# On Fractional Newton-Type Method for Nonlinear Problems

Authors: Mine Aylin Bayrak; Ali Demir; Ebru Ozbilge
Journal: Journal of Mathematics
 (2022)

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## Abstract

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


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## Body

## 1. 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.The theory and applications of the Newton–Raphson method were presented in [1]. Modified versions of Newton’s method were given in [2–4]. Newton’s method has been implemented for solving constrained or unconstrained minimization problems in [5–7]. 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]. The 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 modification 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 [13–22]. Convergence analysis of first- and second-order FNR is provided, and some conditions on initial guess are obtained. Moreover, these conditions are generalized for higher-order FNR. The main advantages of first- and second-order FNR are that they are more effective 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.


## 2. Preliminaries

This section is devoted to fundamental notions in fractional calculus [16–18].Definition 1.
βthβ≥0 order of Riemann–Liouville integral is given by [16](1)Jβfx=1Γβ∫0xx−tβ−1ftdt,β>0,x>0,J0fx=fx.Definition 2.
βth order fractional derivative in Caputo sense is given by [16](2)DβCfx=Jm−βDmfx=∫0xx−tm−β−1dmdtmftdt,m−1<β<m,x>0,where Dm is the ordinary differential operator of order m.Theorem 1.
[23, 24] Let us suppose that DjβCfx∈Ca,b for j=1,2,…,n+1 where β∈0,1, and then, we have(3)fx=∑i=0nDiβCfax−aiβΓiβ+1+Dn+1βCfξx−an+1βΓn+1β+1,with a≤ξ≤x, for all x∈a,b where DnβC=DβC⋅DβC,…,DβCntimes. Notice that the property of Riemann–Liouville derivative of constant is different than zero unlike the Caputo derivative.


## 3. Fractional Newton–Raphson (FNR) Method

In this section, we give the formulation of the first- and second-order FNR [25–28].


### 3.1. First-Order FNR

By taking the first two terms of fractional Taylor series expansion, we have(4)y−fx0=fβx0Γβ+1x−x0β.To solve for thex intercept, we set y=0 and rearrange the terms(5)0=fx0+fβx0Γβ+1x1−x0β.Thus, we have(6)x1=x0+−fx0fβx0Γβ+11/β.Repetition of this algorithm generates a sequence ofx values x0,x1,x2,… which leads to the following recurrence relation:(7)xn+1=xn+−fxnfβxnΓβ+11/β,fβxn≠0.


### 3.2. Second-Order FNR

By taking the first three terms of the fractional Taylor series expansion, we get(8)y−fx0=fβx0Γβ+1x−x0β+f2βx0Γ2β+1x−x02β.To solve for thex intercept, we set y=0 and rearrange terms(9)0=fx0+fβx0Γβ+1x1−x0β+f2βx0Γ2β+1x1−x02β,which leads to the following(10)x1=x0+Γ2β+1−fβx0/Γβ+1+fβx0/Γβ+12−4f2βx0/Γ2β+1fx02f2βx01/β.Repetition of this algorithm generates a sequence ofx values x0,x1,x2,… which are formulated by the following recurrence relation:(11)xn+1=xn+Γ2β+1−fβxn/Γβ+1+fβxn/Γβ+12−4f2βxn/Γ2β+1fxn2f2βxn1/β.


## 3.1. First-Order FNR

By taking the first two terms of fractional Taylor series expansion, we have(4)y−fx0=fβx0Γβ+1x−x0β.To solve for thex intercept, we set y=0 and rearrange the terms(5)0=fx0+fβx0Γβ+1x1−x0β.Thus, we have(6)x1=x0+−fx0fβx0Γβ+11/β.Repetition of this algorithm generates a sequence ofx values x0,x1,x2,… which leads to the following recurrence relation:(7)xn+1=xn+−fxnfβxnΓβ+11/β,fβxn≠0.


## 3.2. Second-Order FNR

By taking the first three terms of the fractional Taylor series expansion, we get(8)y−fx0=fβx0Γβ+1x−x0β+f2βx0Γ2β+1x−x02β.To solve for thex intercept, we set y=0 and rearrange terms(9)0=fx0+fβx0Γβ+1x1−x0β+f2βx0Γ2β+1x1−x02β,which leads to the following(10)x1=x0+Γ2β+1−fβx0/Γβ+1+fβx0/Γβ+12−4f2βx0/Γ2β+1fx02f2βx01/β.Repetition of this algorithm generates a sequence ofx values x0,x1,x2,… which are formulated by the following recurrence relation:(11)xn+1=xn+Γ2β+1−fβxn/Γβ+1+fβxn/Γβ+12−4f2βxn/Γ2β+1fxn2f2βxn1/β.


