Solutions Of Nonlinear Differential Equations Research Articles

Analytical solution to boundary layer flow and convective heat transfer for low Prandtl number fluids under the magnetic field effect over a flat plate

AbstractThe 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.

Asymptotic expansions about infinity for solutions of nonlinear differential equations with coherently decaying forcing functions

This paper studies, in fine details, the long-time asymptotic behavior of decaying solutions of a general class of dissipative systems of nonlinear differential equations in complex Euclidean spaces. The forcing functions decay, as time tends to infinity, in a coherent way expressed by combinations of the exponential, power, logarithmic and iterated logarithmic functions. The decay may contain sinusoidal oscillations not only in time but also in the logarithm and iterated logarithm of time. It is proved that the decaying solutions admit corresponding asymptotic expansions, which can be constructed concretely. In the case of the real Euclidean spaces, the real-valued decaying solutions are proved to admit real-valued asymptotic expansions. Our results unite and extend the theory investigated in many previous works.

Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times

In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the continuous dependence and Gâteaux differentiable of the solution relative to the initial value are also presented in a new topology sense. For the autonomous impulsive system, the periodicity of the solution is given. As an application, the properties of the solution for a type of controlled nonlinear ordinary differential equation with impulses at variable times is obtained. These results are a foundation to study optimal control problems of systems governed by differential equations with impulses at variable times.

The general Bernstein function: Application to χ‐fractional differential equations

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear ‐fractional differential equations ( ‐FDEs) with variable coefficients. The fractional derivative used in this work is the ‐Caputo fractional derivative sense ( ‐CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments.

The Stability and Catastrophic Behavior of Finite Periodic Solutions in Non-Linear Differential Equations

This study focuses on the stability and catastrophic behavior of finite periodic solutions in non-linear differential equations. The occurrence of folding surfaces and their relationship with saddle-node bifurcations are explored, being classified as fold and butterfly types of catastrophes. Additionally, the application of catastrophe theory is discussed to analyze the qualitative changes in solutions with the change in system parameters.

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

A Chebyshev polynomial approach to approximate solution of differential equations using differential evolution

Most of the problems associated with physical, biological and engineering systems are complex and are usually modeled with the help of linear as well as non-linear differential equations. In general, the exact solutions of non-linear differential equations may not be obtained by analytical methods and thus, in this paper, a numerical method is proposed to find approximate solutions of differential equations using Chebyshev polynomials and metaheuristic optimization algorithms. In this paper, Ordinary differential equations (ODEs) are modeled as optimization problems to determine their approximate solutions by considering Chebyshev polynomials as the base approximation function, taking initial and/or boundary conditions as constraints. The unknown coefficients of the polynomials are calculated using Differential Evolution (DE) in order to get the approximate solution with minimal error. The proposed method is applied to find the approximate solution to both Initial and Boundary Value Problems. For all test problems considered in this paper, the algorithms have been programmed using MATLAB software to run on computer. The effectiveness of the method is demonstrated by graphical comparison of the computed approximate solutions with that of the exact ones. Further, the performance of the method proposed in this paper is shown by finding out the Root Mean Square Error (RMSE) between the exact and approximate solution, which is found to be better than that of the related existing papers.

Quasi-periodic development of the economy

The article considers the mathematical definition of the space of states of the macroeconomy, where each point represents a certain distribution of the growth of the main factors of production – labor, capital, and natural resources. We study the violation of the sustainable regime of economic development, described by the system of differential equations in partial derivatives. The phenomenon of stable development of the economy, as well as the presence of more than one stable regime of economic development, depending on political and international factors, is studied. The role of periodic development in the economy and the existence of multiple attractors that create various stable regimes are considered in detail. The various stable regimes that arise are solutions of non-linear differential equations. We study the process of the emergence of instability, when the economic number that defines a particular economy continues to increase after reaching its first critical value. A further increase in the economic number leads to the fact that periodic development becomes unsustainable. The trajectory analysis of production factors is performed by the method presented by L.D. Landau and E.M. Lifshitz. The emerging stabilization of the economy can be broken again as the economic number approaches the next critical value; instability reappears with the advent of a new frequency. After passing through several critical values of the economic number, a stable limit cycle can appear on the torus, which is a geometric interpretation of the trajectories of economic development. The appearance of a stable limit cycle means the manifestation of synchronization of oscillations the disappearance of the quasi-periodic and the establishment of a new periodic motion. The birth of a stable limit cycle prevents the emergence of a regime of superpositions of motions of trajectories with many incommensurable frequencies.

Kashuri Fundo Decomposition Method for Solving Michaelis-Menten Nonlinear Biochemical Reaction Model

In most of the real life problems, we encounter with nonlinear differential equations. Problems are made more understandable by modeling them with these equations. In this way, it becomes easier to interpret the problems and reach the results. In 1913, the basic enzymatic reaction model introduced by Michaelis and Menten to describe enzyme processes is an example of nonlinear differential equation. This model is the one of the simplest and best-known approaches of the mechanisms used to model enzyme-catalyzed reactions and is the most studied. For most nonlinear differential equations, it is very difficult to get an analytical solution. For this reason, various studies have been carried out to find approximate solutions to such equations. Among these studies, those in which two different methods are used by blending attract attention. In this study, a blended form of the Kashuri Fundo transform method and the Adomian decomposition method, so-called the Kashuri Fundo decomposition method, is used to find a solution to the Michaelis-Menten nonlinear biochemical reaction model in this way. This method has been applied to the biochemical reaction model and an approximate solution has been obtained for this model without complex calculations. This shows that the hybrid method is an effective, reliable, simpler and time-saving method in reaching the solutions of nonlinear differential equations.

Study of nonlinear quadratic convection on magnetized viscous fluid flow over a non-Darcian circular elastic surface via spectral approach

The present work investigates quadratic convection on magnetized viscous fluid flow on a circular elastic surface susceptible to Rosseland nonlinear thermal radiation and the nonlinear Boussinesq approximation. The flow is traversing through a Darcy-Brinkman-Forchheimer porous material. The energy equation investigates the implications of heat source/sink, Joule heating, and viscous dissipation. Utilizing suitable similarity variables, the hypothesized mathematical modelling is converted into nonlinear systems. Numerical solutions of nonlinear differential equations are computed using the spectral technique, often recognized as the spectral local linearization method (SLLM). The proposed approach SLLM is based on decoupling and linearizing systems of linear equations using a basic linearization process combined with spectral collocation linearization. The computational findings for nondimensional parameters are reported for the velocity and temperature profiles. As a particular case of our investigation, we also perform a comparison with published results.

New Exact Solutions of Landau-Ginzburg-Higgs Equation Using Power Index Method

In the present study, the optimality approach is applied to find the exact solution of the Landau-Ginzburg-Higgs Equation (LGHE) using new transformations. This method is a direct algebraic method for obtaining exact solutions of nonlinear differential equations. We find suitable solutions of the LGHE in terms of elliptic Jacobi functions by applying transformations of basic functions. Exact solutions of the equations are obtained with the help of symbolic software (Maple) which allows the computation of equations with parameter constants. It is exposed that PIM is influential, suitable, and shortest and offers an exact solution of LGHE.

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.