Vol. 12, No. 1 (2023): Regular Issue

In this paper, we study the boundedness and continuability of the solutions of a generalized Liénard type system, using fractional derivatives of the local type. We obtain sufficient conditions for the solutions to be bounded and continuous by a suitably defined Lyapunov function. We illustrate the results and suggest extensions to asymptotic stability, through various examples adapted from the relevant literature.

In this article, the influence of reabsorption on steady flow of viscous incompressible Newtonian fluid through a permeable channel is presented. Perturbation solution is used to get the expressions for the velocities, mean pressure drop, shear stress at the wall and stream function. The influence of reabsorption on the velocities, wall shear stress, mean pressure drop and streamlines are discussed through graphs. The results of the model is applicable to the physiological problem of flow through renal tubules.

This work made for analyzing the finite-time stability of impulsive nonlinear delay damped system with caputo fractional derivative of orders $\alpha_1\in(1,2]$ and $\alpha_2\in (0,1]$. Sufficient conditions which are derived from extended form of Gronwall's inequality to analyze the stability in the finite range of time for such multi-term fractional-order impulsive control system. The potential of the proposed approach is demonstrated with the support of two numerical examples.

This research uses the stability radius approach to investigate the robust stability of an infinite-dimensional linear discrete-time system subjected to stochastic perturbations. First, we characterize the stability radius in terms of a Lyapunov equation. These characterizations improve a computational formula for calculating the stability radius. The second goal is to study how state feedback can maximize the stability radius. We characterize the maximum attainable stability radius using an infinite-dimensional discrete-time Riccati equation. Examples are provided to demonstrate the achieved outcomes.

In this paper the fractional order smoking model is represent with Caputo and Caputo Fabrizio fractional derivative operator of order $\phi \in (0, 1]$ for dynamical transmission of smoking. Human beings face dangerous diseases caused by smoking, including arms, lungs, stomach, cervix, breast, pancreatic cancer and many others. Stability and qualitative analysis of model is studied to show the dynamical behaviour of the model in feasible region. It's important to note that a more powerful approach for computing convergent solutions is applied for mathematical models based on a fractional order differential equation structure. Study of the convergence is often provided to demonstrate the process's effectiveness. It shows the stability, uniqueness and applicability of the model for the control of smoking in the society. Numerical simulation are established to show the actual behavior of the smoking spread.

In present work, we apply a novel $(\frac{G'}{G})$-formalism to construct more general solitary traveling wave solutions of Nonlinear Evolution Equations (NLEEs) such as Vakhnenko equation (VE), Camassa-Holm equation (CH), Symmetric Regularized Longwave Equation (SRWE). Method that we have chosen, is simple, straightforward and, gives the three types of solutions including trigonometric, exponential, and rational solutions as compared to other existing methods. Distinct periodic and solitary wave solutions are derived witch are rich in structure and gives wide range of solution under different parametric regime. Wolfram Mathematica 11 is used to perform the computation work and their corresponding plots and counter graphs are plotted using Matlab.

This paper addresses the effects of thermal radiation on fully developed flow in the containing nanoparticles in the presence of external magnetic field. A square vertical duct of side L is considered. At four sides of the duct Dirchlet boundary conditions are assumed. The governing nonlinear partial differential equations(PDE) are transformed into dimensionless PDE using suitable nondimensional parameters. Finite element method is employed to solve the highly nonlinear and coupled dimensionless PDE. The results are analyzed in terms of velocity contours and temperature distributions. It is observed that the fully developed nanofluid flow is significantly controlled by radiation and magnetic field.

The space of complete dynamical networks of systems with discontinuous piecewise linear maps is analyzed. The purpose of this paper is to investigate synchronizability and measures of information transmission in this complex network space. The network topologies are regular ring lattices which are characterized by circulant matrices and the conditional Lyapunov exponents are explicitly determined. The mutual information rate and the Kolmogorov-Sinai entropy are characterized and properties of these measures are proved, depending on the topological entropy of the local dynamics and on the synchronization interval. A topological order is defined and monotony properties between the network topological entropy, the mutual information rate and the Kolmogorov-Sinai entropy are established. Numerical simulations are provided to illustrate the theoretical results.

