Novel higher order iterative schemes based on the q − Calculus for solving nonlinear equations

: The conventional inﬁnitesimal calculus that concentrates on the idea of navigating the q − symmetrical outcomes free from the limits is known as Quantum calculus (or q − calculus). It focuses on the logical rationalization of di ﬀ erentiation and integration operations. Quantum calculus arouses interest in the modern era due to its broad range of applications in diversiﬁed disciplines of the mathematical sciences. In this paper, we instigate the analysis of Quantum calculus on the iterative methods for solving one-variable nonlinear equations. We introduce the new iterative methods called q − iterative methods by employing the q − analogue of Taylor’s series together with the inclusion of an auxiliary function. We also investigate the convergence order of our newly suggested methods. Multiple numerical examples are utilized to demonstrate the performance of new methods with an acceptable accuracy. In addition, approximate solutions obtained are comparable to the analogous solutions in the classical calculus when the quantum parameter q tends to one. Furthermore, a potential correlation is established by uniting the q − iterative methods and traditional iterative methods.


Introduction
It is widely acknowledged that the study of the nonlinear equations characterized as f (ũ) = 0 can be used to investigate a broader range of problems that occur in physical sciences.Because of its significance, many scientists have investigated numerous different order multistep methods to explore the solutions of the nonlinear equations using diversified approaches; such as variational iterative methods, homotopy perturbation method, homotopy analysis method, and the decomposition techniques, for details see [1][2][3][4][5][6][7][8][9].These developed approaches are of varying order of convergence.Firstly, Traub [16] has initiated the study of repetitious schemes for solving nonlinear equations who developed a central quadratic convergent Newton iterative method which has much importance in literature.
Later on, to increase the practical usefulness and efficiency index of Newton's method, its various rectifications have been presented by many researchers (see [17][18][19][20]).Daftardar-Gejji and Jafari [21] have proposed a straightforward approach that does not necessitate the derivative evaluation of the Adomian polynomial by making different modifications in the Adomian decomposition method [1].Moreover, this technique helps us to write the nonlinear equation as a combination of both linear and nonlinear components, and plays a remarkable role in developing different iterative schemes to estimate the solution of the nonlinear equations.Saqib and Iqbal [22] have determined the fourth and fifthorder convergent iterative methods for computing roots of the nonlinear equations by using a modified decomposition approach and presented some test examples to check the efficacy and performance of the newly established iterative methods.Ali et al. [23] have established a new class of the iterative methods by implementing the technique [21] and testified the validity of these schemes by considering some mathematical models.Alharbi et al. [2] have introduced some new and efficient iterative methods and implemented a decomposition technique along with an auxiliary function.Variational iteration method (VIM) is another effective tool that is employed to develop effective iterative methods for getting approximate, converging solutions of the nonlinear equations.Based on VIM, Naseem et al. [17] have investigated a new class of iterative methods that are superior in convergence and efficient as compared to other methods.They also elaborated the behavior and dynamical aspects of the suggested iterative schemes by using polynomiographs.The q−calculus is materialized as the composition of Physics and Mathematics in the last twenty-five years of the XX century (see [24][25][26][27]43]).Many researchers have been designated considerable thought due to its diversified choice of the utilization in mathematical spheres; such as mechanics, theory of relativity, basic hypergeometric function, quantum, and number theory.Firstly, the q−analogue of derivative and the q−Taylor's formula were introduced by Jackson [29].Then, by using the differentiation technique in the q−calculus, Jing and Fan [30] have presented some modifications in Taylor's formula together with its remainder in the sphere of the q−calculus.They compared the q-Taylor's formula and the ordinary Taylor's formula and found signified results on the q−remainder.
