Order Nonlinear Differential Equations Research Articles

NEW WAVE SOLUTIONS OF TIME FRACTIONAL CHAFEE-INFANTE EQUATION WITH BETA DERIVATIVE

In this article, we discuss the exact solutions forthe Chafee-Infante equation involving beta fractional derivative. Beta fractional derivative which is a local derivative, is a modification of conformable fractional derivative. Using the Modified Kudryashov Method, we obtain the general solution of the time fractional Chafee-Infante equation with the help of Wolfram Mathematica. We use chain rule and wave transform to convert the equation into integer order nonlinear ordinary differential equation. Hence, we don’t need any discretization, normalization, or reduction. Moreover, 3D graphical representations are given. With the help of these representations, we can have an idea on the physical and geometrical behavior of the solutions.

Wave solutions in (3 + 1)-dimensional generalized fractional mKdV-ZK equation utilizing jacobi elliptic functions

In this research, we investigate the effects of fractional order on the (3 + 1)-dimensional generalized space-time fractional modified KdV-Zakharov-Kuznetsov (mKdV-ZK) equation. We approach the problem by utilizing the conformable fractional derivative. By reducing the mKdV-ZK equation to an integer order nonlinear ordinary differential equation, we apply the Jacobi elliptic function method to find exact solutions. These solutions are specifically tailored for the fractional order of the (3 + 1)-dimensional generalized mKdV-ZK equation, encompassing solitary waves, shock waves, and periodic waves. We also compare these exact solutions with fractional solutions to gain further insights. Notably, our approach demonstrates the feasibility of solving nonlinear time-fractional differential equations with conformable derivatives. Several diagrams have been included to visually depict the behavior of the solutions under fractional order when certain special parameter values are employed.

A mean field game model for optimal trading in the intraday electricity market

AbstractIn this study, we provide a simple one period mean-field-games setting for the joint optimal trading problem for electricity producers in the electricity markets. Based on the Markowitz mean-variance approach from stock trading, we consider a decision problem of an electricity provider when determining the optimal fractions of production that should be traded in the day-ahead and in the intraday markets. Moreover, all such providers are related by a ranking criterion and each one wants to perform as good as possible in this ranking. We first start with a simple model where only the price risk in the intraday market is present and subsequently extend the problem to the cases involving either production and/or demand uncertainty. The key technique is to reduce the optimality conditions to a first order non-linear ordinary differential equation. We will illustrate our findings by various numerical examples. Our findings will in particular be important for electricity producers using renewable resources.

Numerical methods of fourth, sixth and eighth orders convergence for solving third order nonlinear ODEs

In this paper, we design numerical methods of fourth, sixth and eighth orders convergence for solving BVPs of fully third order nonlinear differential equations. The methods are based on the use of high order quadrature formulas for computing integrals containing Green function and its derivatives at each iteration of the iterative method on continuous level for finding the solutions of the BVPs. We prove that the order of the methods is equal to the order of quadrature methods used. Many examples confirm the theoretical conclusion.

Radiation effect on convective boundary layer dusty flow over a stretching surface

The effect of radiation evolved from fluids and particles on a convective balanced boundary layer dusty fluid flow model has been analyzed over a vertical stretching surface. Models are available with the consideration of radiation and viscous dissipation in fluid phase only. However, an equal and opposite reaction/force is also developed in the particle/dust phase, which was not taken into account in the previous literature. So, this paper investigates a model where the radiation and viscous dissipation are coupled through both the fluid and particle phase equations. A linear stretching of surface generates the boundary layer flow. By the help of suitable similar transformation, the prevailing equations are transferred to ordinary non-linear differential equations of order three and are solved numerically by the bvp4c method of MATLAB. An analysis has been performed to show the effectiveness of different thermophysical parameters on different boundary layer characteristics. The numerical results are in good agreement with the past literature, validating the present method and result. Graphs are plotted for different flow characteristics to explore the physics of problem.

