Abstract
Various iterative methods have been introduced by involving Taylor’s series on the auxiliary function to solve the nonlinear equation . In this paper, we introduce the expansion of with the inclusion of weights such that and knots in order to develop a new family of iterative methods. The methods proposed in the present paper are applicable for different choices of auxiliary function , and some already known methods can be viewed as the special cases of these methods. We consider the diverse scientific/engineering models to demonstrate the efficiency of the proposed methods.
1. Introduction
Most of the problems in science and engineering involves nonlinear equation of the form , where is a sufficiently smooth function in the neighborhood of a simple zero . Many physical problems related to diverse areas such as biological applications in population dynamics and genetics where impulses arise naturally, motion of a particle on an inclined plane and projectile motion in physics, Van der Waals problem, and continuous stirred tank reactor equation in chemistry etc., can be modelled by nonlinear equations. Consequently, many numerical methods based on different techniques have been developed for solving nonlinear equations, see for example [1–12] and references therein. The following quadratically convergent Newton’s method [13], which needs function evaluations per iteration, is considered as the fundamental tool for obtaining the numerical solution of nonlinear equations:
Definition 1. (see [12]).
Let be the convergence order of the iterative method and is the number of functional evaluations per iteration required by the method, then the efficiency index (IE) of the method is defined asObviously, the efficiency index (IE) of Newton’s method is 1.4142.
An appropriate selection of an initial guess makes Newton’s method very efficient, whereas this method works ordinarily for an inappropriate initial guess. In second century BC, an ancient Chinese, Ying Buzu Shu, introduced a more useful method, which gives accurate results in some cases even where Newton’s method does not work, for which two initial approximations are chosen. Using and as initial approximations, the next approximation can be obtained asDuring the last two decades, ancient Chinese algorithms have been a scorching topic for many researchers [14–16]. In 2016, another Chinese He [17] introduced two modifications (Bubbfil Algorithms 1 and 2) of the above method with fast convergence.
Several modifications of Newton’s method have also been introduced, e.g., [5, 12, 18, 19]. Weerakon and Fernando [20] constructed the following useful cubically convergent method:The midpoint rule [19, 21],with an order of convergence 3, is also a well-known technique. There exist several other third- and fourth-order methods involving diverse techniques for nonlinear equations, e.g., [5, 18, 22, 23]. Abbasbandy [1] used the Adomian decomposition method (ADM) [2] to find the simple root of nonlinear equations. But employing ADM involves higher order derivatives of Adomian polynomials which is a major weakness of ADM. But this is not the case in the decomposition technique due to Cherruault [24, 25]. The technique of Cherruault has extensively been used to develop some useful algorithms for solving nonlinear equations [3, 9, 26]. Recently, Shah et al. [27], using the same technique and an auxiliary function along with the expansion of the original equation , have introduced a family of iterative methods for nonlinear equations.
In the present paper, making use of the decomposition technique of [24, 25], together with the expansions of both and , we construct a new family of swiftly convergent iterative methods. In order to exhibit the efficiency of proposed methods, we present numerical as well as graphical analysis by considering three mathematical models from different branches of science, i.e., physics, mathematics, and chemistry.
2. Construction of Methods
Assume the nonlinear equation:with a simple zero and the initial guess sufficiently close to . We take as an auxiliary function, such that is sufficiently smooth and ,
Using the quadrature formula and the fundamental law of calculus, the following coupled system can be formed:where and represents and respectively; and are weights and knots, respectively, such that and .
We write equation (8) in terms of nonlinear operator as follows:where
We construct a sequence of higher order iterative methods by decomposing in the following way [24, 25]:
Our purpose is to seek the solution in the form of the following series:
Combining equations (10), (13), and (14), we get
Thus, we have the following iterative scheme:
Thus,
From equations (11), (12), and (16), we have
Using equation (9), we get
We note that is approximated aswhere
For , we have
Thus, the following recurrence relation can be formed by using equation (22).
Algorithm 1. (28).i.e., corresponding to the initial approximation , approximation can be determined. It is notable that the above algorithm is actually Newton’s method applied to .
Now, for , equation (20) givesUsing equation (12), we haveUsing equations (9), (16), (22), and (24), we getThus, the following recurrence relation can be formed using equation (26).
Algorithm 2. which is a new predictor-corrector method, and corresponding to initial approximation , approximation can be determined.
Now, for , equation (20) givesUsing equation (12), we haveUsing equations (9), (16), (26), and (28), we getThus, the following recurrence relation can be formed using equation (30).
Algorithm 3. which is also a new three-step iterative technique.
2.1. Some Special Cases of Algorithm 2
Algorithm 4. Taking , and , Algorithm 2 reduces to the following iterative method:Shah and Noor [28] have established the above formula.
Algorithm 5. Taking , and , Algorithm 2 reduces to the following iterative method.
Algorithm 6. Taking , and , Algorithm 2 reduces to the following iterative method:
Algorithm 7. Taking , and , Algorithm 2 reduces to the following iterative method:
Algorithm 8. Taking , and , Algorithm 2 reduces to the following iterative method:To the best of our knowledge, Algorithms 5–8 are new methods having convergence orders and 4, respectively.