The fundamental objective of this paper is to tackle the time-fractional order transportation equations through two analytical methods, the method of q-homotopy analysis (q-HAM) and the method of reduced differential transform (RDTM) through numerical computation and simulations. The fractional derivative is considered in Caputo's sense. Three examples have been employed to illustrate the preciseness and effectiveness of the proposed methods for theoretical and numerical analysis purpose. The techniques provide series-form solution that converges sharply to the exact solution as the non-integer order approaches the integer order. Also, the graphical depictions of solutions are provided to compare the results of these methods.

An Ostwald-de Waele power-law nanofluid flow over a truncated cone in a non-Darcy porous medium is analysed numerically. Different volume fractions of Ti-alloy and MWCNTs nanoparticles are considered to obtain their complete influence on physical profiles. Error analysis is conducted and the results in special cases are also compared with previously published papers. Higher velocity is noticed for dilatant nanofluid when compared to pseudo-plastic nanofluid and it is decreased with nanoparticle volume fractions increment. This problem is very helpful in aerospace and medical sectors due to involvement of Ti-alloys in making aircraft turbines, orthopaedical instruments etc.

The subject of the present paper is recurrent neural networks with variable impulsive moments. The impact activation functions are specified such that the structure for the jump equations are in full accordance with that one for the differential equation. The system studied in this paper covers the works done before, not only because the impacts have recurrent form, but also impulses are not state-dependent. The conditions for existence and uniqueness of asymptotically stable discontinuous almost periodic solutions are obtained. Through the present study, the possibility of neuron membranes with negative capacitance is involved in neural networks and this is one of the main novelties of the present study. The vector-matrix representation of the system is used for the clarity of the proofs and for making calculations easier.

Dynamical systems' behavior is sometimes varied with some impulse and sudden changes in process. The dynamics of these systems can not be modeled by previous concepts of derivative or fractional derivatives any longer. The short memory concept is a solution and a better choice for fractional modeling of such processes. We apply short memory fractional differential equations for these systems. We propose collocation methods based on piecewise polynomials to approximate solutions of these equations. We provide various examples to demonstrate the application of the short memory derivative for impulse systems and efficiency of the presented numerical methods.

This paper reveals a new way to indicate the presence of chaos in continuous-time models, beside other techniques such as the method of Lyapunov exponents and bifurcation diagrams. The sequential test confirms the existence of an unpredictable solution, and therefore, Poincaré chaos for differential equations. The main part of the study consists of the description of the novel algorithm, demonstrating its convenience for analysis of chaotic dynamics. The procedure facilities are carefully determined, and they are implemented to Lorenz and Rössler systems. The peculiarity of the method lies in the fact that in addition to the indication of just chaos, we clarify the divergence character of a single trajectory, which is based on the unpredictability feature. Potentially it can be more effective than the conventional ways for indication of chaos. The presence of chaos is also approved in the case of zero largest Lyapunov exponent.

In this paper, Hardy-Rogers type common fixed point theorem of self contractive maps in cone 2-metric spaces over Banach algebra is proved. The corresponding conclusions in the literature are improved and generalize by obtained results. Some examples proposed to illustrate our main results.

The parameter estimation of interatomic potentials are considered as a solution of some second order ordinary differential equation. The developed method for the multiple goal function approach can be applied to a wide range of transcendental nonlinear problems. The parameter estimation of several interatomic potentials in classical functional forms are estimated using the objective least squares function method. Potentials such as Lennard-Jones, Classical Rydberg, Classical, Generalized Morse and Biswas-Hamann potential were each considered. Two new interatomic potentials, Modified Generalized Morse and Modified Lennard-Jones potential are proposed in this text. Numerical estimates were obtained using gold atom for numerical simulation in MathCad® software while the potential energy curves are constructed and reconstructed in Mathematica®. The estimated parameters gave good fit to experimental data sets of gold atom as the error plots of the potential energy curves are small. The values obtained for the goal function also shows that the approximated parameters values are good approximations.