An analytical investigation on nonlinear vibrations and stability of Timoshenko pipes conveying two-phase flow

The primary objective of this study is to conduct a comprehensive investigation on the nonlinear free vibration and stability analysis of the multilayer graphene reinforced pipe conveying liquid–gas two–phase flow based on the Timoshenko beam model. The Halpin–Tsai micromechanics model is employed which provides a means to evaluate and analyze the mechanical behavior and performance of the multilayer FG reinforced pipe. According to Von–Kármán strain relations and applying the Hamilton's principle, the governing equations of motion are derived. The Galerkin technique is operated to convert the partial differential equation (PDE) to the ordinary differential equation (ODE). The homotopy analysis method is employed to solve the fourth order nonlinear differential equation. Closed–form expressions for the nth nonlinear frequency and time history as well as the critical flow velocity based on the Timoshenko theory are achieved. The results demonstrate that neglecting the effects of rotary inertia and shear deformation may cause some errors by the increment of the initial amplitude. Furthermore, the critical flow velocity, depends not only on the mechanical and geometrical properties of the pipe but also on the initial mode shape excitation. The results indicate that by the increment of flow velocity, liquid density and slenderness ratio, the nonlinear frequency decreases, on the other hand, by the increment of the initial amplitude, the nonlinear frequency increases.

Existence results for positive periodic solutions to first order neutral differential equations with distributed deviating arguments

We take into account the first order nonlinear neutral differential equation with distributed deviating arguments.
 Using Krasnoselskii's fixed point theorem, we give some new criteria for the existence of positive periodic solutions to this equation. The theorems we have established are illustrated by an example.

Some new Lyapunov-type inequalities for a class of (<i>n</i>+1)th order nonlinear differential equations

Some new Lyapunov-type inequalities for a class of (<i>n</i>+1)th order nonlinear differential equations

Uniform Ultimate Boundedness Results for some System of Third Order Nonlinear Delay Differential Equations

Lyapunov functional, third-order delay differential system, Boundedness.

Global attractivity of a higher order nonlinear difference equation with decreasing terms

In the present paper, we further study the asymptotical behavior of the following higher order nonlinear difference equation x ( n + 1 ) = a x ( n ) + b f ( x ( n ) ) + c f ( x ( n − k ) ) , n = 0 , 1 , … where a , b and c are constants with 0 < a < 1 , 0 ≤ b < 1 , 0 ≤ c < 1 and a + b + c = 1 , f ∈ C [ [ 0 , ∞ ) , [ 0 , ∞ ) ] with f ( x ) > 0 for x > 0 , and k is a positive integer, which has been recently studied in: On global attractivity of a higher order difference equation and its applications [Electron. J. Qual. Theory Diff. Equ. 2022, No. 2, 1–14]. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation, and show the applications of these results to some population models.

New Criteria on the Oscillatory Behaviour of Second Order Nonlinear Differential Equations with Mixed Neutral Terms

Abstract The authors present some new criteria for the oscillation of second order nonlinear differential equations with mixed nonlinear neutral terms and mixed deviating arguments. The approach used is to linearize the equation under consideration and then to deduce the oscillation from that of the linear form. In so doing, the authors obtain new oscillation criteria via a comparison with the first order equations whose oscillatory behaviors are known. They illustrate their results with some examples.

Analytical study on couple stress flow of GO-EG and GO-W nanofluid over an extending cylinder along with variable viscosity

