Order Nonlinear Equations Research Articles

Numerical study of wave disturbances of the high order nonlinear Boussinesq equations arising in engineering

Numerical study of wave disturbances of the high order nonlinear Boussinesq equations arising in engineering

Dynamics of chains of a large number of fully coupled oscillators, as well as oscillators with one-way and two-way delay couplings

This articles considers chains consisting of identical nonlinear equations of second order with couplings in linear elements. It is assumed that there is a large delay in the coupling elements. Moreover, we assume that the chain possesses a large number of elements. Therefore, instead of the original system with many elements, we can study a second order nonlinear integro-differential equation with periodic boundary conditions. The main study is focused on the local dynamics of chains with one-sided, two-sided couplings, as well as fully coupled chains. Also, we study the stability of equilibrium states and identify the critical cases. The condition of a sufficiently large delay helps to determine the parameters for the implementation of critical cases explicitly. Our methodology is based on the infinite-dimensional normalisation method proposed by the author, namely the method of quasinormal forms. We construct quasinormal forms for the chains under consideration that determine the asymptotics of the leading terms of the asymptotic expansions of the solutions to the original system.

Higher order contribution to ion-acoustic Korteweg–de Vries (KdV) soliton in unmagnetized quantum plasmas with arbitrary degeneracy of electrons

The propagation of nonlinear ion-acoustic Korteweg–de Vries (KdV) solitons and dressed solitons (with higher order contribution term) in unmagnetized quantum plasmas with arbitrary degeneracy of electrons is investigated using quantum hydrodynamic model. The reductive perturbation method is used to derive the first order nonlinear ion-acoustic KdV equation and also higher order contribution of inhomogeneous differential equation is obtained for the dressed ion-acoustic wave soliton in arbitrary degenerate electron plasmas. Moreover, the higher order inhomogeneous differential equation is solved (nonsecular solution is obtained) using the renormalization method. The conditions for the validity of the higher order correction are also discussed in detail for ion-acoustic solitons in unmagnetized arbitrary degenerate electrons plasma case. The effects of chosen values of electron temperature, equilibrium fugacity and corresponding electron density of a physical quantum plasma system and quantum diffraction parameter H (for two conditions i.e., H < 2 or H > 2) on the amplitude and width of the KdV and dressed ion-acoustic wave solitons for both electrostatic potential hump (compressive) or dip (rarefactive) structures are investigated. The numerical plots are also presented for illustration with numerical values of a physical partially degenerate electrons plasma system exist in the astrophysical or laboratory conditions.

Inverse problem of recovering a time-dependent nonlinearity appearing in third-order nonlinear acoustic equations *

This paper is devoted to some inverse problems of recovering the nonlinearity for the Jordan–Moore–Gibson–Thompson equation, which is a third order nonlinear acoustic equation. This equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained with the small initial and boundary data. By the second order linearization to the nonlinear equation, and construction of complex geometric optics solutions for the linearized equation, the uniqueness of recovering the nonlinearity is derived.

Modified optimal auxiliary functions method for approximate-analytical solutions in fractional order nonlinear Foam Drainage equations

The study aims to obtain new approximate analytical solutions for the fractional order nonlinear Foam Drainage equation through a novel modification of the well-known Optimal Auxiliary Functions Method (OAFM). This method systematically constructs auxiliary functions, and by optimizing their parameters, it provides approximate solutions closely mirroring the original behavior of the problem. Numerical simulations, presented in tabular form, demonstrate the effectiveness of the proposed approach. Graphical representations of the solutions are utilized to further validate the applicability and accuracy of the suggested technique. Additionally, the results affirm that the modified OAFM stands out as one of the most effective tools for addressing nonlinear problems in the field of science and technology, thereby contributing valuable addition to the field.

Traveling Wave Solutions of Nonlinear Wave Equation Based on Functional Analysis and Multiple Variables as Applied to IoT

Abstract In applied mathematics, physics, and mechanics, many mathematical models are established in the form of nonlinear wave equations. Therefore, a method for studying traveling wave solutions of nonlinear wave equations based on functional analysis is proposed. The boundary value stability of the nonlinear second order nonlinear wave equation is modified by using functional analysis of multiple variables. In this paper, a new integral term is introduced into the organization mapping functional, the concept and construction of the nonlinear wave equation are analyzed, and its continuity is proved. According to the growth order of the traveling wave solution of nonlinear wave equation, the method of finding the traveling wave solution and its stability are studied.

