Solutions Of Nonlinear Differential Equations Research Articles

A hybrid classical-quantum algorithm for solution of nonlinear ordinary differential equations

A hybrid classical-quantum approach for the solution of nonlinear ordinary differential equations using Walsh-Hadamard basis functions is proposed. Central to this hybrid approach is the computation of the Walsh-Hadamard transform of arbitrary vectors, which is enabled in our framework using quantum Hadamard gates along with state preparation, shifting, scaling, and measurement operations. It is estimated that the proposed hybrid classical-quantum approach for the Walsh-Hadamard transform of an input vector of size N results in a considerably lower computational complexity (O(N) operations) compared to the Fast Walsh-Hadamard transform (O(Nlog2(N)) operations). This benefit will also be relevant in the context of the proposed hybrid classical-quantum approach for the solution of nonlinear differential equations. Comparisons of results corresponding to the proposed hybrid classical-quantum approach and a purely classical approach for the solution of nonlinear differential equations (for cases involving one and two dependent variables) were found to be satisfactory. Some new perspectives relevant to the natural ordering of Walsh functions (in the context of both classical and hybrid approaches for the solution of nonlinear differential equations) and the representation theory of finite groups are also presented here.

Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential Equations.

Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf-Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers-Huxley, generalized equation of Camassa-Holm, generalized equation of Swift-Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods.

Mathematical simulation of autonomous wind electric installation with magnetoelectric generator

Purpose. Development of a mathematical model of an autonomous wind power plant based on an end-generator with a double stator and combined excitation to evaluate methods for improving the efficiency of conversion of mechanical wind energy into electricity. Methodology. The research used methods of general theory of wind power plants, methods of mathematical modeling, which are based on the numerical solution of nonlinear differential equations to evaluate methods for correcting the output power in Matlab-Simulink by modifying standard units. Findings. A numerical simulation mathematical model of an autonomous wind power plant consisting of a magnetoelectric generator with an axial magnetic flux with combined excitation and a double stator has been developed. The model was created to study the parameters and characteristics of the installation, as well as to evaluate methods and means to improve the efficiency of conversion of wind energy into electricity. According to the research, it is established that a more effective method for regulating the output power of the generator in the wind turbine is the use of additional winding for magnetization, compared with the use of additional capacity. The latter provides up to 716% increase in output power, while using the magnetizing winding can increase the output power to 3235%. The results obtained by the authors allow further developing a number of methods to increase the efficiency of conversion of mechanical energy of the wind turbine rotor into electrical energy. Originality. The mathematical model developed for the first time, in contrast to the existing ones, takes into account the presence of a double stator, an additional winding for magnetization of the magnetic system and the axial nature of the circuit of the main and additional magnetic flux. The developed model also takes into account the change in the parameters of the electric generator with axial magnetic flux when changing the parameters of the wind, the rotor of the wind turbine and the load. The model is designed to analyze the possibility of adjusting the output power of the generator when the wind speed changes. Practical value. The simulation results indicate the prospects of industrial implementation of wind power plant based on magnetoelectric generator for their use as autonomous electrical installations and as part of shunting power systems.

Dynamical analysis of discrete time equations with a generalized order

Some natural phenomena that arise in nonlinear sciences can be sometimes discussed using discrete-time systems. In this work, we investigate the periodicity, boundedness, oscillation, stability, and some exact solutions of nonlinear difference equations with generalized order. Exact solutions are extracted using the traditional iteration and the modulus operator. The stability of the equilibrium points is examined using some well-known theorems. Some numerical examples are also introduced to confirm the validity of the theoretical work. We use MATLAB to implement the numerical part. The used technique can be easily utilized to deal with other rational recursive problems.

Dynamical analysis and solutions of nonlinear difference equations of twenty-fourth order

This paper discusses the behaviors and solutions of some rational recursive relations of twenty-fourth order using the iteration technique and the modulus operator. The stability of the equilibrium points are comprehensively analyzed. We also present other properties such as periodicity, oscillation and bounded solutions. Some numerical examples are obviously given to ensure the validity of the theoretical work. These examples are plotted using MATLAB. The proposed techniques can be utilized to be applied on other nonlinear equations.