The main goal of this research is to present the concept of enhancing heat transfer within emerging technology. To achieve this, tiny metal and nonmetal particles ranging from 1 to 100 nm in size are introduced into base liquids. These nanoscale particles are utilized to improve the thermal performance of the liquids, leading to what are termed nanofluids. The utilization of these fluids and the examination of the flow of thin films have valuable implications across various sectors such as engineering, technology, and industries. This research focuses on analyzing the convective flow behavior of nanofluids, specifically, graphene oxide-ethylene glycol (GO−EG) and graphene oxide-water (GO−W) on a moving surface. The study investigates the impacts of magnetic fields and varying viscosity. By making use of the thermophysical characteristics of the base fluid and the nanofluid, as well as implementing a similarity transformation within the fundamental equations that govern energy and momentum, we formulate a 5th order nonlinear ordinary differential equation (NODE) to describe the velocity profile. This is combined with a second-order NODE that describes the distribution of temperature. To solve this derived NODE, we employ a method known as the Homotopy Analysis Method (HAM) for analytical solution. The impact of the relevant factors, Prandtl number, including magnetic field parameter, thickness of the liquid, couple stress parameter, temperature distribution, dynamic viscosity, and Eckert number, on the skin friction, velocity profile, and Nusselt's number are interrogated through graphical representation. The velocity field exhibits a decline as the couple stress parameter, magnetic field parameter, liquid thickness, and dynamic viscosity experience an increase. Conversely, the temperature field displays a rise as the Eckert number and dynamic viscosity experience an increase. To ensure the convergence of the issue, dual solutions of the problem are employed, and this is verified through the utilization graphs and tables. Due to the considerable challenge encountered in heat transfer applications for cooling diverse equipment and devices across industries like automotive, microelectronics, defense, and manufacturing, there is a strong expectation that this theoretical methodology could make a favorable contribution towards enhancing heat transfer efficiency. This improvement is sought to meet the requirements of the manufacturing and engineering sectors.

Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow

The present work introduces a deep learning approach to describe the perturbations of the pressure and radius in arterial blood flow. A mathematical model for the simulation of viscoelastic arterial flow is developed based on the assumption of long wavelength and large Reynolds number. Then, the reductive perturbation method is used to derive nonlinear evolutionary equations describing the resistance of the medium, the elastic properties, and the viscous properties of the wall. Using automatic differentiation, the solutions of nonlinear evolutionary equations at different time scales are represented using state-of-the-art physics-informed deep neural networks that are trained on a limited number of data points. The optimal neural network architecture for solving the nonlinear partial differential equations is found by employing Bayesian Hyperparameter Optimization. The proposed technique provides an alternate approach to avoid time-consuming numerical discretization methods such as finite difference or finite element for solving higher order nonlinear partial differential equations. Additionally, the capability of the trained model is demonstrated through graphs, and the solutions are also validated numerically. The graphical illustrations of pulse wave propagation can provide the correct interpretation of cardiovascular parameters, leading to an accurate diagnosis and successful treatment. Thus, the findings of this study could pave the way for the rapid development of emerging medical machine learning applications.

NUMERICAL STUDY OF THE EQUATION ON THE GRAPH FOR THE STEADY STATE NON-NEWTONIAN FLOW IN THIN TUBE STRUCTURE

The dimension reduction for the viscous flows in thin tube structures leads to equations on the graph for the macroscopic pressure with Kirchhoff type junction conditions in the vertices. Non-Newtonian rheology of the flow generates nonlinear equations on the graph. A new numerical method for second order nonlinear differential equations on the graph is introduced and numerically tested.

Retraction notice to ‘‘Oscillation Properties of Third Order Nonlinear Delay Dynamic Equations on Time Scales’’ [J. Amr. Uni. 3 (2023) 75 – 100

In this paper, we shall investigate the oscillatory properties of third order nonlinear delay dynamicequations. Applying suitable comparison theorems and by a Riccati transformation technique, weestablish some new sufficient conditions which insure that every solution of this equation either oscillatesor converges to zero. Our results not only unify the oscillation of third order nonlinear differential anddifference equations but also can be applied to different types of time scales with

Utilizing the extended tanh-function technique to scrutinize fractional order nonlinear partial differential equations

