On Fractional Newton-Type Method for Nonlinear Problems

Te current manuscript is concerned with the development of the Newton–Raphson method, playing a signifcant role in mathematics and various other disciplines such as optimization, by using fractional derivatives and fractional Taylor series expansion. Te development and modifcation of the Newton–Raphson method allow us to establish two new methods, which are called frst-and second-order fractional Newton–Raphson (FNR) methods. We provide convergence analysis of frst-and second-order fractional methods and give a general condition for the convergence of higher-order FNR. Finally, some illustrative examples are considered to confrm the accuracy and efectiveness of both methods.


Introduction
Newton-Raphson method is one of the most powerful techniques to locate the solutions of linear or nonlinear equations in numerous areas of science. Compare to the other methods, the convergence rate of this method is much better. In this method, the neighbourhood of the solution of the equations is determined by utilizing the tangent lines of the curves. However, this method has some drawbacks such as having no solution for some forms of equations in a situation that the initial guess or an iteration coincides with a loop which leads to divergence or oscillation of the method.
Te theory and applications of the Newton-Raphson method were presented in [1]. Modifed versions of Newton's method were given in [2][3][4]. Newton's method has been implemented for solving constrained or unconstrained minimization problems in [5][6][7]. A novel block Newton method was developed for the computation invariant pairs to represent eigenvalues and eigenvectors in [8]. A criterion was established for selecting the appropriate model and its applications in [9]. Various optimization methods such as the Newton-Raphson, bisection, gradient, and secant methods were reviewed and discussed in [10]. Te optimization solution of the estimating function in the regression models was determined in [11,12].
In the current study, we focus on developing a novel modifcation of the Newton-Raphson method by utilizing fractional derivatives and fractional Taylor series expansion which allows us to eliminate the shortcomings of this method. One advantage of this method is that this method can be applied to various fractional derivatives such as Riemann-Liouville and Caputo derivatives [13][14][15][16][17][18][19][20][21][22]. Convergence analysis of frst-and second-order FNR is provided, and some conditions on initial guess are obtained. Moreover, these conditions are generalized for higher-order FNR. Te main advantages of frst-and second-order FNR are that they are more efective and accurate compared to other existing methods. Moreover, the convergency of the obtained solutions is faster than the convergency of the solutions, obtained by other methods.

Preliminaries
Tis section is devoted to fundamental notions in fractional calculus [16][17][18]. Defnition 1. β th (β ≥ 0) order of Riemann-Liouville integral is given by [16] (1) Defnition 2. β th order fractional derivative in Caputo sense is given by [16] where D m is the ordinary diferential operator of order m. Theorem 1. [23,24] Let us suppose that with . . , C D β (n times). Notice that the property of Riemann-Liouville derivative of constant is diferent than zero unlike the Caputo derivative.
3.1. First-Order FNR. By taking the frst two terms of fractional Taylor series expansion, we have To solve for the x intercept, we set y � 0 and rearrange the terms Tus, we have Repetition of this algorithm generates a sequence of x values x 0 , x 1 , x 2 , . . . which leads to the following recurrence relation: 3.2. Second-Order FNR. By taking the frst three terms of the fractional Taylor series expansion, we get To solve for the x intercept, we set y � 0 and rearrange terms which leads to the following Repetition of this algorithm generates a sequence of x values x 0 , x 1 , x 2 , . . . which are formulated by the following recurrence relation:

Convergence
In this section, the convergence analysis of frst-and secondorder FNR is given by the following theorems, respectively.
Under the assumption that x n converges to x * as n ⟶ ∞, we have for n sufciently large. Tus, x n converges to x * quadratically.
Proof. Let e n � x n − x * , so that x n − e n � x * . Setting x � x n and h � −e n in fractional Taylor's Teorem, we get for some ξ n ∈ (x n , x * ).
Having the condition that β th derivative of f is continuous with f (β) (x n ) ≠ 0 as long as x n is close enough to x * allows us to divide by (f β (x n )/Γ(β + 1)) which leads to the following: As a result, the formulation of frst-order FNR gives the following: . (17) After rearrangement, we have Finally, By continuity, if for sufciently large n.

□
Theorem . Assume that f is three times continuously diferentiable on an open interval (a, b) and that there exists Implementing the frst-order FNR method, we have the following recurrence relation: Under the assumption that x n converges to x * as n ⟶ ∞, we have for sufciently large n. Tus, x n converges to x * quadratically.
Proof. Let e n � x n − x * , so that x n − e n � x * . Setting x � x n and h � −e n , in fractional Taylor's Teorem, we get Journal of Mathematics 3 Having the condition that β th derivative of f is continuous with as long as x n is close enough to x * allows us to divide by f 2β (x n )/Γ(2β + 1) which leads to the following: As a result, the formulation of frst-order FNR gives the following: After rearrangement, we have Finally, By continuity, if for sufciently large n.
In general, for the convergence of higher-order FNR, we obtain the following condition: for sufciently large n.

Numerical Examples
In this section, some illustrative examples are presented to show the implementation of frst-and second-order FNR which allows us to confrm the obtained results given in the previous section. Matlab R2016b with stopping criterion |x n+1 − x n | < 10 − 8 and a maximum of 500 iterations are           It can be observed from Tables 1 and 2 that the estimation of second-order FNR is better than the one of the frst-order FNR when the order of the derivative is close to one in both Caputo and Riemann-Liouville derivatives. In Figures 1-4, the convergence plane of the polynomial function f 1 (x) is given when x 0 ∈ [−3, 3] for various values of β. Example 2. Let us consider the function f 2 (x) � exp(x) − 1 , whose only root is x 1 � 0.
It can be observed from Tables 3 and 4 that the estimation of second-order FNR is better than the one of the frst-order FNR when the order of the derivative is close to one in both Caputo and Riemann-Liouville derivatives. In Figures 5-8, the convergence plane of the polynomial function f 2 (x) is given when x 0 ∈ [−10, 10] for various values of β.

. Conclusion
First-and second-order FNR are developed, and analyzed and applied in this study. Moreover, the convergence of both methods is established. It is shown that second-order FNR gives better results compare to frst-order FNR when the order of fractional derivative is close to one in both Caputo and Riemann-Liouville derivatives. It is also shown that the order of convergence for frst-order FNR is quadratic while one of the second-order FNR is 3/2. It is clear from tables that as the fractional parameter increases to one, the number of iterations decreases for both developed methods. Moreover, fgures depict that the convergence of approximate solutions is better for β ∈ (0.7, 1] which can be seen also from the tables. Generally, it is obvious from the obtained formulation that the order of convergence for k th order FNR is (k + 1/k). Te obtained results are verifed by presented examples, too.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.  Journal of Mathematics 9