THE EXACT MEROMORPHIC SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS

We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions.

THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS

In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.

On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations

The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense.

Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

Thermomechanical Dynamics (TMD) and Bifurcation-Integration Solutions in Nonlinear Differential Equations with Time-Dependent Coefficients

The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.

INITIAL VALUE PROBLEM FOR A CLASS OF NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS

The existence of global weak solutions to the periodic boundary problem or the initial value problem for the nonlinear Pseudo-hyperbolic equation u_(tt)-[a_1+a_2(u_x)^(2m)]u_(xx)-a_3u_(xxt)=f(x,t,u,u_x) is proved by the method of the vanishing of the additional diffusion terms, Leray-Schauder's fixedpoint argument and Sobolev's estimates,where m≥1 is a natural number and a_i>0(i=1,2,3)are constants.

BLOW－UP AND DIE－OUT OF SOLUTIONS OF NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS OF GENERALIZED NERVE CONDUCTION TYPE

This paper considers the initial boundary value problems with three types of the boundary conditions for nonlinear pseudo-hyperbolic equations of generalized nerve conduction type, using foe eigenfunction method, the conditions for which the solutions blow-up and die-out in the finile time are got.

AD GALERKIN ANALYSIS FOR NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS

AD (Alternating direction) Galerkin schemes for d-dimensional nonlinear pseudo-hyperbolic equations are studied. By using patch approximation technique, AD procedure is realized, and calculation work is simplified. By using Galerkin approach, highly computational accuracy is kept. By using various priori estimate techniques for differential equations, difficulty coming from non-linearity is treated, and optimal H1 and L2 convergence properties are demonstrated. Moreover, although all the existed AD Galerkin schemes using patch approximation are limited to have only one order accuracy in time increment, yet the schemes formulated in this paper have second order accuracy in it. This implies an essential advancement in AD Galerkin analysis.

Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential

In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L2-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.

Coupled-generalized nonlinear Schr¨odinger equations solved by adaptive step-size methods in interaction picture

We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schr ¨odinger equation(GNLSE): one is the conservation quantity error adaptive step-control method(RK4IP-CQE), and the other is the local error adaptive step-control method(RK4IP-LEM). The methods are developed in the vector form of fourthorder Runge–Kutta iterative scheme in the interaction picture by converting a vector equation in frequency domain. By simulating the supercontinuum generated from the high birefringence photonic crystal fiber, the calculation accuracies and the efficiencies of the two adaptive step-size methods are discussed. The simulation results show that the two methods have the same global average error, while RK4IP-LEM spends more time than RK4IP-CQE. The decrease of huge calculation time is due to the differences in the convergences of the relative photon number error and the approximated local error between these two adaptive step-size algorithms.

Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method

How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations(NAEs).This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear terms.We decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear system.Through the maximal orthogonal projection concept,to minimize a merit function within a selected interval of splitting parameters,the optimal parameters can be quickly determined.In each step,a linear system is solved by the Gaussian elimination method,and the whole iteration procedure is convergent very fast.Several numerical tests show the high performance of the optimal split-linearization iterative method(OSLIM).

Efficient Numerical Scheme for Solving Large System of Nonlinear Equations

A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper.Convergence analysis demonstrates that the local order of convergence of the numerical method is five.The computer algebra system CAS-Maple,Mathematica,or MATLAB was the primary tool for dealing with difficult problems since it allows for the handling and manipulation of complex mathematical equations and other mathematical objects.Several numerical examples are provided to demonstrate the properties of the proposed rapidly convergent algorithms.A dynamic evaluation of the presented methods is also presented utilizing basins of attraction to analyze their convergence behavior.Aside from visualizing iterative processes,this methodology provides useful information on iterations,such as the number of diverging-converging points and the average number of iterations as a function of initial points.Solving numerous highly nonlinear boundary value problems and large nonlinear systems of equations of higher dimensions demonstrate the performance,efficiency,precision,and applicability of a newly presented technique.

Nondegenerate solitons of the(2+1)-dimensional coupled nonlinear Schrodinger equations with variable coefficients in nonlinear optical fibers

We derive the multi-hump nondegenerate solitons for the(2+1)-dimensional coupled nonlinear Schrodinger equations with propagation distance dependent diffraction,nonlinearity and gain(loss)using the developing Hirota bilinear method,and analyze the dynamical behaviors of these nondegenerate solitons.The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers,varying diffraction and nonlinearity parameters.In addition,when all the variable coefficients are chosen to be constant,the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons.Finally,it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.

