Semi-Local Analysis and Real Life Applications of Higher-Order Iterative Schemes for Nonlinear Systems

: Our aim is to improve the applicability of the family suggested by Bhalla et al. (Computational and Applied Mathematics, 2018) for the approximation of solutions of nonlinear systems. Semi-local convergence relies on conditions with ﬁrst order derivatives and Lipschitz constants in contrast to other works requiring higher order derivatives not appearing in these schemes. Hence, the usage of these schemes is improved. Moreover, a variety of real world problems, namely, Bratu’s 1D, Bratu’s 2D and Fisher’s problems, are applied in order to inspect the utilization of the family and to test the theoretical results by adopting variable precision arithmetics in Mathematica 10. On account of these examples, it is concluded that the family is more efﬁcient and shows better performance as compared to the existing one.

Bhalla et al. [21] have also presented the following scheme: where Here α 1 and α 2 are non-real parameters provided both parameters do not equal zero simultaneously. In particular, for global convergence schemes for algebraic systems can be found in [22][23][24]. However, here we work on Banach spaces to determine only a locally unique solution.
Let us consider as a motivational example the nonlinear mixed Hammerstein integral type appearing in numerous studies [9][10][11][12] given by where Solving Equation (7) is equivalent to solving Q(ϑ) = 0, with Q : D ⊆ C[0, 1] → C[0, 1] given as It follows that The local convergence analysis given in [21] uses Taylor expansions and derivatives up to the eighth order. Consequently, these results cannot be used on Q(ϑ) = 0. These types of suppositions confine the usage of Scheme (3) and related schemes. Motivated by these problems, we develop a semi-local analysis for Scheme (5) but with suppositions only for the first Fréchet derivative as well as divided differences of order one (see, in particular, the continuation of this example in Example 1). Equation (10) is used to determine how close the initial guess should be to the solution to assure convergence of the method. Moreover, the (H) conditions that follow indicate the constraints on the involved operators. Notice also that the work in [22][23][24] cannot be used to solve the motivational example.
Let the (H) conditions be used in the semi-local convergence in the following way Function v defined by Equation (10) has the smallest positive zero denoted by R. (9) and (11)

Conditions
Let U(γ, ρ),Ū(γ, ρ) be, respectively, open and closed balls in B centered at γ ∈ B and of radius ρ > 0. Next, let the local convergence analysis of Scheme (5) be under the (H) conditions as follows: Theorem 1. Let us assume that the (H) conditions are satisfied. Then, the sequence achieved by Scheme (5) is valid, remains inŪ(ϑ 0 , R) and converges to the required solution ϑ * of expression Γ(ϑ) = 0.
Next, a uniqueness result is presented.
Proposition 1. By adopting hypotheses (H), we consider that there will be an R 1 ≥ R such that w 0 (R, R 1 ) < 1.

Numerical Results
Several real life problems are provided to illustrate the convergence nature of the schemes shown in Scheme (5). Therefore, we consider all five higher-order methods out of our proposed scheme, namely (ψ 2 for α 1 = ±10 20 , denoted by (OM1 4 ), (OM2 4 ), (OM3 6 ), (OM4 6 ), (OM5 8 ), (OM6 8 ) and (OM7 8 ), respectively, to test the convergence criteria. Out these seven schemes first two are of order four; the third and fourth are of order six; and the last three are of order eight. We compare them with fourth-order schemes proposed by Sharma and Arora [25], out of them we consider Equations (2) and (3) denoted by (SA1 4 ) and (SA2 4 ), respectively. In addition, we consider two higher-order methods of order four and six presented by Grau et al. [18], out of them we choose Equations (12) and (14), denoted by (GM1 4 ) and (GM2 6 ), respectively. Finally, we also compared them with seventh-order methods given by Wang and Zhang [26]; out them we consider Equations (56) and (57), denoted WZ1 7 and WZ2 7 , respectively.
For fair contrast of our schemes, we have depicted the error among the exact and estimated zero ϑ m − ϑ * ; COC stands for computational order of convergence, and CPU stands for time in Tables 1-4. We used the succeeding formulas [14] or COC and the approximate computational order of convergence (ACOC), respectively. It is vital to note that ρ or ρ * do not need any kind of higher-order derivatives to compute the error bounds. The notation a 1 (±a 2 ) employs a 1 × 10 (±a 2 ) .
The computational works are performed with programming software Mathematica-10 (Wolfram Research, Champaign, IL, USA) [27] and the configurations of our computer are given below: We consider at least 1000 digits of mantissa in order to minimize the round-off errors. In addition, all the problems are first transformed into nonlinear systems of equations and then solved using the proposed iterative scheme. (7), for [ϑ, Λ : Γ] = 1 0 Γ (Λ + v(ϑ − Λ))dv, we see that our results can apply, if we choosew

