A different approach for the fractional chemical model

This article analyzes and compares the two algorithms for the numerical solutions of the fractional isothermal chemical equations (FICEs) based on mass action kinetics for autocatalytic feedback, involving the conversion of a reactant in the Liouville-Caputo sense. The first method is based upon the spectral collocation method (SCM), where the properties of Legendre polynomials are utilized to reduce the FICEs to a set of algebraic equations. We then use the well-known method like Newton-Raphson method (NRM) to solve the set of algebraic equations. The second method is based upon the properties of Newton polynomial interpolation (NPI) and the fundamental theorem of fractional calculus. We utilize these methods to construct the numerical solutions of the FICEs. The accuracy and effectiveness of these methods is satisfied graphically by combining the numerical results and plotting the absolute error. Also, the absolute errors are tabulated, and a good agreementfound in all cases.

The modelling of nonlinear distance sensor using piecewise newton polynomial with vertex algorithm

The Sharp GP2Y0A02YK0F is categorized as a nonlinear sensor for distance measurement. This sensor is also categorized as a low-cost sensor. The higher resolution, cheap, high accuracy and easy to install are the advantages. The accuracy level of this sensor depends on the type of the measured object materials, requires an additional device unit and further processing is required since the output is non-linear. The distance determination is not easy for this type of sensor since the characteristic of this sensor fulfills non-injective function. The modelling process is one of methods to convert the output voltage of the sensor to a distance unit. The advantages of polynomial modelling are simple form model, moderate in flexibilities of shape, well known and understood properties, and easy to use for computational matters. The obstacle of polynomial-based modelling is the presence of Runge’s phenomenon. The minimization of Runge’s phenomenon can be done with decreasing the model order. The piecewise Newton polynomials with vertex determination method have been succeeded to generate a nonlinear model and minimize the occurrence of Runge’s phenomenon. The low level of MSE by 0.001 and error percentage of 2.38% has been obtained for the generated model. The low MSE level leads to the high accuracy level of the generated model.

ATANGANA–SEDA NUMERICAL SCHEME FOR LABYRINTH ATTRACTOR WITH NEW DIFFERENTIAL AND INTEGRAL OPERATORS

In this paper, we present a new numerical scheme for a model involving new mathematical concepts that are of great importance for interpreting and examining real world problems. Firstly, we handle a Labyrinth chaotic problem with fractional operators which include exponential decay, power-law and Mittag-Leffler kernel. Moreover, this problem is solved via Atangana-Seda numerical scheme which is based on Newton polynomial. The accuracy and efficiency of the method can be easily seen with numerical simulations.