## 4. Convergence

In this section, the convergence analysis of first- and second-order FNR is given by the following theorems, respectively.Theorem 2.
Assume thatf is twice continuously differentiable on an open interval a,b and that there exists a point x∗∈a,b with fβx∗≠0. Implementing the first-order FNR method, we have the following recurrence relation:(12)xn+1=xn+−fxnfβxnΓβ+11/β,n=1,2,….
Under the assumption thatxn converges to x∗ as n⟶∞, we have(13)xn+1−x∗≤Mxn−x∗2,M>f2βx∗fβx∗Γβ+1Γ2β+11/β,for n sufficiently large. Thus, xn converges to x∗ quadratically.Proof.
Leten=xn−x∗, so that xn−en=x∗. Setting x=xn and h=−en in fractional Taylor’s Theorem, we get(14)fxn−en=fxn−enβΓβ+1fβxn+en2βΓ2β+1f2βξn,for some ξn∈xn,x∗. Since fxn−en=fx∗=0, we have(15)0=fxn−xn−x∗βΓβ+1fβxn+en2βΓ2β+1f2βξn.
Having the condition thatβth derivative of f is continuous with fβxn≠0 as long as xn is close enough to x∗ allows us to divide by fβxn/Γβ+1 which leads to the following:(16)0=fxnfβxn−xn−x∗β+en2βf2βξnΓβ+1Γ2β+1.
As a result, the formulation of first-order FNR gives the following:(17)xn+1−x∗β=en2βf2βξnfβxnΓβ+1Γ2β+1.
After rearrangement, we have(18)xn+1−x∗=en2f2βξnfβxnΓβ+1Γ2β+11/β.
Finally,(19)xn+1−x∗=f2βξnfβxnΓβ+1Γ2β+11/βxn−x∗2.
By continuity,fβxn converges to fβx∗. Convergence of ξn∈xn,x∗ to x∗ leads to convergence of f2βξn to f2βx∗. As a result, we have(20)xn+1−x∗≤Mxn−x∗2,if(21)M>f2βx∗fβx∗Γβ+1Γ2β+11/βorMβΓ2β+1Γβ+1>f2βx∗fβx∗,for sufficiently large n.Theorem 3.
Assume thatf is three times continuously differentiable on an open interval a,b and that there exists x∗∈a,b with f2βx∗≠0. Implementing the first-order FNR method, we have the following recurrence relation:(22)xn+1=xn+Γ2β+1−fβxn/Γβ+1+fβxn/Γβ+12−4f2βxn/Γ2β+1fxn2f2βxn1/β,n=1,2,….
Under the assumption thatxn converges to x∗ as n⟶∞, we have(23)xn+1−x∗≤Mxn−x∗3/2,M>f3βx∗f2βx∗Γ2β+1Γ3β+11/2β,for sufficiently large n. Thus, xn converges to x∗ quadratically.Proof.
Leten=xn−x∗, so that xn−en=x∗. Setting x=xn and h=−en, in fractional Taylor’s Theorem, we get(24)fxn−en=fxn−enβΓβ+1fβxn+en2βΓ2β+1f2βxn−en3βΓ3β+1f2βξn,for some ξn∈xn,x∗. Since fxn−en=fx∗=0, we have(25)0=fxn−xn−x∗βΓβ+1fβxn+xn−x∗2βΓ2β+1f2βxn−xn−x∗3βΓ3β+1f3βξn.
Having the condition thatβth derivative of f is continuous with as long as xn is close enough to x∗ allows us to divide by f2βxn/Γ2β+1 which leads to the following:(26)0=Γ2β+1fxnf2βxn−Γ2β+1Γβ+1xn−x∗βfβxnf2βxn+xn−x∗2β−en3βf3βξnf2βxnΓ2β+1Γ3β+1.
As a result, the formulation of first-order FNR gives the following:(27)xn+1−x∗2β=en3βf3βξnf2βxnΓ2β+1Γ3β+1.
After rearrangement, we have(28)xn+1−x∗=en3/2f3βξnf2βxnΓ2β+1Γ3β+11/2β.
Finally,(29)xn+1−x∗=f3βξnf2βxnΓ2β+1Γ3β+11/2βxn−x∗3/2.
By continuity,f2βxn converges to f2βx∗. Convergence of ξn∈xn,x∗ to x∗ leads to convergence of f3βξn to f3βx∗. As a result, we have(30)xn+1−x∗≤Mxn−x∗3/2,if(31)M>f3βx∗f2βx∗Γ2β+1Γ3β+11/2βorM2βΓ3β+1Γ2β+1>f3βx∗f2βx∗,for sufficiently large n.
In general, for the convergence of higher-order FNR, we obtain the following condition:(32)xn+1−x∗≤Mxn−x∗k+1/k,M>fk+1βx∗fkβx∗Γkβ+1Γk+1β+11/kβ,k=1,2,3,…,for sufficiently large n.


