Modelling of Chaotic Processes with Caputo Fractional Order Derivative.

Chaotic dynamical systems are studied in this paper. In the models, integer order time derivatives are replaced with the Caputo fractional order counterparts. A Chebyshev spectral method is presented for the numerical approximation. In each of the systems considered, linear stability analysis is established. A range of chaotic behaviours are obtained at the instances of fractional power which show the evolution of the species in time and space.

Optimal Control of Time-Delay Fractional Equations via a Joint Application of Radial Basis Functions and Collocation Method.

A novel approach to solve optimal control problems dealing simultaneously with fractional differential equations and time delay is proposed in this work. More precisely, a set of global radial basis functions are firstly used to approximate the states and control variables in the problem. Then, a collocation method is applied to convert the time-delay fractional optimal control problem to a nonlinear programming one. By solving the resulting challenge, the unknown coefficients of the original one will be finally obtained. In this way, the proposed strategy introduces a very tunable framework for direct trajectory optimization, according to the discretization procedure and the range of arbitrary nodes. The algorithm's performance has been analyzed for several non-trivial examples, and the obtained results have shown that this scheme is more accurate, robust, and efficient than most previous methods.

Heat Transfer Coefficients Analysis in a Helical Double-Pipe Evaporator: Nusselt Number Correlations through Artificial Neural Networks.

In this study, two empirical correlations of the Nusselt number, based on two artificial neural networks (ANN), were developed to determine the heat transfer coefficients for each section of a vertical helical double-pipe evaporator with water as the working fluid. Each ANN was obtained using an experimental database of 1109 values obtained from an evaporator coupled to an absorption heat transformer with energy recycling. The Nusselt number in the annular section was estimated based on the modified Wilson plot method solved by an ANN. This model included the Reynolds and Prandtl numbers as input variables and three neurons in their hidden layer. The Nusselt number in the inner section was estimated based on the Rohsenow equation, solved by an ANN. This ANN model included the numbers of the Prandtl and Jackob liquids as input variables and one neuron in their hidden layer. The coefficients of determination were R 2 > 0.99 for both models. Both ANN models satisfied the dimensionless condition of the Nusselt number. The Levenberg-Marquardt algorithm was chosen to determine the optimum values of the weights and biases. The transfer functions used for the learning process were the hyperbolic tangent sigmoid in the hidden layer and the linear function in the output layer. The Nusselt numbers, determined by the ANNs, proved adequate to predict the values of the heat transfer coefficients of a vertical helical double-pipe evaporator that considered biphasic flow with an accuracy of ±0.2 for the annular Nusselt and ±4 for the inner Nusselt.

Chaotic Attractors with Fractional Conformable Derivatives in the Liouville-Caputo Sense and Its Dynamical Behaviors.

This paper deals with a numerical simulation of fractional conformable attractors of type Rabinovich-Fabrikant, Thomas' cyclically symmetric attractor and Newton-Leipnik. Fractional conformable and ß -conformable derivatives of Liouville-Caputo type are considered to solve the proposed systems. A numerical method based on the Adams-Moulton algorithm is employed to approximate the numerical simulations of the fractional-order conformable attractors. The results of the new type of fractional conformable and ß -conformable attractors are provided to illustrate the effectiveness of the proposed method.