Abstract
We introduce a new family of iterative methods for solving mathematical models whose governing equations are nonlinear in nature. The new family gives several iterative schemes as special cases. We also give the convergence analysis of our proposed methods. In order to demonstrate the improved performance of newly developed methods, we consider some nonlinear equations along with two complex mathematical models. The graphical analysis for these models is also presented.
1. Introduction
Solving nonlinear equations is one of the important problems in mathematical sciences, especially in numerical analysis. There is a vast literature available to find the solution of nonlinear equations; see, for example, [1–24] and references therein. The construction of numerical methods is usually based on diverse techniques such as Taylor series, quadrature formulas, homotopy perturbation, and decomposition. One of the most powerful and well-known techniques for finding the solution of nonlinear equations is Newton’s method which converges quadratically [19].To improve the efficiency, several modified higher order methods have been presented in the literature by using different techniques [1–6, 9–12, 14–20, 22–24]. Recently, some useful methods have been introduced in [7, 8, 13, 21]. Abbasbandy [1] used Adomian decomposition method (ADM) [2] to find the simple root of nonlinear equations. It is worth mentioning here that the involvement of higher order derivatives of Adomian polynomials is a major weakness of ADM. This weakness was eradicated by Daftardar-Gejji and Jafari [5], when they introduced a new decomposition technique. The said decomposition technique is quite simple as compared to the ADM as it does not need the higher order derivatives of functions. This technique has extensively been used to develop some useful algorithms for solving nonlinear equations [4, 5, 10, 18]. In , Noor et al. [18], using the same decomposition technique along with the idea of coupled system, proposed two fourth-order iterative methods ([18], Algorithms 2.12 and 2.13) with efficiency index of each.
In this work, using the quadrature formula along with the fundamental law of calculus, truncating the series of at quadratic level and decomposition technique of [5], we construct a family of new iterative methods for solving nonlinear equations. As special cases, we propose two fourth-order and two sixth-order methods. The number of evaluations per iteration for the third-order methods is and ; the fourth-order methods and ; and sixth-order methods and ; thus the efficiency indices of our methods are , and The convergence criteria of newly constructed families are also presented. In order to demonstrate the validity and better efficiency of our proposed methods, we solve the nonlinear equations arising in the population model and in the motion of a particle on an inclined plane [18]. We also present the graphical analysis for the endorsement of numerical results.
2. Iterative Methods
Let be the simple root of the nonlinear equation:We rewrite (2) in the form of the following coupled system using the quadrature formula and fundamental law of calculus:where denotes the initial approximation sufficiently close to the exact root of (2), are knots in , and are the weights satisfyingEquation (3) can be written as follows:whereHere is a nonlinear operator which can be decomposed as follows using the decomposition technique mainly due to Daftardar-Gejji and Jafari [5].The purpose of the above decomposition is to find the solution of (2) in the form of a series:Combining (7), (10), and (11), we getThus, we have the following iterative scheme:Therefore, we haveSince , (11) givesFrom (8) and (13), we haveFrom (4), one can easily computeUsing (9), (13), and (16), we haveUsing condition (5) and (17), the above equation yieldsWe note that is approximated asFor , in (20) and using (16) and (19), we haveThis formulation allows us to propose the following iterative method for solving nonlinear equation (2).
Algorithm 1. For a given , compute the approximate solution by the following iterative scheme:which is the well-known Newton’s method for solving nonlinear (2). Now from (21), we haveFrom (4) and using (23), the following can easily be obtained:Similarly, from (9), by using (14) and (24), we haveFor , in (20) and by using (13), (14), (16), and (25), we getUsing above relation, we can suggest the following two-step iterative method for solving nonlinear equation (2).
Algorithm 2. For a given , compute the approximate solution by the following iterative scheme:From (26), we obtainedFrom (4) and by using (28), we haveFrom (9), by using (14) and (29), we obtainFor , in (20), we haveUsing this formulation, we suggest the following three-step iterative method for solving nonlinear equations (2).
Algorithm 3. For a given , compute the approximate solution by the following iterative scheme:According to our knowledge, Algorithms 2 and 3 are new to solve nonlinear equation (2).
2.1. Some Special Cases of Algorithm 2
Now, we present some special cases of Algorithm 2. For and , Algorithm 2 reduces to the following iterative method for solving nonlinear equations.
Algorithm 4. For a given , compute the approximate solution by the following iterative schemeUsing and , Algorithm 2 reduces to the following iterative method.