Ernst [31] has investigated some novel modifications in the q−Taylor's formula with the help of q− integration by parts and developed its varient formulations.Firstly, some novel recursive schemes under the q−analysis were suggested and analyzed by Singh et al. [32] and introduced some varied forms of the q−iterative schemes by opting several values of the q−parameter.The stability and reliability of the q−iterative methods are checked by presenting comparative analysis of several nonlinear algebraic equations with some classical methods.The q−difference equation plays a vital role in the realm of the q−calculus.For solving the partial differential equations in the q−calculus, Jafari et al. [33] have applied an iterative method called the Daftardar-Jafari decomposition method.It is demonstrated that the proposed procedure's computational outcome converges to the true solution of the q−difference equations subject to specific constraints.The study of q−integro differential equation with three criteria was investigated by Abdeljawad and Samei [12] and checked its solution existence by applying the q−calculus.Sadik and Orie [11] have introduced a convenient and efficient method based on q−calculus known as q-differential transform method for solving partial q-differential equations.The solution obtained by this method is expressed in terms of convergent power series and the validity of this method is checked by computing several examples.Liang and Samei [13] have determined the existence of solutions for non-linear problems regular and singular fractional q−differential equation subject to certain constraints.They have presented some results with the support of numerical examples and by applying definitions of the fractional q-derivative of Riemann-Liouville & Caputo type.Many real-life problems can be modeled in the form of q−fuzzy differential equations.Noeiaghdam et al. [15] have introduced two fuzzy numerical methods based on the generalized Hukuhara q−differentiability named as the fuzzy q−Euler's and the local q−Taylor's expansion method for solving q−fuzzy initial value differential equations.In an attempt of transformation of the classical results towards the q−calculus considered by Srivastava et al. [45], the two subclasses of normalized analytic functions are investigated by using various operators of q−calculus and fractional q−calculus in the complex z−plane.Sana et al. [10] have transformed the classical iterative methods over the q−iterative methods and presented a comparative analysis of these methods with the classical methods.They also presented the generalized formulation of new methods and test their reliability, effectiveness & convergence speed via various numerical examples.
Motivated and inspired by the research going on in this direction, we have restructured some new multistep iterative methods for computing zeros of the nonlinear equations in the context of the q−calculus.First, we find some new q−analogues of the iterative methods initiated and advanced by Shah and Noor [9].Then, to obtain the needed results, we rephrase the supposed nonlinear equation accompanied by an auxiliary function and apply the q−Taylor's formula.For the best implementation of the results and the derivations of recursive schemes, we utilize the decomposition methodology [21] under the q−paradigm.It is essential to mention that the new suggested algorithms can reduce the number of computing costs compared to conventional iterative methods while good numerical accuracy is maintained by appropriately choosing the parameter q ∈ [0, 1].Now, we recollect some of the basic ideas in the q−analysis [34] that are prerequisites and reinforce the construction of our novel q−iterative schemes for computing solutions of the nonlinear equations.Let the q−integer, for q ∈(0,1) is described such as: [m] q = m for q = 1. (1.3) For 0 ≤ p ≤ m, the q−factorial and the q−binomials are defined as: Definition 1 (see [34]).A q−analogue of classical exponential function e ũ q is defined as The derivative of the classical exponential function remains unchanged under differentiation.The q−analogue of exponential function also remains the same in the q−calculus such as: (1.6) Definition 2 (see [34]).Let f (ũ) is a real valued continuous function and its q−derivative is prescribed as follows: where (D q f )(ũ) represents q−derivative is known as Jackson derivative.It reduces to the standard derivative when q approaches to one.The q−derivative with higher-order for the function f (ũ) is prescribed as: Definition 3 (see [34]).The q−derivative of product and quotient of function f (ũ) and g(ũ) is defined as follows: such that g(ũ)g(qũ) 0.
Then, the q−Taylor's formula for the function f (ũ) instigated by Jackson is explained as: where D q , D 2 q , . . .are all q−derivatives, where 0 < q < 1.

Evolvement of the iterative schemes
This section deals with the construction of some novel iterative schemes by employing Taylor's formula and Daftardar-Jafari decomposition technique [21] in the paradigm of the quantum calculus.

