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This paper addresses an important method for finding traveling wave solutions of nonlinear partial differential equations, solutions that correspond to a specific symmetry reduction of the equations. The method is known as the simplest equation method and it is usually applied with two a priori choices: a power series in which solutions are sought and a predefined auxiliary equation. Uninspired choices can block the solving process. We propose a procedure that allows for the establishment of their optimal forms, compatible with the nonlinear equation to be solved. The procedure will be illustrated on the rather large class of reaction–diffusion equations, with examples of two of its subclasses: those containing the Chafee–Infante and Dodd–Bullough–Mikhailov models, respectively. We will see that Riccati is the optimal auxiliary equation for solving the first model, while it cannot directly solve the second. The elliptic Jacobi equation represents the most natural and suitable choice in this second case. [ABSTRACT FROM AUTHOR]

In this work, we extend the theory of differential equations to differential-anti-differential equations and solve several types of such equations. To do this, first, we define a derivative-antiderivative operator with a base function and express derivatives and antiderivatives in the form of this operator. Next, we come to differential-anti-differential equations. To solve some of these equations, first, we investigate the Auxiliary equation(s) and find the roots. Roots are then used in the base function to get the exact solutions of the differential-anti-differential equation. The process can be used to solve several well-known differential equations such as Continuity, Heat, Wave, Laplace, Schrodinger, Euler, Blasius, etc. Our technique shows that every elementary function can solve several types of linear and non-linear ordinary as well as partial differential-anti-differential equations. [ABSTRACT FROM AUTHOR]

Abstract In the present paper, we firstly improve the results on traveling wave solution that were established in (Liu and Weng in J. Differ. Equ. 258:3688–3741, 2015) for a neutral reaction–diffusion equation with quasi-monotone reaction. Secondly, by constructing two auxiliary equations and using Schauder’s Fixed Point Theorem, we further establish the existence and the asymptotic properties of the traveling wave solution for the equation with non-monotone reaction. Two examples are also given as the application of our results.

The aim of this paper is to compute the exact solutions and conservation of a generalized (1 + 1) dimensional system. This can be achieved by employing symbolic manipulation software such as Maple, Mathematica, or MATLAB. In theoretical physics and in many scientific applications, the mentioned system naturally arises. Time, space, and scaling transformation symmetries lead to novel similarity reductions and new exact solutions. The solutions obtained include solitary waves and cnoidal and snoidal waves. The familiarity of closed-form solutions of nonlinear ordinary and partial differential equations enables numerical solvers and supports stability analysis. Although many efforts have been dedicated to solving nonlinear evolution equations, there is no unified method. To the best of our knowledge, this is the first time that Lie point symmetry analysis in conjunction with an ansatz method has been applied on this underlying equation. It should also be noted that the methods applied in this paper give a unique solution set that differs from the newly reported solutions. In addition, we derive the conservation laws of the underlying system. It is also worth mentioning that this is the first time that the conservation laws for the equation under study are derived.

• Model-based Experiment design for nonlinear state-delay dynamic system. • The experiment design problem is formulated as a time-delay optimal control problem. • An efficient numerical scheme is designed to solve the optimal control problem. • The approach largely increases the information contained by the experiment data. This paper studied the model-based design of experiment (MBDoE) for nonlinear dynamical systems with unknown state delay and continuous state inequalities (SDCSI). The design problem is formulated as a time-delay optimal control problem of a dynamic system governed by augmented sensitivity-state equations (ASSE). Consider the co-existence of control delay and multiple state delays in the ASSE, each continuous state inequality constraint is approximated by an integral constraint using a constraint-handling technique based on local smoothing approximation and constraint transcription, which guaranteed the satisfaction of state inequality constraints during the experiment. In addition, an efficient numerical procedure is derived to determine the gradients of the objective function and constraints, which involved integrating an auxiliary impulsive time-delay system backward in time. The procedure is combined with standard gradient-based optimization methods to solve the MBDoE-SDCSI problem. The performance of the proposed scheme is illustrated through the experimental design of a fed-batch biomass fermentation process. [ABSTRACT FROM AUTHOR]

By constructing auxiliary equations and combining the expansion method of ( G ′ / G , 1 / G ), we study a class of nonlinear Schrödinger equation with perturbation terms which describes the propagation of the waves in optical metamaterials. More types of exact solutions, particularly solitary wave solutions, are obtained for the first time. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation theory of a delayed periodic equation without quasimonotonicity. Because of time delay, it does not generate a monotone semiflow. A threshold is obtained, which is the spreading speed as well as the minimal wave speed. To estimate the spreading speed, some auxiliary equations are given, which also implies the nonexistence of traveling wave solutions. The existence of traveling wave solutions is investigated by Schauder's fixed point and regularity of analytic semigroup, and is confirmed by showing the existence of generalized upper and lower solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with asymptotic spreading for a time-periodic predator-prey system where both species synchronously invade a new habitat. Under two different conditions, we show the bounds of spreading speeds of the predator and the prey, which is proved by the theory of asymptotic spreading of scalar equations, comparison principle and generalized eigenvalue. We show either the predator or the prey has a spreading speed that is determined by the linearized equation at the trivial steady state while the spreading speed of the other also depends on the interspecific nonlinearity. From the viewpoint of population dynamics, our results imply that the predator may play a negative effect on the spreading of the prey while the prey may play a positive role on the spreading of the predator. [ABSTRACT FROM AUTHOR]