Decision-Making on the Solution of a Stochastic Nonlinear Dynamical System of Kannan-Type in New Sequence Space of Soft Functions

In this paper, we construct and investigate the space of weighted Gamma matrix of order r in Nakano sequence space of soft functions. The idealization of the mappings has been achieved through the use of extended s -soft functions and this sequence space of soft functions. This new space’s topological and geometric properties, the multiplication mappings that stand in on it, and the mappings’ ideal that correspond to them are discussed. We construct the existence of a fixed point of Kannan contraction mapping acting on this space and its associated prequasi ideal. Interestingly, several numerical experiments are presented to illustrate our results. Additionally, some successful applications to the existence of solutions of nonlinear difference equations of soft functions are introduced.

Some Results on Fractional Differential Equation With Mixed Boundary Condition via S-Iteration

The present paper discuss the existence, uniqueness and other properties of solutions of nonlinear differential equation of fractional order involving the Caputo fractional derivative with mixed boundary condition. The analysis of obtained results is based on application of S-iteration method. Since the study of qualitative properties in general required differential and integral inequalities, but here S-iteration method itself has equally important contribution to study various properties such as dependence on boundary data, closeness of solutions and dependence on parameters and functions involved therein. The results obtained are illustrated through example.

New Numerical Approach for Solution of Nonlinear Differential Equations

Nonlinear differential equations extensively used mathematical models for many interesting and important phenomena observed in numerous areas of science and technology. They are inspired by problems in diverse fields such as economics, biology, fluid dynamics, physics, differential geometry, engineering, control theory, materials science, and quantum mechanics. This special issue aims to highlight recent developments in methods and applications of nonlinear differential equations. In addition, there are papers analyzing equations that arise in engineering, classical and fluid mechanics, and finance. In this paper, we present a new numerical approach, which is concerned with the solutions of Nonlinear Differential Equations determined by a new approximation system based on inverse Laplace transforms using Chebyshev polynomials functional matrix of integration. The obtained solutions are novel, and previous literature lacks such derivations. The reliability and accuracy of our approach were shown by comparing our derived solutions with solutions obtained by other existing methods. The efficiency of the proposed numerical technique is exhibited through graphical illustrations and results drafted in tabular form for specific values of the parameters to validate the numerical investigation. The system capability is clarified through several standard nonlinear differential equations: Duffing, Van der Pol, Blasius, and Haul. The numerical results illustrate that the estimated result is in good agreement with exact or numerical styles available in literature whenever the exact results are unknown. Errors estimation to the corresponding numerical scheme also is carried out.

Governed optical solitons of the generalized Schrödinger equation with dual-power law of refractive index

Objective:Partial differential equation in a dual-power law nonlinear medium with variable coefficients is considered. The objective is to find solitary wave solution that can be considered as a governed optical solitons. Methods:Exact solutions of nonlinear differential equation are found using the direct method. Constraints on variables of coefficients are used for reducing equation to the nonlinear ordinary differential equation having analytical solutions. Results:Periodic and solitary wave solutions are obtained of nonlinear partial differential equation with dual-power law of nonlinearity and coefficients depending on variable. Two simple examples of acceleration and deceleration of solitary wave solutions are discussed.

The Approximate Solution of Nonlinear Flexure of a Cantilever Beam with the Galerkin Method

For the optimal design and accurate prediction of structural behavior, the nonlinear analysis of large deformation of elastic beams has broad applications in various engineering fields. In this study, the nonlinear equation of flexure of an elastic beam, also known as an elastica, was solved by the Galerkin method for a highly accurate solution. The numerical results showed that the third-order solution of the rotation angle at the free end of the beam is more accurate and efficient in comparison with results of other approximate methods, and is perfectly consistent with the exact solution in elliptic functions. A general procedure with the Galerkin method is demonstrated for efficient solutions of nonlinear differential equations with the potential for adoption and implementation in more applications.

On an approach to the analysis of oscillatory solutions of nonlinear delay difference equations

The objective of this article is to establish an oscillation criterion for all solutions of nonlinear difference equation where is a sequence of nonnegative real numbers and is not necessarily monotone sequence. Also, an example is presented to illustrate the significance of the main result.