Higher order mKdV breathers: nonlinear stability

We are interested in stability results for breather solutions of the 5th, 7th and 9th order mKdV equations. We show that these higher order mKdV breathers are stable in $H^2(\R)$, in the same way as \emph{classical} mKdV breathers. We also show that breather solutions of the 5th, 7th and 9th order mKdV equations satisfy the same stationary fourth order nonlinear elliptic equation as the mKdV breather, independently of the order, 5th, 7th or 9th, considered.

New Oscillation Criteria for Higher Order Nonlinear Dynamic Equations

AbstractThis paper deals with some new criteria for the oscillation of higher-order nonlinear dynamic equations with mixed deviating arguments. The purpose of the present paper is the linearization of the equation under consideration. Specifically, we will infer the oscillation of the studied equation from its linear form and establish new oscillation criteria by comparing it with first-order equations whose oscillatory behaviour is known. The obtained results are new, improve, and correlate many of the known oscillation criteria appearing in the literature. The results are illustrated by two examples.

Variable-order porous media equations: Application on modeling the S&P500 and Bitcoin price return.

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.

Soliton equations: admitted solutions and invariances via B\"acklund transformations

A couple of applications of B\"acklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on proving a new invariance admitted by a third order nonlinear evolution equation and, on the other one, on the construction of solutions. Indeed, via B\"acklund transformations, a {\it B\"acklund chart}, connecting Abelian as well as non Abelian equations can be constructed. The importance of such a net of links is twofold since it indicates invariances as well as allows to construct solutions admitted by the nonlinear evolution equations it relates. The present study refers to third order nonlinear evolution equations of KdV type. On the basis of the Abelian wide B\"acklund chart which connects various different third order nonlinear evolution equations an invariance admitted by the {\it Korteweg-deVries interacting soliton} (int.sol.KdV) equation is obtained and a related new explicit solution is constructed. Then, the corresponding non-Abelian {\it B\"acklund chart}, shows how to construct matrix solutions of the mKdV equations: some recently obtained solutions are reconsidered.

IMPLEMENTATION AND ASSESSMENT OF THE SIMPLE EQUATION TECHNIQUE FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

In this paper, the simple equation method is especially used to solve two Nonlinear Partial Differential Partial Equations NLPDEs, the Kodomstev-Petviashvili (KP) equation and the (2+1)-dimensional breaking soliton equation. The modified Benjamin-Bona-Mahony equation and the Klein-Gordon equation in (1+2) dimensions are two illustrations of second order nonlinear equations that can benefit from using this approach. The Bernoulli equation acts as the trial condition and aids in the mathematical a description of the nonlinear wave equation in the simple equation.

THE EXISTENCE OF GENERALIZED SOLUTION TO ONEDIMENSIONAL MIXED PROBLEM FOR ONE CLASS OF THIRD ORDER NONILINEAR PSEUDOPARABOLIC EQUATIONS

Abstract. This work is dedicated to the study of existence in small of generalized solution of onedimensional mixed problem for one class of third order nonlinear pseudoparabolic equations. Conception of generalized solution for mixed problem under consideration is introduced. After applying Fourier method, the solution of original problem is reduced to the solution of some countable system of nonlinear inteqro-differential equations in unknown Fourier coefficients of the sought solution. Besides, existence theorem in small of generalized solution of the mixed problem is proved by contracted mappings principle.

Lower bounds on the radius of spatial analyticity for the higher order nonlinear dispersive equation on the real line

Lower bounds on the radius of spatial analyticity for the higher order nonlinear dispersive equation on the real line

A Qualitative Study on Fractional Logistic Integro-differential Equations in an Arbitrary Time Scale

This manuscript deals with the investigation related to uniqueness and existence of solution of fractional order nonlinear pantograph integro-differential equation in arbitrary time scale. The fractional derivatives are defined in Riemann-Liouville sense, the primary tools are taken as Banach contraction principle and Schauder’s fixed point theory to establish the theoretical outcomes. Finally, we give examples to show the efficiency of our results.