A numerical simulation of a moving vehicle on the rigid roadway pavement with discontinuities is presented in this paper by using the data from one of the toll road segments in Indonesia. The analysis procedure considers the dynamic interaction between the vehicle and the rigid pavement. The rigid roadway pavement is modeled using the Kirchhoff theory of thin slab on elastic foundations. The aim of the numerical analysis is to obtain the vertical deflection in the middle of the slab, to obtain the time history of the slab deflection; and to identify the parameters of the sub-grade that has a significant effect on the dynamic response of the rigid roadway pavement. The equations of motion are derived in the form of partial differential equations of the fourth order. The mode shapes of the slab deflection are determined from the method of first and the second auxiliary equations of Levy-type in the x- and y-directions, which is also known as the Modified Bolotin Method (MBM). The equations of motion are solved numerically by using Mathematica. The influence of various parameters including the peak vertical velocity due to the moving vehicle, the stiffness of subgrade, the slab thickness and the road profile to the dynamic behavior of the rigid roadway pavement are studied in detail. The results obtained from the plate computing model in this paper are then compared and analyzed with the results computed using the finite element method by Strand 7 to show that the solution obtained using the MBM method is accurate. In addition, the paper aims to show that the thickness of the plate plays a significant role in the distribution of the bending stresses in plate discontinuities and is necessary for engineers to design rigid roadway pavements. [ABSTRACT FROM AUTHOR]

Abstract When deriving nodal methods, the known process of transverse integration introduces additional unknowns to the problem, related to the leakage terms. Additional auxiliary equations must be proposed in order to close the system and to allow its solution. In this work, a study on approximations to those unknowns, fluxes on the contours of the nodes, is carried out. In particular, two-dimensional fixed-source problems are analyzed. Therefore, three alternative proposals of approximations are considered: constant, linear and exponential functions. Such approaches are employed along with of the ADO method, which is used to solve the one-dimensional transversely integrated equations. Numerical results for section averaged scalar fluxes are presented and compared with other nodal schemes, showing some differences among the various approximations, mainly for coarse meshes, although satisfactory comparisons are achieved in the source region. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation theory of a predator‐prey system in a periodic environment. We consider the invasion process that the prey is the aboriginal, whereas the predator is the invader. If the initial habitat size of the predator is finite, then the asymptotic speed of spreading of the predator is given. Furthermore, the existence and nonexistence of traveling wave solutions are obtained, which defined a minimal wave speed of traveling wave solutions. Our results imply that the minimal wave speed of traveling wave solutions is equal to the asymptotic speed of spreading of the predator. [ABSTRACT FROM AUTHOR]

Assigning the boundary conditions for water flow and sediment transport simulation in a looped river network can be challenging. Possible difficulties are presented on the example of a five year simulation period in a river network, and they appear as a result of the extensive model domain (both spatial and temporal sense). In the considered example, greater difficulties arose while defining the boundary conditions for the sediment transport model. These challenges were a result of incomplete measurements. Since the successful calibration and verification of the model requires complete measurements, additional equations were defined that enabled the authors to complement the deficient data.

This paper is concerned with the propagation modes of a second order integrodifference equation without monotonicity. The equation cannot generate monotone semifolws. By constructing auxiliary functions/equations and applying some known results, the minimal wave speed of traveling wave solutions and asymptotic speeds of spread are established. [ABSTRACT FROM PUBLISHER]

In this paper, an analytical method is proposed to construct explicitly exact and approximate solutions for nonlinear evolution equations. By using this method, some new traveling wave solutions of the Kuramoto-Sivashinsky equation and the Benny equation are obtained explicitly. These solutions include solitary wave solutions, singular traveling wave solutions and periodical wave solutions. These results indicate that in some cases our analytical approach is an effective method to obtain traveling solitary wave solutions of various nonlinear evolution equations. It can also be applied to some related nonlinear dynamical systems. [ABSTRACT FROM AUTHOR]

Abstract: The Auxiliary equation method is used to find analytic solutions for the Kawahara and modified Kawahara equations. It is well known that different types of exact solutions of the given auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, new exact solutions of the auxiliary equation are presented. Using these solutions, many new exact travelling wave solutions for the Kawahara type equations are obtained. [Copyright &y& Elsevier]

The aim of this paper is to compute the exact solutions and conservation of a generalized (1 + 1) dimensional system. This can be achieved by employing symbolic manipulation software such as Maple, Mathematica, or MATLAB. In theoretical physics and in many scientific applications, the mentioned system naturally arises. Time, space, and scaling transformation symmetries lead to novel similarity reductions and new exact solutions. The solutions obtained include solitary waves and cnoidal and snoidal waves. The familiarity of closed-form solutions of nonlinear ordinary and partial differential equations enables numerical solvers and supports stability analysis. Although many efforts have been dedicated to solving nonlinear evolution equations, there is no unified method. To the best of our knowledge, this is the first time that Lie point symmetry analysis in conjunction with an ansatz method has been applied on this underlying equation. It should also be noted that the methods applied in this paper give a unique solution set that differs from the newly reported solutions. In addition, we derive the conservation laws of the underlying system. It is also worth mentioning that this is the first time that the conservation laws for the equation under study are derived. [ABSTRACT FROM AUTHOR]

In the present paper, we firstly improve the results on traveling wave solution that were established in (Liu and Weng in J. Differ. Equ. 258:3688–3741, 2015) for a neutral reaction–diffusion equation with quasi-monotone reaction. Secondly, by constructing two auxiliary equations and using Schauder's Fixed Point Theorem, we further establish the existence and the asymptotic properties of the traveling wave solution for the equation with non-monotone reaction. Two examples are also given as the application of our results. [ABSTRACT FROM AUTHOR]