Fractional Newton-Raphson Method Accelerated with Aitken's Method

The Newton-Raphson (N-R) method is characterized by the fact that generating a divergent sequence can lead to the creation of a fractal, on the other hand the order of the fractional derivatives seems to be closely related to the fractal dimension, based on the above, a method was developed that makes use of the N-R method and the fractional derivative of Riemann-Liouville (R-L) that has been named as the Fractional Newton-Raphson (F N-R) method. In the following work we present a way to obtain the convergence of the F N-R method, which seems to be at least linearly convergent for the case where the order $\alpha$ of the derivative is different from one, a simplified way to construct the fractional derivative and fractional integral operators of R-L is presented, an introduction to the Aitken's method is made and it is explained why it has the capacity to accelerate the convergence of iterative methods to finally present the results that were obtained when implementing the Aitken's method in F N-R method.Comment: Newton-Raphson Method, Fractional Calculus, Fractional Derivative of Riemann-Liouville, Method of Aitken. arXiv admin note: substantial text overlap with arXiv:1710.0763

A Modified Differential Transform Method for Solving Nonlinear Systems

Many natural phenomena are now being modelled by fractional calculus, however, there is still need for improvements of the present numerical approaches due to the non-local property of the fractional derivative and difficulty in solving problems related to physical units. In this paper, we present an improved numerical method based on Newton-Raphson fractional method (NFM) for solution of some nonlinear systems in complex space. Unlike in the Newton-Raphson fractional method, the commands of fractional derivatives are replaced with functions which is valid for one and several variables. The conformable fractional derivative of fractional order was employed to replace the first order derivative in the Newton method. Numerical results have been presented which show that the proposed numerical approach is efficient and promising

Mechanical Attributes of Fractal Dragons

Fractals are ubiquitous natural emergences that have gained increased attention in engineering applications, thanks to recent technological advancements enabling the fabrication of structures spanning across many spatial scales. We show how the geometries of fractals can be exploited to determine their important mechanical properties, such as the first and second moments, which physically correspond to the center of mass and the moment of inertia, using a family of complex fractals known as the dragons

Topological texture-fractal processing of signals and fields in radiophysics, radio engineering and radiolocation: developed methods and technologies (1979 – 2022) - fractal engineering

The report discusses the main directions of the introduction of textures, fractals, fractional operators, dynamic chaos and methods of nonlinear dynamics to create new information (breakthrough) technologies. The research is carried out in the fundamental scientific direction "Fractal radio physics and fractal radio electronics: design of fractal radio systems", initiated and developed by the author in V. A. Kotel’nikov IREE RAS from 1979 to the present. The relevance of these studies is related to the need for a more accurate description of all real processes occurring in radio physical and radio engineering systems: taking into account the hereditarity (memory), non-Gaussianity and scaling of physical signals and fields. The use of fractal systems, sensors and nodes is a fundamentally new solution that significantly changes the principles of building intelligent radio engineering systems and devices. The performed studies are priority ones in the world and serve as a basis for further development and justification of the practical application of fractal-scaling and texture methods in the synthesis of fundamentally new topological texture-fractal methods for detecting signals in the space-time channel of waves propagation with scattering (a new type of radar). The concepts of fractal engineering are introduced for the first time