OPTIMAL PORTFOLIO CHOICE WITH CRASH AND DEFAULT RISK

We consider the worst-case scenario portfolio approach, introduced by Korn & Wilmott (2002), in a multi-asset setting, where asset defaults can occur in addition to asset crashes. In our model, the strictly risk-averse investor does not know which asset is affected by the worst-case scenario. Based on a reformulation of the value function we define the set of minimum constant portfolio processes. We prove the existence of such a process that solves the worst-case crash/ default portfolio problem. In particular, we prove the existence of a portfolio process that is optimal for the investor and at the same time makes him/ her indifferent to the time of occurrence of the worst possible crash or default scenario. The optimal portfolios are derived from solutions of non-linear differential equations. Therefore, we construct an algorithmic framework to analyze selected examples. These examples show the importance of adapting the portfolio process when considering such worst-case scenarios in the investment problem.

Mixed Convection Flow over an Elastic, Porous Surface with Viscous Dissipation: A Robust Spectral Computational Approach

A novel computational approach is developed to investigate the mixed convection, boundary layer flow over a nonlinear elastic (stretching or shrinking) surface. The viscous fluid is electrically conducting, incompressible, and propagating through a porous medium. The consequences of viscous dissipation, Joule heating, and heat sink/source of the volumetric rate of heat generation are also included in the energy balance equation. In order to formulate the mathematical modeling, a similarity analysis is performed. The numerical solution of nonlinear differential equations is accomplished through the use of a robust computational approach, which is identified as the Spectral Local Linearization Method (SLLM). The computational findings reported in this study show that, in addition to being simple to establish and numerically implement, the proposed method is very reliable in that it converges rapidly to achieve a specified goal and is more effective in resolving very complex models of nonlinear boundary value problems. In order to ensure the convergence of the proposed SLLM method, the Gauss–Seidel approach is used. The SLLM’s reliability and numerical stability can be optimized even more using Gauss–Seidel approach. The computational results for different emerging parameters are computed to show the behavior of velocity profile, skin friction coefficient, temperature profile, and Nusselt number. To evaluate the accuracy and the convergence of the obtained results, a comparison between the proposed approach and the bvp4c (built-in command in Matlab) method is presented. The Matlab software, which is used to generate machine time for executing the SLLM code, is also displayed in a table.

On the domain of generalized mean in Nakano sequence space, and its prequasi ideal with some applications

Some topological and geometric behavior of the space of all sequences whose generalized mean transforms are in Nakano sequence space, the multiplication mappings acting on it, and the eigenvalue distribution of mappings ideal generated by this space and s-numbers are discussed. We construct the existence of a fixed point of Kannan contraction mapping on these spaces. Several numerical experiments are presented to illustrate our results. Moreover, some successful applications to the existence of solutions of nonlinear difference equations are explained. The strength here is that the current results are constructed under flexible setups given by controlling the weight and power of these spaces.

Numerical computation of magnetized bioconvection nanofluid flow with temperature-dependent viscosity and Arrhenius kinetic

The current study focused on the impact of activation energy and magnetic field on immiscible steady nanofluid moving over an elastic stretched surface containing motile gyrotactic microorganisms. We use the Reynolds exponential model to develop the mathematical modeling since the viscosity of the fluid is temperature-dependent. Under the presence of uniformly disseminated nanoparticles, the base fluid is electrically conductive. The Buongiorno model, which incorporates thermophoretic forces and Brownian motion, has been used. Using the spectral local linearization approach (SLLM), the looming nonlinear, coupled differential equations are numerically addressed. Because it is based on a smooth Uni-variant linearization of non-linear functions, the SLLM method is simple to construct and use. The numerical performance of SLLM is even better since it grows a collection of equations that are solved sequentially by operating the findings from one equation into the next. The successive over-relaxation technique was used to accelerate and enhance the convergence of the SLLM system. The reliability of the SLLM will be verified through the use of well-known techniques and comparisons with other data We have illustrated and analyzed in detail the graphical behavior of all emergent variables for temperature, velocity, and concentration distributions, and also the Nusselt number, Sherwood number, skin friction, and density number of motile microorganisms. It is significant to claim that the spectral local linearization approach has been demonstrated to be extremely stable and adaptive for the solution of non-linear differential equations.

