Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

Realistic scenarios typically involve several interacting dependent variables, leading to nonlinearity in mathematical models. Nonlinear differential equations are a crucial mathematical phenomenon for describing real-world problems. On the other hand, fuzzy set theory is a valuable tool for modelling the uncertainty present in a system. So, nonlinear mathematical models under uncertainty need an explanatory study on a system of nonlinear fuzzy differential equations. In this paper, we analyse nonlinear fuzzy differential equations with possible applications. This paper advances the existing literature by introducing a definition of a fuzzy sequence and developing the existence and uniqueness theory for fuzzy solutions of nonlinear differential equations using the fixed-point theorem. The component-wise analysis of a fuzzy nonlinear system has been conducted. Furthermore, the analysed theory precedes an aptly fitted application of a fuzzy nonlinear dynamical system using a numerical scheme.

Seven common errors in finding exact solutions of nonlinear differential equations

The process of hiding texts or images into images and signals has been remained contemporary topics over more than decades in Information Sciences. An attempt has been made to hide information or images into the solutions of non-linear differential equations of some especial type like Lorezn and Rossler’s equations, these equations have great importance in almost all practical applications from engineering, geography (weather forecasting) and even in stock markets due to its chaotic behavior of solutions which is dependent on initial conditions. The chaotic nature and the aperiodicity of such solutions have become the main source of attraction for encryptions. The 3D parametric (time) series of solutions of Lorenz and Rossler’s equations are produced phase vectors, and one of these phase vectors is chosen for the purpose of text or image embedding into its error (noise). The error (noise) is a part of chaotic solutions behaved aperiodicity in phase vectors. This attracted the embedding of manipulated text or image into it. Modified or reconstructed phase vector embedded with text or image when reproduced in the form of phase surfaces are quite similar to the original solutions surfaces. So Lorenz and Rossler solutions are used for embedding benchmark images like Lena, Fruits and Vegetables. Having embedded these images into series solutions, we found Lorenz solutions are more appropriate over the Rossler solutions. A thorough statistical analysis together with histograms of both original images and embedded images are reported. Experiments with RGB values of embedded images are also reported. In the present research, we have further extended this technique by using Markov processes to our target images and converted into encrypted images before embedding into solution phase vector of Lorenz and Rossler’s equations. Markov processes are made more confusing/diffusing the target images and has become difficult for intruders to attack for retrieval.

The present study aims to quantify the flow field, flow velocity, and heat transfer features over a horizontal flat plate under the influence of an applied magnetic field, with a particular emphasis on low Prandtl number fluids. Nonlinear partial differential expressions can be incorporated into the ordinary differential framework with the use of appropriate transformations. This research utilizes the variational iteration method (VIM) to approximate solutions for the system of nonlinear differential equations that define the problem. The objective is to demonstrate superior flexibility and broader application of the VIM in addressing heat transfer issues, compared to alternative approaches. The results obtained from the VIM are compared with numerical solutions, revealing a significant level of accuracy in the approximation. The numerical findings strongly suggest that the VIM is effective in providing precise numerical solutions for nonlinear differential equations. The analysis includes an examination of the flow field, velocity, and temperature distribution across various parameters. The study found that improving temperature patterns, velocity distribution, and flow dynamics were all positively impacted by increasing the Prandtl numbers. As a result, this leads to the thickness of the boundary layer to decrease and improves heat transfer at the moving surface. Thus, the convection process becomes more efficient. When the strength of the magnetic field is increased, the velocity of the fluid decreases. This observation aligns with expectations since the magnetic field hampers the natural flow of convection. Notably, the convection process can be precisely controlled by carefully applying magnetic force.

A new analytical study of MHD stagnation-point flow in porous media with heat transfer

The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find any elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is being paid to the case of elliptic solutions with several poles in a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.

The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find an elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is paid to the case of elliptic solutions with several poles inside a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.