## 5. Numerical Examples

In this section, some illustrative examples are presented to show the implementation of first- and second-order FNR which allows us to confirm the obtained results given in the previous section. Matlab R2016b with stopping criterionxn+1−xn<10−8 and a maximum of 500 iterations are utilized. In the tables of corresponding examples, the reached root xn+1−xn, fxn+1, and the number of iterations are shown.Example 1.
Let us consider the functionf1x=x3−10x2+34x−40 with roots x0=4,x1=3−i and x2=3+i.
It can be observed from Tables1 and 2 that the estimation of second-order FNR is better than the one of the first-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 f1x is given when x0∈−3,3 for various values of β.Table 1
First-order FNR and second-order FNR results forf1x with Caputo derivative and initial guess x0=1.
βxn+1xn+1−xnfxn+1n1st N-R14.000000001.46549e − 142.84217e − 14102nd N-R14.000000008.38330e − 180.00000e + 0081st N-R0.984.000000009.65937e − 091.23761e − 08172nd N-R0.984.000000008.93729e − 091.14804e − 08171st N-R0.963.999999996.31251e − 092.16859e − 08282nd N-R0.963.999999998.36562e − 092.82381e − 08271st N-R0.943.999999977.76010e − 095.26373e − 08412nd N-R0.943.999999978.85140e − 095.94304e − 08401st N-R0.923.999999949.40705e − 091.13137e − 07582nd N-R0.923.999999958.97743e − 091.08442e − 07571st N-R0.93.999999909.68422e − 091.98513e − 07822nd N-R0.93.999999909.48429e − 091.94823e − 07801st N-R0.883.999999849.37726e − 093.20376e − 071162nd N-R0.883.999999839.85311e − 093.34523e − 071121st N-R0.863.999999739.78559e − 095.41971e − 071622nd N-R0.863.999999739.66965e − 095.36477e − 071581st N-R0.843.999999569.88726e − 098.80428e − 072272nd N-R0.843.999999569.82503e − 098.75790e − 072221st N-R0.823.999999309.84707e − 091.40112e − 063192nd N-R0.823.999999309.85142e − 091.40163e − 063131st N-R0.83.999998889.90469e − 092.23380e − 064482nd N-R0.83.999998889.93908e − 092.23999e − 064411st N-R0.783.999996002.83959e − 088.00875e − 065002nd N-R0.783.999996202.65292e − 087.59583e − 065001st N-R0.763.999982751.10103e − 073.44952e − 055002nd N-R0.763.999983671.02423e − 073.26542e − 055001st N-R0.743.999938963.53606e − 071.22080e − 045002nd N-R0.743.999942403.26873e − 071.15193e − 045001st N-R0.723.999815219.78747e − 073.69503e − 045002nd N-R0.723.999826458.96989e − 073.47038e − 045001st N-R0.73.999506712.40749e − 069.86099e − 045002nd N-R0.73.999539652.18121e − 069.20282e − 04500Table 2
First-order FNR and second-order FNR results forf1x with Riemann–Liouville derivative and initial estimation x0=1.
βxn+1xn+1−xnfxn+1n1st N-R14.000000001.46549e − 142.84217e − 14102nd N-R14.000000008.38330e − 180.00000e + 0081st N-R0.984.000000004.68492e − 094.77773e − 09162nd N-R0.984.000000006.74956e − 096.72255e − 09151st N-R0.963.999999999.31417e − 092.26436e − 08202nd N-R0.963.999999996.15820e − 091.54268e − 08211st N-R0.943.999999998.41656e − 093.90606e − 08282nd N-R0.943.999999989.57249e − 094.39009e − 08281st N-R0.923.999999979.83530e − 097.65395e − 08352nd N-R0.923.999999978.10233e − 096.42359e − 08381st N-R0.93.999999949.40993e − 091.18506e − 07492nd N-R0.93.999999949.37709e − 091.18142e − 07471st N-R0.883.999999919.61954e − 091.88983e − 07652nd