain 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.
Scheme . Consider Note that corresponds to the QLM scheme which is quadratically convergent.
Scheme . Consider
Scheme . Consider
Scheme . Consider By calculating computational order of convergence, we show that the order of convergence of is .
Consider the following nonlinear coupled boundary value problem: where and are linear derivative operators. Equation (27) can be rewritten as where and . Let be an initial guess, which satisfies boundary conditions, for problem (28). Taylor’s expansion of around is 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.
Scheme . Consider Note that is the QLM scheme which is quadratically convergent.
Scheme . Consider
Scheme . Consider
Scheme . Consider
2.2. Coupled Boundary Value with Many Variables
Consider the following nonlinear coupled boundary value problem with many variables: where . The compact form of (43) is where , , , . Let be an initial guess for (44) which satisfies the boundary conditions where and . The expansion of around is From (44), we obtain We decompose the solution into infinite series sum Equation (48) implies By adding (51) and (53), we get We denote From (54) and (46), After renaming the variables, we get the following iterative schemes.
Scheme . Consider
Scheme . Consider
Scheme . Consider
Scheme . Consider
3. 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] where , , and are successive iterations closer to the solution of boundary value problem and such that defines the partition of domain of the BVP. For BVPs with infinite domain, for instance, , , or , one could replace infinity by a suitable large number to make the domain compact. The iterative scheme for Bratu problem with boundary conditions is and is an initial guess for (62). Similarly, we can obtain the scheme as follows: The and schemes are respectively. Tables 1 and 2 show the infinity norms of error and rates of convergence for (62), (63), (64), and (65) for , and runs over 50, 100, 150, and 200. We denote and and . The construction of scheme for (2) is given below and others are similar. Consider The initial guess for FrankKamenetzkii problem is used to start the proposed schemes. The infinity norms of error and rates of convergence for FrankKamenetzkii problem () are depicted in Table 4. In order to plot bifurcation diagram for the onedimensional Bratu problem and the FrankKamenetzkii problem, we rewrite the both problems as follows: If , , and , then the Bratu problem has two solutions, unique solution and no solution, respectively, and similarly, for the FrankKamenetzkii problem statement is valid if and . We define B1problem and FK1problem 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 B1problem 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 B1problem and FK1problem.


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 scheme0 in [13] and 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 (Scheme0, , 6.66e−49), (Scheme2, , 6.34e−49), (Scheme2, , 6.34e−49), and (Scheme3, , 6.34e−49) in Table 1 [13] and in this article, in Table 1 (, , 7.7e−49), (, , 1.67e−69), (, , 1.67e−69), (, , 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 FrankKamenetzkii 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 FrankKamenetzkii 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.
