Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlinear Systems

Torres-Hernandez, A.; Brambila-Paz, F.

doi:10.5121/mathsj.2020.7102

Cited by 20 publications

(51 citation statements)

References 9 publications

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“…We begin this section by mentioning that, although the interest in fractional calculus has mainly focused on the study and development of techniques to solve differential equation systems of order non-integer [8][9][10][11][12][13][14]. Over the years, iterative methods have also been developed that use the properties of fractional derivatives to solve algebraic equation systems [4,5,[20][21][22][23][24][25]. These methods may be called fractional iterative methods; recently, these methods have been useful in the search for solutions to algebraic equation systems related to hybrid solar receivers [4,16].…”

Section: Fractional Newton-raphson Methodsmentioning

confidence: 99%

“…These methods may be called fractional iterative methods; recently, these methods have been useful in the search for solutions to algebraic equation systems related to hybrid solar receivers [4,16]. It should be noted that, depending on the definition of fractional derivative used, fractional iterative methods have the particularity that they may be used of local form [20] or of global form [5].…”

Section: Fractional Newton-raphson Methodsmentioning

confidence: 99%

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Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Torres-Hernandez

Brambila-Paz

Iturrarán-Viveros

et al. 2021

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In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

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Section: Fractional Newton-raphson Methodsmentioning

confidence: 99%

Section: Fractional Newton-raphson Methodsmentioning

confidence: 99%

Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Torres-Hernandez

Brambila-Paz

Iturrarán-Viveros

et al. 2021

Axioms

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show abstract

“…Although the interest in fractional calculus has mainly focused on the study and development of techniques to solve differential equation systems of order non-integer [5,6,7,8,9,10,11]. Over the years, iterative methods have also been developed that use the properties of fractional derivatives to solve algebraic equation systems [4,12,13,14,15,16,17,18,19,20]. These methods may be called fractional iterative methods, which under certain conditions, may accelerate their speed of convergence with the implementation of the Aitken's method [2,4], recently these methods have been useful in the search for solutions to algebraic equation systems related to hybrid solar receivers [3,19].…”

Section: Riemann-liouville Fractional Operatorsmentioning

confidence: 99%

“…These methods may be called fractional iterative methods, which under certain conditions, may accelerate their speed of convergence with the implementation of the Aitken's method [2,4], recently these methods have been useful in the search for solutions to algebraic equation systems related to hybrid solar receivers [3,19]. It should be noted that depending on the definition of fractional derivative used, fractional iterative methods have the particularity that they may be used of local form [12] or of global form [18]. Furthermore, for some definitions of fractional derivative, it is fulfilled that the derivative of the order α of a constant is different from zero, that is, An Approximation to Zeros of the Riemann Zeta Function Using Fractional Calculus…”

Section: Riemann-liouville Fractional Operatorsmentioning

confidence: 99%

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An Approximation to Zeros of the Riemann Zeta Function Using Fractional Calculus

Torres-Hernandez¹,

Brambila-Paz²

2021

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In this paper an approximation to the zeros of the Riemann zeta function has been obtained for the first time using a fractional iterative method which originates from a unique feature of the fractional calculus. This iterative method, valid for one and several variables, uses the property that the fractional derivative of constants are not always zero. This allows us to construct a fractional iterative method to find the zeros of functions in which it is possible to avoid expressions that involve hypergeometric functions, Mittag-Leffler functions or infinite series. Furthermore, we can find multiple zeros of a function using a singe initial condition. This partially solves the intrinsic problem of iterative methods, which in general is necessary to provide N initial conditions to find N solutions. Consequently the method is suitable for approximating nontrivial zeros of the Riemann zeta function when the absolute value of its imaginary part tends to infinity. Some examples of its implementation are presented, and finally 53 different values near to the zeros of the Riemann zeta function are shown.

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