Algorithm 5. For a given , compute the approximate solution by the following iterative scheme:Now, using , and , Algorithm 2 reduces to the following iterative method.
Algorithm 6. For a given , compute the approximate solution by the following iterative scheme:Now by using , and , Algorithm 2 reduces to the following iterative method for solving .
Algorithm 7. For a given , compute the approximate solution by the following iterative scheme:To the best of our knowledge, Algorithms 4–7 are new iterative methods for solving nonlinear equation (2).
2.2. Some Special Cases of Algorithm 3
Now, we present some special cases of Algorithm 3. For , and , Algorithm 3 reduces to the following iterative method for solving nonlinear equations.
Algorithm 8. For a given , compute the approximate solution by the following iterative scheme:Using , and , Algorithm 3 reduces to the following iterative method for solving .
Algorithm 9. For a given , compute the approximate solution by the following iterative scheme:Now, using , and , Algorithm 3 reduces to the following iterative method for solving .
Algorithm 10. For a given , compute the approximate solution by the following iterative scheme:Using , , , and , Algorithm 2 reduces to the following iterative method for solving .
Algorithm 11. For a given , compute the approximate solution by the following iterative scheme:
To the best of our knowledge, Algorithms 8–11 are new iterative methods for solving nonlinear equation (2).
3. Convergence Analysis
In this section, convergence criteria of newly suggested methods are studied in the form of the following theorem.
Theorem 12. Assume that the function on an open interval has a simple root . Let be sufficiently differentiable in the neighborhood of ; then the convergence orders of the methods defined by Algorithms 2–11 are and , respectively.
Proof. Let be a simple zero of . Since is sufficiently differentiable, the Taylor series expansions of , , and about are given bywhere and
From (41) and (42), we getUsing (44), we haveFrom (45), we getNow, the Taylor’s series expansions of and are given byUsing (45), (47), and (48), we obtain the error term for Algorithm 2 as follows:From (49), we haveThe Taylor’s series expansions of and are given byUsing (49) and (51), the error term for Algorithm 3 is obtained asNow, we prove the convergence orders of the special cases of Algorithms 2 and 3.
The Taylor’s series expansion of is given byUsing (45), (46), and (53), the error term for Algorithm 4 is obtained asNow, expending by Taylor’s series, we getUsing equations (45), (46), and (55), the error term for Algorithm 5 is obtained asThe Taylor’s series expansions of and are given byUsing (45), (46), (57), and (58), we obtain the error term for Algorithm 6 as follows:Expanding and by Taylor’s series, we haveUsing (42), (43), (45), (46), and (60), the error term for Algorithm 7 is obtained asUsing (54), we haveUsing (42), (43), (54), and (62), the error term for Algorithm 8 is obtained asNow, from (56), we haveUsing (56) and (64), the error term for Algorithm 9 is obtained asUsing (58), we getTaylor’s series expansions of and are given byTherefore, using (59), (66), and (67), the error term for Algorithm 10 is obtained asUsing (61), we haveTaylor’s series expansions of and are given byUsing (61), (69), and (70), the error term for Algorithm 11 is obtained asThis complete the proof.
4. Numerical Examples
In this section, we demonstrate the validity and efficiency of our proposed iterative schemes by considering the nonlinear equations obtained from population model and the motion of a particle on an inclined plane, that is,We take for (72) and for (73) as initial guess for computer program. Tables 1 and 2 and Figures 1 and 2 give the comparison of our newly proposed iterative methods, that is, Algorithm 6 (AG ), Algorithm 7 (AG ), Algorithm 10 (AG ), and Algorithm 11 (AG ) with standard Newton’s method (NM, (1)), Noor et al. methods ([18], Algorithm 2.6) (NR ), ([18], Algorithm 2.7) (NR ), ([18], Algorithm 2.12) (NR ), and ([18], Algorithm 2.13) (NR ). In Table 3, we present numerical results for various nonlinear equations which also indicate that our methods are more efficient.
We use Maple for all numerical computations and MATLAB to plot the graphs. For the computer programs, we use the following stopping criteria:
5. Conclusions
We have introduced a new family of iterative methods (Algorithms 2 and 3), based on decomposition technique, for solving nonlinear equations using coupled system of equations. Several new iterative methods have been established as special cases of newly established family. We have explored the convergence criteria of our new methods and investigated for convergence order and efficiency index. We present the comparative study both numerically and graphically (Tables 1 and 2 and Figures 1 and 2) of our newly constructed methods with some known methods by considering two real life models. In Table 3, we present numerical results by considering various nonlinear equations.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.