Blow-up upper and lower bounds for solutions of a class of higher order nonlinear pseudo-parabolic equations

&lt;abstract&gt;&lt;p&gt;In this paper, we study the initial boundary value problem for a class of higher-order nonlinear pseudo-parabolic equations with a memory term. First, the blow-up results of the solution when the initial energy is negative or positive are obtained by using concavity analysis, and an upper bound on the blow-up time $ T^* $ is given. Second, a lower bound on the blow-up time $ T^* $ is obtained by applying differential inequalities when the solutions blow up.&lt;/p&gt;&lt;/abstract&gt;

Asymptotic behaviour of the non-oscillating solutions of first order nonlinear impulsive functional-differential equations of neutral type

This paper deals with nonlinear first-order impulsive functional-differential equations of neutral type with variable coefficients. The asymptotic behaviour of the non-oscillating solutions of such equations is investigated as a function of the values of the variable coefficient in the neutral term to serve as a basis for further studies of the oscillation of the solutions of such equations.

Explore dynamical soliton propagation to the fractional order nonlinear evolution equation in optical fiber systems

Explore dynamical soliton propagation to the fractional order nonlinear evolution equation in optical fiber systems

Shock waves with irrotational Rankine-Hugoniot conditions

Shock wave stability for isentropic irrotational flow is studied for Euler system but with shock front conditions corresponding to the second order nonlinear wave equation. It is shown that the usual Lax’ shock condition still guarantees the uniform linear stability and therefore the existence of the shock waves solution.

A hybrid fourth order time stepping method for space distributed order nonlinear reaction-diffusion equations

A new fourth order highly efficient Exponential Time Differencing Runge-Kutta type hybrid time stepping method is developed by hybriding two fourth order methods. The new hybrid method takes advantages of a positivity preserving L-stable method to damp unnatural oscillations due to low regularity of the initial data and of a computationally efficient A-stable method to achieve optimal order convergence. Computational efficiency and stability of the new method is further enhanced by applying a splitting technique which makes it possible to implement the method on parallel processors. A hybriding criteria is presented and an algorithm based on the hybrid method is developed. The method is implemented to solve two two-dimensional problems having Riesz space distributed order diffusion and nonlinear reaction terms, an Allen-Cahn equation with a cubic nonlinearity and an Enzyme Kinetics equation with a rational nonlinearity. Fourth-order temporal convergence is obtained through numerical experiments. A third test problem with exact solution available is also considered and both spatial as well as temporal orders of convergence are computed. High computational efficiency of the method is shown by recording the CPU time. Numerical solutions using different order strength distribution functions are also presented.

Applying Adomian Decomposition Method for solving the Covid-19 epidemic with Vaccine

Abstract— A pandemic Covid-19 is an epidemic that spreads over a big region, Crosse international borders, and often affects a lot of people. Only a few pandemics result in severe illness in a subset of people or in an entire community. The virus has mainly affects the elderly population. The virus, that causes Covid-19, has mainly been transmitted through droplet generate once an infected persons exhales, sneezes and coughs. These symptoms are too heavy; to hang in air, and quickly, fall on surface or floor. The COVID-19 pandemic model including the Vaccination Campaign is of natural phenomenon which can be represented as a system of differential equations for the first order; the mathematical models include a system of several second order nonlinear equations. We applied the Adomian decomposition methods to the mathematical models of Covid-19. The main advantage of this method is that it can be directly applied to all kinds of linear and nonlinear differential equations, homogeneous or nonhomogeneous, with constant or variable coefficients. The derivatives of all compartments of the coronavirus model are continuous at t ≥ 0. The solutions of the model are non-negativity. It indicates that the, infection, will be gradually the epidemic and disappear will, stop. If, R_0&gt;1, the average of each affected individually. More than one person has infected, and the incidence of infection is in wrinkles. That means the epidemic, will not be end, while maintain the existence of the disease, the R_0=1 means that each infected patient results in an average infections.