AI-Enhanced Performance Evaluation of Python, MATLAB, and Scilab for Solving Nonlinear Systems of Equations: A Comparative Study Using the Broyden Method

This research extensively evaluates three leading mathematical software packages: Python, MATLAB, and Scilab, in the context of solving nonlinear systems of equations with five unknown variables. The study’s core objectives include comparing software performance using standardized benchmarks, employing key performance metrics for quantitative assessment, and examining the influence of varying hardware specifications on software efficiency across HP ProBook, HP EliteBook, Dell Inspiron, and Dell Latitude laptops. Results from this investigation reveal insights into the capabilities of these software tools in diverse computing environments. On the HP ProBook, Python consistently outperforms MATLAB in terms of computational time. Python also exhibits a lower robustness index for problems 3 and 5 but matches or surpasses MATLAB for problem 1, for some initial guess values. In contrast, on the HP EliteBook, MATLAB consistently exhibits shorter computational times than Python across all benchmark problems. However, Python maintains a lower robustness index for most problems, except for problem 3, where MATLAB performs better. A notable challenge is Python’s failure to converge for problem 4 with certain initial guess values, while MATLAB succeeds in producing results. Analysis on the Dell Inspiron reveals a split in strengths. Python demonstrates superior computational efficiency for some problems, while MATLAB excels in handling others. This pattern extends to the robustness index, with Python showing lower values for some problems, and MATLAB achieving the lowest indices for other problems. In conclusion, this research offers valuable insights into the comparative performance of Python, MATLAB, and Scilab in solving nonlinear systems of equations. It underscores the importance of considering both software and hardware specifications in real-world applications. The choice between Python and MATLAB can yield distinct advantages depending on the specific problem and computational environment, providing guidance for researchers and practitioners in selecting tools for their unique challenges.

Actuator Fault Diagnosis for a Class of One-Dimensional Nonlinear Heat Equations via Backstepping Method

In this paper, an actuator fault diagnosis scheme based on the backstepping method is proposed for a class of nonlinear heat equations. The fault diagnosis scheme includes fault detection, fault estimation and time to failure (TTF) prediction. Firstly, we achieve fault detection by comparing the detection residual with a predetermined threshold, where the detection residual is defined as the difference between the observer output and the system measurement output. Then, we estimate the fault function through the fault parameter update law and calculate the TTF using only limited measurements. Finally, the numerical simulation is performed on a nonlinear heat equation to verify the effectiveness of the proposed fault diagnosis scheme.

Ostrowski’s Method for Solving Nonlinear Equations and Systems

The dynamic characteristics and the efficiency of the Ostrowski’s method allow it to be crowned as an excellent tool for solving nonlinear problems.This article shows different versions of the classic method that allow it to be applied to a wide range of engineering problems.Among them stands out the derivative-free definition applying divided differences,the introduction of memory and its extension to the resolution of nonlinear systems of equations.All of these versions are compared in a numerical simulations section where the results obtained are compared with other classic methods.

Nonconforming <i>H</i>¹-Galerkin Mixed Finite Element Method for Pseudo-Hyperbolic Equations

Based on H1-Galerkin mixed finite element method with nonconforming quasi-Wilson element, a numerical approximate scheme is established for pseudo-hyperbolic equations under arbitrary quadrilateral meshes. The corresponding optimal order error estimate is derived by the interpolation technique instead of the generalized elliptic projection which is necessary for classical error estimates of finite element analysis.

OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

In this paper,the oscillation criteria for the solutions of the nonlinear differential equations of neutral type of the forms:[x(t)+p(t)x(σ(t))]″+q(t)f(x(τ(t)))g(x′(t))=0and[x(t)+p(t)x(σ(t))]″+q(t)f(x(t),x(τ(t)))g(x′(t))=0are obtained.

A CLASS OF NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

A class of nonlinear singularly perturbed problems for reaction diffusion equations are considered. Under suitable conditions, by using the theory of differential inequalities, the asymptotic behavior of solutions for the initial boundary value problems are studied, reduced problems of which possess two intersecting solutions.

OSCILLATORY BEHAVIOUR OF A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS OF THIRD ORDER

This paper investigates the oscillatory and nonoscillatory behaviour of solu- tions of a class of third order nonlinear differential equations. Results extend and improve some known results in the literature.