Main results
We consider the nonlinear algebraic equation of the general form: Let g(ũ) be an auxiliary function.Suppose κ is an initial guess in the neighbourhood of β which is the simple root of nonlinear equation (2.1).
Using q−Taylor's series about κ and the technique of He [28], we rewrite the nonlinear equation (2.2) as a parallel coupled system of the equation : Eq (2.4) can be rewritten as: where and The term N q (ũ) is treated as nonlinear and c as a constant.Let ũ0 be an initial guess then from relation (2.4), we can easily compute a key equation that is helpful in the development of new q−iterative methods : (2.9) We now carry out a decomposition technique primarily due to Daftardar-Gejji and Jafari [21], known as the Daftardar-Jafari decomposition technique, to set up arrangements of higher-order iterative methods.The central idea behind using this methodology is to seek the solution of the q−basic equation (2.6) in the series form.
(2.10) Now, we deteriorate the nonlinear operator N q (ũ) which is defined in (2.8) such as: From the equations (2.6), (2.10) and (2.11), we have finally, we obtain the following iterative procedure: It follows that Note that ũ is approximated by and thus lim x→∞ ũn = ũ.Theorem 2.1 (see [33]).If N q is a contraction, then the series specified in (2.10) is absolutely convergent.
Proof.Let N q is a contraction mapping, then by definition we can write: then in view of (2.13), we have then the series ũ= ∞ k=0 ũk is uniformly and absolutely convergent to an answer of the equation (2.6) (see [36]).This completes the proof.Now, we construct the following iterative schemes to find the solution of the nonlinear algebraic equation (2.1) Algorithm A: From (2.13), we have for n = 0 : .16)This composition permits us to put forward the subsequent recursive approach for solving the nonlinear equation (2.1), and the iterative schema computes the approximate solution ũn+1 for a given starting guess ũ0 : This represents main q−analogue of iterative scheme which is prospected by He [28] and Shah [9].This main iterative scheme helps us to generate different q−algorithms for solving the nonlinear equation (2.1).Now, with the help of (2.4) and (2.13), we get:

.18)
Algorithm B: From (2.13), we have for n = 1 : By using (2.16) and (2.18), we have .20)This composition permits us to put forward the subsequent iterative approach for solving the nonlinear equation (2.1), and the iterative schema computes the approximate solution ũn+1 for a given starting guess ũ0 : This is q−analogue of Algorithm 2.2 which is investigated by [9] and the error equation of Algorithm B is determined in Theorem 3.1.By using (2.4), (2.13) and (2.18), we can obtain

.24)
Algorithm C: Now, considering from (2.13), we have for n = 2 : By using (2.16) and (2.23), we get . (2.26) This composition permits us to put forward the subsequent iterative approach for solving the nonlinear equation (2.1), and the iterative schema computes the approximate solution ũn+1 for a given starting guess ũ0 : ) This is q−analogue of Algorithm 2.3 which is investigated by [9] and the error equation of Algorithm C is determined in Theorem 3.1.
Algorithm A, Algorithm B, Algorithm C are the main and general iterative schemes that are used to generate some new algorithms by considering different choices of auxiliary functions that are the main attractiveness of modification of this technique.To convey the idea, we consider the following auxiliary functions.
Case : Let g(ũ n ) = e −βũ n and D q g(ũ n ) = −βe −βũ n .Using these values we obtain the following iterative methods for the solving nonlinear equations.
Algorithm D: For a given ũ0 (initial guess), approximate solution ũn+1 is computed by the following iterative scheme:

.30)
Algorithm E: For a given ũ0 (initial guess), approximate solution ũn+1 is computed by the following iterative scheme: Algorithm F: For a given ũ0 (initial guess), approximate solution ũn+1 is computed by the following iterative scheme: ) To the best of our knowledge, the new schemes Algorithm D, Algorithm E, Algorithm F appear to be new ones.