N-R0.883.999999909.84849e − 091.92880e − 07641st N-R0.863.999999869.60193e − 092.89676e − 07862nd N-R0.863.999999859.96806e − 092.99047e − 07851st N-R0.843.999999789.53412e − 094.36870e − 071142nd N-R0.843.999999789.64049e − 094.40927e − 071131st N-R0.823.999999669.94288e − 096.80455e − 071502nd N-R0.823.999999669.91663e − 096.78992e − 071491st N-R0.83.999999499.98970e − 091.02156e − 061992nd N-R0.83.999999509.78692e − 091.00502e − 061991st N-R0.783.999999249.96299e − 091.51847e − 062652nd N-R0.783.999999249.98659e − 091.52126e − 062641st N-R0.763.999998889.90426e − 092.24483e − 063542nd N-R0.763.999998879.98421e − 092.25855e − 063531st N-R0.743.999998339.92379e − 093.33049e − 064732nd N-R0.743.999998339.94614e − 093.33603e − 064731st N-R0.723.999995512.29296e − 088.98833e − 065002nd N-R0.723.999995482.30961e − 089.03519e − 065001st N-R0.73.999987625.73722e − 082.47620e − 055002nd N-R0.73.999987505.81547e − 082.49976e − 05500Figure 1
Convergence planes of first-order FNR onf1x for Caputo derivative.Figure 2
Convergence planes of second-order FNR onf1x for Caputo derivative.Figure 3
Convergence planes of first-order FNR onf1x for Riemann–Liouville derivative.Figure 4
Convergence planes of second-order FNR onf1x for Riemann–Liouville derivative.Example 2.
Let us consider the functionf2x=expx−1, whose only root is x1=0.
It can be observed from Tables3 and 4 that the estimation of second-order FNR is better than the one of the first-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 f2x is given when x0∈−10,10 for various values of β.Table 3
First-order NR and second-order NR results forf2x with Caputo derivative and initial estimation x0=0.2.
βxn+1xn+1−xnfxn+1n1st N-R1−0.000000000.00000e + 000.00000e + 0052nd N-R1−0.000000006.66798e − 100.00000e + 0031st N-R0.98−0.000000002.42649e − 091.64532e − 1082nd N-R0.98−0.000000001.07859e − 097.31359e − 1181st N-R0.96−0.000000003.91885e − 095.46557e − 10102nd N-R0.96−0.000000002.58893e − 093.61075e − 10101st N-R0.94−0.000000005.70944e − 091.22846e − 09122nd N-R0.94−0.000000004.32641e − 099.30883e − 10121st N-R0.92−0.000000002.76674e − 098.16363e − 10152nd N-R0.92−0.000000008.17968e − 092.41352e − 09141st N-R0.9−0.000000007.94065e − 093.01220e − 09172nd N-R0.9−0.000000006.59632e − 092.50224e − 09171st N-R0.88−0.000000004.85622e − 092.27344e − 09212nd N-R0.88−0.000000009.82148e − 094.59793e − 09201st N-R0.86−0.000000007.02051e − 093.94299e − 09252nd N-R0.86−0.000000005.90703e − 093.31762e − 09251st N-R0.84−0.000000006.69774e − 094.42000e − 09312nd N-R0.84−0.000000009.87756e − 096.51845e − 09301st N-R0.82−0.000000018.16846e − 096.23332e − 09392nd N-R0.82−0.000000016.60392e − 095.03942e − 09391st N-R0.8−0.000000017.97894e − 096.95120e − 09522nd N-R0.8−0.000000018.52364e − 097.42575e − 09511st N-R0.78−0.000000019.79061e − 099.63595e − 09742nd N-R0.78−0.000000018.49008e − 098.35597e − 09731st N-R0.76−0.000000019.85552e − 091.08611e − 081252nd N-R0.76−0.000000019.15524e − 091.00893e − 081211st N-R0.74−0.000000019.63986e − 091.18037e − 083602nd N-R0.74−0.000000019.78282e − 091.19788e − 083171st N-R0.72−4.313653785.71734e − 075.23966e − 055002nd N-R0.72−0.143932921.11043e − 011.43663e − 015001st N-R0.7−4.784603642.45965e − 011.47957e + 005002nd N-R0.7−1.294223127.32260e − 017.50525e − 01500Table 4
