Abstract
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions. The origin of these methods is the fractional Newton-Raphson method but unlike the latter, the orders of fractional derivatives proposed here are functions. In the first method, a function is used to guarantee an order of convergence (at least) quadratic, and in the others, a function is used to avoid the discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible that the methods have at most an order of convergence (at least) linear.
Highlights
When starting to study the fractional calculus, the first difficulty is that, when wanting to solve a problem related to physical units, such as determining the velocity of a particle, the fractional derivative seems to make no sense, this is due to the presence of physical units such as meter and second raised to non-integer exponents, opposite to what happens with operators of integer order
The aforementioned led to the study of Newton-Raphson method and a particular problem related to the search for roots in the complex space for polynomials: if it is needed to find a complex root of a polynomial using Newton-Raphson method, it is necessary to provide a complex initial condition x0 and, if the right conditions are selected, this leads to a complex solution, but there is the possibility that this leads to a real solution
The Newton-Raphson method is useful for finding the roots of a function f. This method is limited because it cannot find roots ξ ∈ C\R, if the sequence {xi}∞i=0 generated by (13) has an initial condition x0 ∈ R. To solve this problem and develop a method that has the ability to find roots, both real and complex, of a polynomial if the initial condition x0 is real, we propose a new method called fractional Newton-Raphson method, which consists of Newton-Raphson method with the implementation of the fractional derivative
Summary
When starting to study the fractional calculus, the first difficulty is that, when wanting to solve a problem related to physical units, such as determining the velocity of a particle, the fractional derivative seems to make no sense, this is due to the presence of physical units such as meter and second raised to non-integer exponents, opposite to what happens with operators of integer order. The second difficulty, which is a recurring topic of debate in the study of fractional calculus, is to know what is the order “optimal” α that should be used when the goal is to solve a problem related to fractional operators. To face these difficulties, in the first case, it is common to dimensionless any equation in which non-integer operators are involved, while for the second case different orders α are used in fractional operators to solve some problem, and it is chosen the order α that provides the “best solution” based about an established criteria. Fractional derivative that fits the function with which one is working This change, in essence simple, allows us to find roots in the complex space using real initial conditions because fractional operators generally do not carry polynomials to polynomials
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