Convergence analysis
In this part, the order of convergence of the primary q−iterative methods made out by Algorithm A, Algorithm B, and Algorithm C is investigated.In the same approach, the rest of the iterative procedures can be established.
which is the root of f (ũ) = 0 then the iterative methods Algorithm A, Algorithm B and Algorithm C are convergent algorithms of order at least 2, 3, 4 respectively and we format it as follows: [2; q], [3; q] and [4; q], where parameter q corresponds to the quantum calculus.Error equations for these newly established algorithms are given as: Proof.Let f is adequately differentiable function and β is root of f (ũ).Now, expanding f (ũ n ) and D q f (ũ n ) in the q−Taylor's series about β we obtain By expanding g(ũ n ), D q g(ũ n ) in the q−Taylor's series, we obtain where , for k = 2, 3, . . .and e n = ũn − β (3.6) ), in the q−Taylors series about β, we obtain From (3.7), (3.9), (3.8), we get Now, using (3.10) into (2.17),we get the error term of the Algorithm A: Choosing (3.12), we have By expanding f (ṽ n ) in the q−Taylor's series about β and using (3.14), we have From (3.4), (3.9), (3.8) and (3.15) we have (3.17) By expanding wn , f ( wn ) in terms of the q−Taylor's series about β From (3.4), (3.9), (3.8) and (3.19) we have Using (3.18) and (3.20) into (3.12),we obtain the error equation for the Algorithm C: Equation (3.20) shows the error equation for the Algorithm C and has at least fourth-order convergence.
It is noted that Algorithm C is the main iterative scheme and all other schemes investigated from this scheme are at least fourth-order convergent.
Remark 3.1.Based on the study of convergence analysis of proposed iterative methods, it can be easily observed that various order iterative methods can be developed by choosing appropriately multiple choices of the auxiliary function in Algorithm A, Algorithm B and, Algorithm C respectively.
then Algorithm A generates the following iterative method with the initial guess ũ0 .
Algorithm G: For a given initial guess ũo , approximate solution ũn+1 is computed by the following iterative scheme: This is q−analogue of well known Halley method [32] which has cubic convergence i.e. [3, q], where q represents the q−calculus.Now, again using the above stated specified value of an auxiliary function then Algorithm B and Algorithm C reduces to the following iterative procedures.
Algorithm H: For a given initial guess ũo , approximate solution ũn+1 is computed by the following iterative scheme: This method is fourth-order convergent for solving nonlinear equations and appears to be a novel one.
Algorithm I: For a given initial guess ũo , approximate solution ũn+1 is computed by the following iterative scheme: This method emerges as a new method that has fifth-order of convergence.
This completes the proof.

Numerical examples and applications
This section discusses some nonlinear equations.With the support of these examples, we elaborate on the efficacy and performance of newly established methods initiated in this paper.The general algorithm for finding the estimated solution of the given nonlinear function is given as: in Algorithm A, Algorithm B, Algorithm C, we consider ε = 10 −100 as tolerance.We obtain an approximate solution relatively than the exact lean on the computational accuracy ε.We adopt the following stopping criterium for computational performance: For convergence criteria, it is prerequisite that the space of two successive estimations for the zero must be less than 10 −100 .We make use of abbreviations QG & CG for the q−iterative methods and traditional iterative methods respectively.We symbolize Algorithm D, Algorithm E and Algorithm F by QG1, QG2 and QG3 respectively and phrase div served as the divergence of methods.We develop a comparative analysis between the standard Newton's method (NM) [35], Chun method (CM) [4], Noor method (NR) [8], CG1, CG2 and CG3 [9] and our newly proposed q−iterative methods QG1, QG2 and QG3.The computational results of comparative analysis are presented in Tables (4,8,12,14).We exhibit the number of iterations, the final estimated solution and the corresponding functional value by the symbols IT, ũn and f (ũ n ), whereas, the distance in the middle of two successive estimates is shown by ∆.It is necessary to mention that for the best implementation of results, we choose the value of q = 0.9999.We use Maple software to perform all the numerical computations.