First-order NR and second-order NR results forf2x with Riemann–Liouville derivative and initial estimation x0=0.2.
βxn+1xn+1−xnfxn+1n1st N-R10.000000000.00000e + 000.00000e + 0052nd N-R10.000000006.66798e − 100.00000e + 0031st N-R0.98−0.000000002.42649e − 091.64532e − 1082nd N-R0.98−0.000000009.29969e − 105.95854e − 1181st N-R0.96−0.000000003.91885e − 095.46557e − 10102nd N-R0.96−0.000000002.40538e − 093.14007e − 10101st N-R0.94−0.000000005.70944e − 091.22846e − 09122nd N-R0.94−0.000000004.82173e − 099.60702e − 10121st N-R0.92−0.000000002.76674e − 098.16363e − 10152nd N-R0.92−0.000000003.43115e − 099.26171e − 10151st N-R0.9−0.000000007.94065e − 093.01220e − 09172nd N-R0.9−0.000000005.89937e − 092.01913e − 09181st N-R0.88−0.000000004.85622e − 092.27344e − 09212nd N-R0.88−0.000000004.71244e − 091.95934e − 09231st N-R0.86−0.000000007.02051e − 093.94299e − 09252nd N-R0.86−0.000000005.93748e − 092.90861e − 09301st N-R0.84−0.000000006.69774e − 094.42000e − 09312nd N-R0.84−0.000000007.64084e − 094.30750e − 09431st N-R0.82−0.000000018.16846e − 096.23332e − 09392nd N-R0.82−0.000000018.48988e − 095.40319e − 09791st N-R0.8−0.000000017.97894e − 096.95120e − 09522nd N-R0.8−0.005060181.02936e − 027.19943e − 035001st N-R0.78−0.000000019.79061e − 099.63595e − 09742nd N-R0.78−0.070240161.47568e − 019.83535e − 025001st N-R0.76−0.000000019.85552e − 091.08611e − 081252nd N-R0.76−0.141472262.83238e − 011.86505e − 015001st N-R0.74−0.000000019.63986e − 091.18037e − 083602nd N-R0.74−1.571717902.40309e + 001.13046e + 005001st N-R0.72−4.313653785.71734e − 075.23966e − 055002nd N-R0.72−0.000000009.36668e − 095.10297e − 091571st N-R0.7−4.784603642.45965e − 011.47957e + 005002nd N-R0.70.000000009.82670e − 095.30280e − 091011st N-R0.68−5.710707985.01626e − 011.03562e + 015002nd N-R0.680.000000008.29674e − 094.54705e − 09851st N-R0.66−754.278925547.80349e + 012.58569e + 185002nd N-R0.66−0.000000008.34916e − 094.76503e 09821st N-R0.64−1411860.536574821.42625e + 053.96270e + 445002nd N-R0.640.000000009.54192e − 095.80404e − 09891st N-R0.62−1063648380.576397201.04707e + 084.10943e + 675002nd N-R0.62−0.000000008.55254e − 095.65282e − 091131st N-R0.6−875382037053.402830008.39601e + 108.59364e + 905002nd N-R0.6−3.228390262.60655e − 071.73936e − 04500Figure 5
Convergence planes of first-order NR onf2x for Caputo derivative.Figure 6
Convergence planes of second-order NR onf2x for Caputo derivative.Figure 7
Convergence planes of first-order NR onf2x for Riemann–Liouville derivative.Figure 8
Convergence planes of second-order NR onf2x for Riemann–Liouville derivative.


## 6. 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 first-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 first-order FNR is quadratic while one of the second-order FNR is3/2. It is clear from tables that as the fractional parameter increases to one, the number of iterations decreases for both developed methods. Moreover, figures 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 kth order FNR is k+1/k. The obtained results are verified by presented examples, too.



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*Source: 7070253-2022-11-21.xml*