Frequency formula for a class of fractal vibration system

Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.

A trend extraction method based on logistic functions and envelopes

A method of step characteristic trend extraction based on logistic functions and envelopes (LFEs) is proposed in this paper. Compared with the existing trend extraction methods, the LFE method can determine the starting position of the step trend using a logistic function and extract the local trend using upper and lower envelopes. This method enhances the extraction accuracy and reduces the computation time. To verify the effectiveness of the LFE method, a simulated signal with a step trend feature was compared with the five-spot triple smoothing method, wavelet transform method and empirical mode decomposition-based method. All of these methods were applied to a real shock signal. The results demonstrate that the LFE method can reliably and accurately extract the trends with step characteristics based on less prior knowledge. Moreover, the proposed technique is suitable for industrial online applications.

Identification and Modeling of College Students’ Psychological Stress Indicators for Deep Learning

Aiming at the problems of low accuracy of recognition results, long recognition time, and easy interference in traditional methods, a deep learning-oriented recognition modeling method of college students' psychological stress indicators is proposed. First, the ECG signal is collected by the ECG signal acquisition system, and the wavelet transform method is used to denoise the collected ECG signal. Then, the sequential backward selection algorithm is used to select the features of psychological stress indicators to reduce the feature dimension. Finally, based on the convolutional neural network in deep learning technology, a mental pressure indicator recognition model is established and the model parameters are optimized to realize the recognition of college students’ mental pressure indicators. Experimental results show that the method in this paper has high recognition accuracy, has high recognition efficiency, is not susceptible to interference, and has certain feasibility and effectiveness.

Induced magnetic field and viscous dissipation on flows of two immiscible fluids in a rectangular channel

The unsteady, magneto-hydrodynamic generalized Couette flows of two immiscible fluids in a rectangular channel with isothermal walls under the influence of an inclined magnetic field and an axial electric field have been investigated. Both fluids are considered electrically conducting and the solid boundaries are electrically insulated. Approximate analytical solutions for the velocity, induced magnetic, and temperature fields have been determined using the Laplace transform method along with the numerical Stehfest's algorithm for the inversion of the Laplace transforms. Also, for the nonlinear differential equation of energy, a numerical scheme based on the finite differences has been developed. A particular case has been numerically and graphically studied to show the evolution of the fluid velocity, induced magnetic field, and viscous dissipation in both flow regions.

Novel Analysis of Fuzzy Physical Models by Generalized Fractional Fuzzy Operators

The present article correlates with a fuzzy hybrid technique combined with an iterative transformation technique identified as the fuzzy new iterative transform method. With the help of Atangana-Baleanu under generalized Hukuhara differentiability, we demonstrate the consistency of this method by achieving fuzzy fractional gas dynamics equations with fuzzy initial conditions. The achieved series solution was determined and contacted the estimated value of the suggested equation. To confirm our technique, three problems have been presented, and the results were estimated in fuzzy type. The lower and upper portions of the fuzzy solution in all three examples were simulated using two distinct fractional orders between 0 and 1. Because the exponential function is present, the fractional operator is nonsingular and global. It provides all forms of fuzzy solutions occurring between 0 and 1 at any fractional-order because it globalizes the dynamical behavior of the given equation. Because the fuzzy number provides the solution in fuzzy form, with upper and lower branches, fuzziness is also incorporated in the unknown quantity. It is essential to mention that the projected methodology to fuzziness is to confirm the superiority and efficiency of constructing numerical results to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

A Metaheuristic Optimization Algorithm for Solving Higher-Order Boundary Value Problems

An effective metaheuristic algorithm to solve the higher-order boundary value problems, called a genetic programming technique is presented. In this paper, a genetic programming algorithm, which depends on the syntax tree representation, is employed to obtain the analytical solutions of higher- order differential equations with the boundary conditions. The proposed algorithm can be produce an exact or approximate solution when the classical methods lead to unsatisfactory results. To illustrate the efficiency and accuracy of the designed algorithm, several examples are tested. Finally, the obtained results are compared with the existing methods such as the homotopy analysis method, the B-Spline collocation method and the differential transform method.

A Reliable Iterative Transform Method for Solving an Epidemic Model

The main purpose of the work is to apply a new method, so-called LTAM, which couples the Tamimi and Ansari iterative method (TAM) with the Laplace transform (LT). This method involves solving a problem of non-fatal disease spread in a society that is assumed to have a fixed size during the epidemic period. We apply the method to give an approximate analytic solution to the nonlinear system of the intended model. Moreover, the absolute error resulting from the numerical solutions and the ten iterations of LTAM approximations of the epidemic model, along with the maximum error remainder, were calculated by using MATHEMATICA® 11.3 program to illustrate the effectiveness of the method.

Numerical investigation of fractional model of Phytoplankton–Toxic Phytoplankton–Zooplankton system with convergence analysis

In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This model is constituted by three components: phytoplankton, toxic phytoplankton and zooplankton. Phytoplankton species are self-feeding members of the plankton community and play a very significant role in ecosystems. A wide range of sea creatures get food through phytoplankton. This paper focuses on the implementation of the three above-mentioned computational methods for a nonlinear time-fractional phytoplankton–toxic phytoplankton–zooplankton (PTPZ) model with a perception to study the dynamics of a model. This study shows that the solutions obtained by employing the suggested computational methods are in good agreement with each other. The computational procedures reveal that the HASTM solution generates a more general solution as compared to RPSM and HPSTM and incorporates their results as a special case. The numerical results presented in the form of graphs authenticate the accuracy of computational schemes. Hence, the implemented methods are very appropriate and relevant to handle nonlinear fractional models. In addition, the effect of variation of fractional order of a time derivative and time [Formula: see text] on populations of phytoplankton, toxic–phytoplankton and zooplankton has also been studied through graphical presentations. Moreover, the uniqueness and convergence analyses of HASTM solution have also been discussed in view of the Banach fixed-point theory.