Algorithms D, E, F : General roots' finding Algorithm
Input: f ∈ R-nonlinear function, l-maximal number of iterations, I-recursive method, ε accuracy Output: Approximate root of given nonlinear function for ũ0 ∈ A do Now, we recollect the classical Algorithm 2.1 (CG1) in [9], elucidated as: and the classical Algorithm 2.2 (CG2) in [9], described as: , n = 0, 1, 2, . . ., and the classical Algorithm 2.3 (CG3) in [9], described as: We present some examples of nonlinear equations (4.1-4.4) to illustrate the efficiency of the newly developed one-step, two-step and three-step iterative methods in this article.Firstly, for the sake of simplicity, we investigate the efficacy and credibility of the q−recursive schemes for multiple values of q up to three iterations that can be extended to any number of iterations until we achieve the desired accuracy.The results in the Tables (1-3, 5-7, 9-11) demonstrate the calculations of ũi and f (ũ i ), i = 1, 2, 3 by employing QG1, QG2, QG3 for multiple values of q and β = 0.5.We choose β = 0.5 for both q and ordinary iterative methods.
We take ũo = −1.2 as an initial guess for computational evaluation.The numerical findings for equation (4.6) are calculated in the Tables (9-11) by using QG1, QG2, QG3 for multiple values of q and β = 0.5.
The numeric values in Table 9, illustrate that one can obtain more precise values of ũi with the constraint q → one and f (ũ i ) attain zero value, where 1 ≤ i ≤ 3. It is also noted that the values of f (ũ 1 ) = 1.562110e − 01, f (ũ 2 ) = 5.384093e − 02, f (ũ 3 ) = 1.734706e − 02 computed by QG1 at q = 0.9999 exist in the neighbourhood of zero more nearly in comparison to the values f (ũ 1 ) = 1.562383e − 01, f (ũ 2 ) = 5.386087e − 02, f (ũ 3 ) = 1.735750e − 02 calculated by CG1.Following the steps of the Table 9, the equation (4.6) converges towards the root ũ9 = 0.3170617745 and f (ũ 9 ) = 8.400168e − 13, for q = 0.9999.The results in Table 10, elaborate that one can obtain more precise values of ũi with the constraint q → one and f (ũ i ) attain zero value, where 1 ≤ i ≤ 3.By choosing q=0.9999, it is also noted that the values of f (ũ 1 ) = 9.698942e − 02, f (ũ 2 ) = 1.973729e − 02, f (ũ 3 ) = 3.342707e − 03 computed by QG2 are nearer to zero in comparison to the values f (ũ 1 ) = 9.700799e − 02, f (ũ 2 ) = 1.974547e − 02, f (ũ 3 ) = 3.345242e − 03 calculated by CG2.Continuing the iterative procedure as presented in Table 10, the equation (4.6) converges to the root ũ6 = 0.3170617746 for q = 0.9999 and f (ũ 6 ) = 4.953192e − 14.Columns in the Table 11 demonstrate that one can obtain more precise values of ũi with the constraint that q → 1 and f (ũ i ) attain value zero, where 1 ≤ i ≤ 3. It is also noted that the values of f (ũ 1 ) = 6.887648e − 02, f (ũ 2 ) = 9.422356e − 03, f (ũ 3 ) = 8.107219e − 04 computed by QG3 at q = 0.9999 are more adjacent to zero in comparison to the values f (ũ 1 ) = 6.889025e − 02, f (ũ 2 ) = 9.426571e − 03, f (ũ 3 ) = 8.115710e − 04 calculated by CG3.Following the steps of Table 11 and for q = 0.9999, β=0.5, the equation (4.6) converges to the root ũ5 = 0.3170617746 and parallel functional values are obtained as f (ũ 5 ) = 4.850310e − 15.The second column (IT) in Table 12 illustrates the comparsion of different iterative methods with proposed methods in terms of number of iterations.It is clear from the computational results that new methods need less number of iterations as compared to other methods (NM, CM, NR, CG1) to meet the stopping criteria (4.1) or same in some cases when comparing with (CG2, CG3).Remark 4.1.It is worthy to mention that when we evaluate the errors for the q−iterative schemes then it oscillate for various values of q.The error reduces when q tends to the highest values uniting 0 and 1.Therefore, in Table 13 error for equations (4.2), (4.5), (4.6) are computed by using q=0.9999 and β=0.5 which will estimate the classical methods.Example 4.4 (Algebraic and Transedental equations).This example comprises of a few nonlinear equations which help us to examine the reliability and effectiveness of our new q−iterative methods.[32] to validate the theoretical results.The last three numerical equations namely; f 7 (ũ), f 8 (ũ), f 9 (ũ) represent some realworld applications of nonlinear equations.These nonlinear equations are the transformations of some mathematical models that appeared in science and engineering.The first one nonlinear equation f 7 is generated as a solution of mathematical modeling of the growth of population over short periods of time that can be written as in the form of differential equation: where λ denotes the constant birth rate of population and M( t) denotes the number in the population at time t, for details (see [35]).The second nonlinear equation f 8 represents the physical constraint Therefore, (4.9) gives the following iterative scheme D q f (κ)g(κ)+D q g(κ) f (qκ) , . . .ũ0 + ũ1 + ũ2 + ... + ũn−1 + ũn = ũ0 + ũ1 + ũ2 + ... + ũn−1 − f (ũ 0 +ũ 1 +ũ 2 +...+ũ n−1 )g(κ) D q f (κ)g(κ)+D q g(κ) f (qκ) .
This relation enables and allows us to propose the subsequent iterative method.