ВЗАИМОДЕЙСТВИЕ ПАРАМЕТРИЧЕСКИХ И АВТОКОЛЕБАНИЙ ПРИ ЗАПАЗДЫВАНИИ В ДЕМПФИРОВАНИИ

The interaction of parametric and self-oscillations in the presence of a delay in damping and a source of energy of limited power is considered. For the analysis, a widespread model of a mechanical frictional self-oscillating system was used. The solution of nonlinear differential equations describing the dynamics of the system is carried out using the method of direct linearization of nonlinearity, which has a number of advantages over the known methods of nonlinear mechanics (labor and time costs are reduced by several orders of magnitude; it is possible to obtain final design relations regardless of the specific type and degree of nonlinearity; there are no time-consuming and complex approximations of various orders inherent in known methods). Equations of non-stationary and stationary motions are obtained and conditions for stability of stationary oscillations are de-rived. Calculations were carried out to obtain information on the effect of the delay in damping on the dynamics of oscillations. It is established that, depending on the magnitude of the delay, the amplitude-frequency curves change their appearance, shift up/down.

Generalized Sub-Equation Method for the (1+1)-Dimensional Resonant Nonlinear Schrodinger’s Equation

Interest in studying nonlinear models has been increasing in recent years. Dynamical systems, in which the state of the system changes continuously over time, have nonlinear interactions. The use of unique nonlinear differential equations is inescapable in the evaluation of such systems. In mathematical point of view, for obtaining analytical solutions of nonlinear differential equations, it must be fully integrable. Consequently, the importance of fully integrable nonlinear differential equations for nonlinear science has become indisputable. Among these equations, one of the most studied by physicists and mathematicians is the nonlinear Schrödinger equation. This equation has undergone many modifications to evaluate different phenomena. In this study, the resonant nonlinear Schrödinger equation, which is the most important of these physical equations in terms of explaining many physical phenomena, is solved analytically with the generalized sub-equation method.

The analogy of nanoparticle shapes on the theory of convective heat transfer of Au–Fe3O4 Casson hybrid nanofluid

AbstractIn recent times, Au nanoparticles have been commonly used for delivering the drug especially in the case of hypothermia of tumors, but low absorption of IR light does not solve destruction of tumor cells. However, nanoparticles such as Fe3O4 coated with Au could be used to deliver the drug to a specific spot due to applied external magnetic field. Due to these applications, boundary layer approximation is invoked to simplify the mathematical model. This paper presents the nanoparticle shape analysis and heat transfer features of the Au–Fe3O4–blood hybrid nanofluid flowing past a stretching surface on a magnetohydrodynamic medium. Numerical solutions of nonlinear differential equations are obtained by RKF‐45 method with the help of shooting technique. The behavior of emerging parameters is described graphically for velocity and temperature profiles. It is found that the blade‐shaped Au and Fe3O4 nanoparticles have better thermal conductance than brick, sphere, cylinder, needle, and platelet shapes. It is also observed that the Lorentz force generated due to magnetic field helps in controlling the flow and enhance the thermal conductivity of hybrid nanofluid.

New optical soliton solutions of fractional perturbed nonlinear Schrödinger equation in nanofibers

In this article, the space-time fractional perturbed nonlinear Schrödinger equation (NLSE) in nanofibers is studied using the improved [Formula: see text] expansion method (ITEM) to explore new exact solutions. The perturbed nonlinear Schrodinger equation is a nonlinear model that occurs in nanofibers. The ITEM is an efficient method to obtain the exact solutions for nonlinear differential equations. With the help of the modified Riemann–Liouville derivative, an equivalent ordinary differential equation has been obtained from the nonlinear fractional differential equation. Several new exact solutions to the fractional perturbed NLSE have been devised using the ITEM, which is the latest proficient method for analyzing nonlinear partial differential models. The proposed method may be applied for searching exact travelling wave solutions of other nonlinear fractional partial differential equations that appear in engineering and physics fields. Furthermore, the obtained soliton solutions are depicted in some 3D graphs to observe the behaviour of these solutions.

Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity.

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example.