2.2. Some Special Cases of Algorithm 3
Algorithm 9. Taking , and , Algorithm 3 reduces to the following iterative method:
Algorithm 10. Taking , and , Algorithm 3 reduces to the following iterative method:
Algorithm 11. Taking , and , Algorithm 3 reduces to the following iterative method:
Algorithm 12. Taking , and , Algorithm 3 reduces to the following iterative method:
Algorithm 13. Taking , and , Algorithm 3 reduces to the following iterative method:To the best of our knowledge, Algorithms 10–13 are new iterative methods for solving nonlinear equation (6) with convergence orders and 4, respectively.
Obviously, different iterative methods can be obtained from Algorithms 2 and 3 by using distinct values of the auxiliary function . Suitable selection of auxiliary function plays a significant role for the better performance of these methods. To demonstrate the efficiency of our proposed algorithms, we choose . Thus, corresponding to this choice, algorithm , , is the special case of algorithm .
Algorithm 14. The convergence order of the method described in Algorithm 14 is 3, and the total number of evaluations per iteration is 3. Thus, .
Algorithm 15. The convergence order of the method described in Algorithm 15 is 3, and the total number of evaluations per iteration is 4. Thus, .
Algorithm 16. The convergence order of the method described in Algorithm 16 is 3, and the total number of evaluations per iteration is 4. Thus, .
Algorithm 17. The convergence order of the method described in Algorithm 17 is 3, and the total number of evaluations per iteration is 4. Thus, .
Algorithm 18. The convergence order of the method described in Algorithm 18 is 3, and the total number of evaluations per iteration is 4. Thus, .
Algorithm 19. The convergence order of the method described in Algorithm 19 is 4, and the total number of evaluations per iteration is 4. Thus, .
Algorithm 20. The convergence order of the method described in Algorithm 20 is 4, and the total number of evaluations per iteration is 6. Thus, .
Algorithm 21. The convergence order of the method described in Algorithm 21 is 4, and the total number of evaluations per iteration is 6. Thus, .
Algorithm 22. The convergence order of the method described in Algorithm 22 is 4, and the total number of evaluations per iteration is 6. Thus, .
Algorithm 23. The convergence order of the method described in Algorithm 23 is 4, and the total number of evaluations per iteration is 6. Thus, .
3. Convergence Analysis
In this section, convergence criteria of proposed algorithms are studied in the form of the following theorem.
Theorem 1. 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 and 3 are three and four, respectively.
Proof. Let be a simple zero of . Since is sufficiently differentiable, the Taylor series of and about are given bywhere and ,
Expanding and by Taylor series, we getFrom equations (53), (54), and (55), we getUsing equation (56), we haveThe Taylor series of and are given aswhereLet . Taylor’s expansions of and are given byNow, expanding and by Taylor series, we getUsing equations (61)–(63), we obtainUsing equation (64) and the fact , the error term for Algorithm 2 is as follows:The last expression shows that the convergence order of Algorithm 2 is 3.
The Taylor series expansions of and are given bywhereLet . Thus, Taylor’s expansions of and are given byNow, expanding and by Taylor series, we getUsing equations (69), (70), and (71), we obtainThus, by using equation (72) and the fact , the error term for Algorithm 3 is as follows:This completes the proof.
4. Applications in Science and Engineering
In this section, we consider three scientific and engineering models related to mathematics, physics, and chemistry which include motion of a particle on an inclined plane, Van der Waal’s equation of state, and continuous stirred tank rector equation. We make use of Maple software for computational work and MATLAB for graphical analysis. We apply the methods proposed in Algorithm 15 (AG1), Algorithm 16 (AG2), Algorithm 20 (AG3), and Algorithm 21 (AG4) along with Weerakon and Fernando method (WF, equation (3)), midpoint method (MP, equation (4)), Shah et al.’s [27] methods (Algorithm 11 (SH1), Algorithm 13 (SH2), and Algorithm 14 (SN3)), Noor et al.’s [9] method (Algorithm 2.13 (NR1)), Shah and Noor’s [28] method (Algorithm 2.6 (NR2)), and basic Newton’s method (NM, equation (1)) to show that the methods proposed in the present article work more efficiently (see the following Tables 1–3 and Figures 1–3).
We use the following stopping criteria for computer programmes:(1)Motion of particle on an inclined plane [9]. We solve the nonlinear model formed due to the motion of a particle on an inclined plane whose angle of inclination changes at a constant rate . with(2)Van der Waals equation of state [29]. Now, we consider the governing nonlinear equation for the special case of well-known Van der Waals equation of state, i.e., Particularly, we solve the above equation in the following form: with(3)Continuos stirred tank reactor (CSTR) [30]. We consider the nonlinear model of continuous stirred tank reactor, i.e., with
5. Conclusions
In this paper, introducing the weights and knots in the expansion of the auxiliary function , i.e., different from usual cases, we have developed a new family of iterative methods for nonlinear equations, i.e., Algorithms 15, 16, 20, and 21. We have compared our methods with some existing methods both numerically and graphically. The results obtained clearly reveal that the proposed methods are more rapidly convergent methods. These methods, besides giving some more efficient new techniques, are the generalized shape of some well-known methods.
Data Availability
All the data and supplementary material are available, if needed.
Conflicts of Interest
The authors declare that there are no conflicts of interest.