Conclusions and observations
Study and formulation of numerical results in quantum calculus induce interest due to the high demand in mathematics and easy implementation.This manuscript introduces some novel iterative schemes to find the estimated solution of nonlinear equations with success in quantum calculus.The key motivation of proposing q−iterative schemes is to overcome differentiability and convergence issues while getting solutions of algebraic equations.These new iterative schemes are applicable for different choices of an auxiliary function and derived by considering the valuable Daftardar-Jafari decomposition technique.We develop the comparative analysis of newly proposed methods with the traditional iterative methods to demonstrate the performance and efficiency of q−iterative schemes.Moreover, it is shown that the numerical results obtained for both conventional and q−iterative methods remain identical.Also, the errors connected with the suggested schemes are relatively marginal by selecting the value of q approaches to one.Hence, it is evident that the transformation of iterative methods in the q−calculus framework which we referred to as q−analogue of iterative schemes, is better than classical methods, and in limited cases when the parameter q → 1, these methods reduces to the classical iterative methods.The significant challenge of dealing with these schemes which necessitate more exploration is that to get results with high accuracy, we must estimate the value of q in (0, 1).
Our utilization here of the q−calculus in the development of the iterative methods are supposed to promote and motivate major future breakthroughs in Mathematical analysis.It is noticed that in (p, q) analysis the extra parameter p is clearly redundant, Srivastava (see [42, p. 340] and [43, pp. 1511-1512]; see also the related recent works [45,46]) revealed that the so-called (p, q) variations of the suggested q−results which are obtained by inconsequentially and trivially adding a redundant parameter p as quite simple and insignificant modification of the standard q−calculus.Along these lines, while we reinforce and revitalize the q−results introduced in this paper, together with potential q−extensions of other similar developments in physical and engineering sciences, we do not encourage and support the so-called (p, q)-variations of the suggested q−results which are obtained by inconsequentially and trivially adding a redundant or superfluous parameter p.

Table 1 .
The Computational results of Example 4.1 by adopting QG1.

Table 2 .
The Computational results of Example 4.1 by adopting QG2.

Table 3 .
The Computational results of Example 4.1 by adopting QG3.

Table 4 .
Numerical comparison between different algorithms for Example 4.1.
Example 4.2 (Van der Waal's Equation see

Table 5 .
The Computational results of Example 4.2 by adopting QG1.

Table 6 .
The Computational results of Example 4.2 by adopting QG2.

Table 7 .
The Computational results of Example 4.2 by adopting QG3.

Table 8 .
Numerical comparison between different algorithms for Example 4.2.

Table 9 .
The Computational results of Example 4.3 by adopting QG1.

Table 10 .
The Computational results of Example 4.3 by adopting QG2.

Table 11 .
The Computational results of Example 4.3 by adopting QG3.

Table 12 .
Numerical comparison between different algorithms for Example 4.3.