In a range of nonlinear fields, for example molecular biology, physics in plasma, quantum mechanics, elastic media, nonlinear optics, the surface of water waves, and others, many complicated nonlinear behaviors can be pronounced using nonlinear fractional partial differential equations. The nonlinear traveling waves, including nonlinear space-time fractional order differential equations such as the time fractional Sharma-Tasso-Olver (STO) and the space-time fractional Kowerteg-de Vries-Burgers (KdV-Burgers) equations, are analysis through a resourceful approach, namely the extended tanh-function by the definition of the conformable derivative. The stated equations are prevalently used in marine and coastal infrastructure, such as fluid flow, plasma engineering, fiber optics, fusion and fission phenomena, and so on. By renovating fractional order differential equations to ordinary differential equations, fractional order differential transformation simplifies the solution process. Several types of solutions, such as soliton types, bell types, kink types, and some other kinds of results, are developed and portrayed for definite values of free parameters that are plotted in terms of 3D, contour, and spherical shapes. The method is an algebraic approach that is both direct and efficient for dealing with nonlinear equations and yield a distinct category of solutions known as solitons. The suggested method is being used in this study to generate attractive, and effective results that are flexible, easy, and quicker to simulate using general mathematical techniques, namely Maple. It is worth declaring that all resultant solutions are verified for accuracy by exchanging them directly within the original equation through the software package and originating them accurately.MSC: 35C25, 35C07, 35C08, 35Q20, 76B25

Numerical Investigation of MHD Carreau Nanofluid Flow with Nonlinear Thermal Radiation and Joule Heating by Employing Darcy–Forchheimer Effect over a Stretching Porous Medium

Heat transfer in fluid mechanisms has a stronghold in everyday activities. To this end, nanofluids take a leading position in the advent of the betterment of thermal conductivity. The present study examines numerical investigations of incompressible magnetohydrodynamic (MHD) flow of Carreau nanofluid by considering nonlinear thermal radiation, Joule heating, temperature-dependent heat source/sink, and chemical reactions with attached Brownian movement and thermophoresis above a stretching sheet that saturates the porous medium. Pertaining similarity assumptions are used to change the flow equations into tractable forms of higher order nonlinear ordinary differential equations (ODEs). The continuation technique is adopted in the MATLAB bvp4c package for the numerical outcomes. The velocity, temperature, and nanoparticle concentration distributions in contrast to the leading parameters are availed in graphical and tabular descriptions. Among the many outcomes, increasing the radiation parameter from 0.2 to 0.8 surged the heat transfer rate by 47.78 % at n = 1.5 and lifted it only by 8.5 % at n = 0.5. By boosting the magnetic parameter from 0 to 1.5, respective 37.64 % and 20.17 % rises in local drag forces are achieved in shear-thickening and thinning regions. On top of that, chemical reactions and Brownian motion parameters decay the concentration field. The distinctiveness of this method is that a solution is secured for the problem, which is highly sensitive to initial and boundary conditions. It will be worth mentioning that these fluid flow models will be applicable in various fields, such as engineering, petroleum, nuclear safety processes, and medical science.

Exact solutions of classes of second order nonlinear partial differential equations reducible to first order

This paper illustrates the successful implementation of the method of variation of parameters in combination with the method of characteristics and other techniques to obtain exact solutions for a wide range of partial differential equations. The proposed approach reduces partial differential equations (PDEs) to first-order differential equations, referred to as classical equations, including Bernoulli, Ricatti, and Abel equations. In addition, the techniques proposed have the ability to produce precise solutions for nonlinear second order PDEs. For each PDE class, the method's effectiveness is demonstrated through illustrative examples.

Solution of System of First Order Nonlinear Non-Homogeneous Fuzzy Ordinary Differential Equations by Embedding Method

In this study, a system of first order nonlinear non-homogeneous fuzzy ordinary differential equations will be examine in fuzzy environment and solved using embedding method. The results of nonlinear non-homogeneous fuzzy ordinary differential equations are established which followed the form of SxS matrices and all the components of the matrices are real functions of time denoted by t. The accuracy of the results obtained is tested on some constructed example and recommended that further study should consider odd and even system of nonlinear non-homogeneous fuzzy ordinary differential equations by embedding method.

Developing the Third Order Linear and Nonlinear Fuzzy Ordinary Differential Equations in Hilbert Space by the Extension Principle.

In this study, a third nonlinear fuzzy ordinary differential equation is defined and established in Hilbert space using extension principle. The results obtained gives clear distinction between this work and the existing ones in the literature an example is formulated which shows that a closed subset of a Hilbert space is a Hilbert Space. It is recommended that future study should consider the development of system of third order linear and nonlinear fuzzy ordinary differential equations in Hilbert space by the extension principle.