Example 1. Returning back to Equation
for ϑ 0 sufficiently close to the solution ϑ * = 0.

Example 2. Bratu Problem:
Here, we assume the well known Bratu Problem [28], which is given by It has a wide area of application, for example, in radioactive heat transfer, thermal reaction, the fuel ignition model of thermal combustion, chemical reactor theory, the Chandrasekhar model of the expansion of the universe and nanotechnology [28][29][30][31].
The finite difference discretization is used to convert the above boundary value problem, Equation (42), into a non-linear system of size 40 × 40 with step size h = 1/41. For second derivative central difference has been used and is as follows The computational comparison of the solution of this problem is shown in Table 1 and the graphical solution in Figure 1.

(43)
Let us assume that Γ i,j = u(x i , t j ) is a numerical result over the grid points of the mesh. In addition, we consider that τ 1 and τ 2 are the number of steps in the direction of x and t, respectively. Moreover, we choose that h and k are the respective step sizes in the direction of x and y, respectively. In order to find the solution of partial differential equation (PDE) (43), we adopt the following approach which further yields the succeeding system of nonlinear equation (SNE) 2, 3, . . . , τ 1 , j = 1, 2, 3, . . . , τ 2 .

Example 4. Fisher's Equation:
Here, we assume another typical non-linear Fisher's equation [34], which is given by where D is the diffusion coefficient. Let us assume that Γ i,j = u(x i , t j ) is a numerical result over the grid points of the mesh. In addition, we consider τ 1 and τ 2 to be the number of steps in the direction of x and t, respectively. Moreover, we choose that h and k are the respective step sizes in the direction of x and y, respectively. In order to find the solution of PDE (46), we adopt the following approach which further yields the succeeding SNE By choosing τ 1 = τ 2 = 21, h = 1 τ 1 and k = 1 τ 2 , we get a large SNE of order 400 × 400. The starting point is ϑ 0 = (i/(τ 1 − 1) 2 ) T , i = 1, 2, . . . , τ 1 − 1 and results are mentioned in Table 3. The approximate solution has been plotted in Figure 3.
Remark 1. It follows from Tables 1-4 that our methods have a very small error difference between the exact and approximated root as compared to the other mentioned methods. In addition, they have a more stable computational order of convergence and take less CPU time for better accuracy in the required zero.
In some tables the values of error approximations ϑ (m) − ϑ * appear the same for different α 1 and α 2 , but actually they are different; if we mention the errors in more significant digits in those tables then we can see the clear difference. However, due to limited page space only three significant digits are depicted for different α 1 and α 2 .       (4).

Concluding Remarks
In this work, semi-local convergence analysis of the family proposed by [21] has been proved by adopting Lipschitz constants and order one divided differences on a Banach space setting under weak conditions, so that we can expand the applicability of Scheme (5) and other related schemes. The use of this family on real life problems, namely Bratu's 1D (SNE 40 × 40), Bratu's 2D (SNE 100 × 100), Fisher's problem (SNE 400 × 400) and polynomial equations (SNE 100 × 100), also confirms the applicability of this family.