A theoretical analysis of steady and unsteady magnetohydrodynamic (MHD) flow of blood through a stenosed artery with variable viscosity and slip conditions is presented in this work. Slip conditions and variable viscosity dependent on hematocrit for both steady and unsteady blood flow models are taken into consideration in this study. The governing equations have been derived by treating blood as incompressible magnetohydrodynamic (MHD) Newtonian fluid. Galerkin Weighted Residual method, Newton Raphson method and fourth order Runge-Kutta method are used to solve the non-linear differential equations for velocity profiles of the blood flow, flow rate, shear stress and flow resistance for different values of hematocrit parameter, magnetic field parameter and slip velocity. The Galerkin Weighted Residual method is applied to transform the partial differential equations describing the flow into systems of non-linear equations and system of non-linear first order differential equations for the time independent and time dependent MHD flow models respectively. Newton Raphson’s method with the aid of maple software is used to solve the system of non-linear equations while the fourth order Runge-Kutta method is used to solve the system of non-linear first order differential equations. The research analyzed how the flow velocity, flow rate, shear stress and resistance to fluid flow respond to changes in some key parameters such as hematocrit parameter, magnetic field parameter and slip velocity. The results as presented graphically reveal that flow velocity and flow rate decrease while shear stress and resistance to flow increase when the hematocrit parameter increases. Also, flow velocity, flow rate and shear stress increase while resistance to flow decrease when the slip velocity increases. Finally, the increase in magnetic field parameter led to a decrease in flow velocity, flow rate and shear stress but increased the flow resistance. The flow velocity of the unsteady blood flow model is higher when compared to steady blood flow model.

This volume is an on-line reprint of the original book published 1972 by Springer-Verlag which was intended to provide a comprehensive treatment of contemporary developments in methods of perturbation for nonlinear systems of ordinary differential equations. In this respect, it appeared to be a unique work, with hundreds of citations. Even today is a basic reference in the approximate solution of non-linear differential equations, specially appearing in problems of Celestial Mechanics. The original goal was to describe perturbation techniques, discuss their advantages and limitations and give some examples. The approach was founded on analytical and numerical methods of nonlinear mechanics. Attention had been given to the extension of methods to high orders of approximation, required now by the increased accuracy of measurements in all fields of science and technology. The main theorems relevant to each perturbation technique were outlined, but they only provided a foundation and were not the objective of the original book. Each chapter concluded with a detailed survey of the contemporary literature, supplemental information and more examples to complement the text, when necessary, for better comprehension.

The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].

A New Multistep Technique Based on the Nonuniform Rational Basis Spline Curves for Nonlinear Transient Heat Transfer Analysis of Functionally Graded Truncated Cone

The work presents the analysis of the free boundary value problem related to the one-dimensional invasion model of new species in biofilm reactors. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of non-linear hyperbolic partial differential equations governing the microbial species growth and a system of semi-linear elliptic partial differential equations describing the substrate trends. The model is completed with a system of elliptic partial differential equations governing the diffusion and reaction of planktonic cells, which are able to switch their mode of growth from planktonic to sessile when specific environmental conditions are found. Two systems of non-linear differential equations for the substrate and planktonic cells mass balance within the bulk liquid are also considered. The free boundary evolution is governed by a differential equation that accounts for detachment. The qualitative analysis is performed and a uniqueness and existence result is presented. Furthermore, two special models of biological and engineering interest are discussed numerically. The invasion of Anammox bacteria in a constituted biofilm inhabiting the deammonification units of the wastewater treatment plants is simulated. Numerical simulations are run to evaluate the influence of the colonization process on biofilm structure and activity.

Sufficient conditions for the existence of one-sided integral manifolds of solutions of systems of nonlinear difference equations are obtained. One-sided nonlinear projections, defining these manifolds, are constructed.

Travelling wave solutions of nonlinear partial equations by using the first integral method

Existence and stability of stationary solutions of nonlinear random difference equations are studied in this note. Firstly, we give the weak conditions that guarantee the continuity of Lypanunov exponents under small random perturbations. Secondly, we find out the conditions under which the ratio of the random norm and the standard Euclidean norm has deterministic bounds. Based on these new results, we provide easy-to-use conditions that guarantee the existence and almost sure stability of a stationary solution. In addition, we also prove that the stationary solution converges with probability one to the fixed point of the corresponding deterministic system as the noise intensity tends to zero.