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This paper presents a modified metaheuristic algorithm named the modified Firefly algorithm. Any metaheuristic algorithm will have exploration and exploitation steps, and the goal of modification is to maintain a balance between them. The improvement relies on movement equations, alterations to the algorithm's structure by introducing a single loop, and a selection of movement equations at random. Two movement equations are included in the improved method and are randomly selected. This guarantees both regionally and globally focused solution-finding. This prevents the algorithm from getting stuck at a local minimum. Comparing the modified version to the original Firefly method, just one for loop is used, reducing the algorithm's complexity. The algorithm's performance is evaluated with 35 traditional benchmark test functions and 10 CEC2019 test functions. According to the findings, the suggested method performed optimally in 24 traditional benchmark test functions and best in the six remaining benchmark test functions. The improved algorithm produced the best outcomes in seven of the 10 CEC2019 test functions. In contrast, the Firefly algorithm produced optimal results in 18 classical benchmark test functions and the best results in 6 CEC2019 test functions. The proposed algorithm is compared with other variants of the Firefly algorithm for common test functions in the literature. The results show that the proposed algorithm outperforms other variants in most test functions. [ABSTRACT FROM AUTHOR]

In this paper, an efficient Twice Finite Point-set Method (TFPM) coupled with difference scheme is proposed to solve the time-fractional Cahn–Hilliard (TF-CH) equation, and then it is extended to predict the phase separation process under nonlocal memory dominated by two-component TF-CH equations for the first time. The proposed meshless schemes are motivated by the following: (a) a high-order accuracy difference scheme is employed to approximate the time Caputo fractional derivative; (b) the fourth-order spatial derivative is divided into two second-order derivatives, and it is discretized by the FPM scheme continuously twice based on Taylor expansion and weighted least squares; (c) the Neumann boundary can be accurately imposed on the FPM scheme. In the numerical experiments, the error and numerical convergence of the proposed meshless method are first tested, which has near second-order convergent rate. Subsequently, the evolution of phase separation under memory dominated by single CH equation versus time is numerically investigated by the proposed method, and compared with the results in other literatures. The influence of fractional parameter on the separation phenomena is also discussed. Finally, the proposed method is used to predict the phase separation process under different parameters dominated by coupled TF-CH equations. All the numerical results show that the proposed coupled method is accurate in solving the TF-CH and efficient in predicting the phase separation evolution. [ABSTRACT FROM AUTHOR]

There are very few integration methods satisfying the explicit displacement formulation, unconditional stability, and higher-order accuracy, simultaneously. This paper proposes a new predictor–corrector explicit (NPCE) integration method with unconditional stability and higher-order accuracy. The NPCE is developed by matching its temporal discrete equation (TDE) with that of a recently developed fourth-order new dual-explicit (NDE) integration method. The accuracy and stability of the NPCE scheme are analyzed. It is found that the NPCE method possesses the same stability property as the NDE method. In addition, the NPCE method can achieve the same accuracy level as the NDE method for the undamped system, and it is more accurate than the NDE method for the damped system. The performance of the NPCE method is illustrated by four numerical examples. Some representative integration methods are adopted for comparison. The results indicate that the NPCE method has excellent accuracy with low relative absolute and squared errors. [ABSTRACT FROM AUTHOR]

The important character of the applied moment with a fixed line of direction as implied in deriving the governing differential equations of LTB for elastic beams, however, is not represented in stating the boundary conditions based on conventional energy approaches. This paper sets out to consider the boundary conditions derived by conventional potentials in order to identify the ignored second-order moments that need to be added for maintaining the initial direction of the applied moment during LTB. By taking into account the non-zero work done by these ignored moments for modifying original potentials, the nonconservative nature of the elastic LTB of beam under external moments is clarified in two aspects — by discovering the path-dependent property of the non-zero work and by demonstrating that only trivial solution of LTB mode is expected from the analytical solution of the governing differential equations. Using the concept of dynamic instability, an analysis procedure combining the numerical solution of an eigen-value problem and the bisection method is proposed to identify the first appearance of imaginary or conjugate complex frequencies and to determine the value of the critical moment. Demonstrative examples are considered for illuminating the elastic LTB behavior using conventional energy methods and revised virtual work equation. It has been found that the critical moments given by conventional potentials are generally not consistent with each other unless the work done by the ignored second-order moments vanishes. However, consistent critical moment, under which the free vibrating amplitude of elastic beam is increased unboundedly with time, can be obtained by the kinetic approach based on any revised virtual work equation, indicating the necessity of considering the nonconservative nature of non-follower applied moment. [ABSTRACT FROM AUTHOR]

In this paper, using smooth invertible variable transformations and smooth invertible parameter changes, we construct a jerk normal form for the cusp bifurcation of a jerk system with two parameters which displays a nondegenerate fold bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, the second-order scalar auxiliary variable approach combined with finite difference method is employed for the Allen–Cahn equation that represents a phenomenological model for antiphase domain coarsening in a binary mixture. The second-order backward differentiation formula is used in time. The error estimation of the semi-discrete scheme is derived in the sense of L 2 -norm. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, two methods are applied to solve the regularized Schamel equation. Firstly, by using the singular planar dynamical system method, we discover its peakon structure which was not reported before. We also derive some new explicit traveling wave solutions of this equation, including various solitary wave solutions, periodic wave solutions and compactons. Especially, for the first time, we find the W-shape solitary wave solutions and W-shape periodic wave solutions of the equation. Then, in order to discover more wave phenomena, a deep learning framework is introduced to solve complicated initial boundary value problems of this equation. In comparison with the exact solutions given previously, our deep learning framework is reliable and highly accurate in capturing the dynamical behavior of the traveling wave solutions of the equation. As an application, we use it to solve a specific initial boundary value problem and obtain a new data-driven solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of inequalities associated with circular cones. By constructing a new smoothing function, the problem is reformulated as a system of parameterized smooth equations. In addition, we suggest a Newton-type algorithm for solving the smooth equations so that a solution of the problem concerned is found. In particular, the algorithm is proved to be globally and locally quadratically convergent under suitable conditions. The preliminary numerical results demonstrate that the algorithm is effective. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a coupled system of nonlocal fractional q-integro-differential equations. Under certain assumptions, we prove the existence and uniqueness of solutions for a coupled system of fractional q-integro-differential equations. We also study continuous dependence. We solve this system numerically using the finite–Simpson's and cubic spline–Simpson's methods. Finally, three examples are provided to demonstrate the efficacy of the methods employed. [ABSTRACT FROM AUTHOR]

This paper discusses a nonlinear optimization problem with the system of max-Archimedean bipolar fuzzy relation equations as constraints. Some results related to the structure of the solution set of max-Archimedean bipolar fuzzy relation equations are proved. Using these results, a genetic algorithm is proposed to solve the problem for obtaining optimal or converging solutions. The effectiveness of the algorithm is also compared with other methods found in the literature. The previous methods require conversion of the problem into 0-1 mixed integer optimization problem solvable by some nonlinear optimization solvers and thereby, the computational work may increase with the size of the problem. Some test problems are developed to evaluate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a perturbed quintic BBM equation with weak backward diffusion and dissipation effects is investigated. By applying geometric singular perturbation theory and analyzing the perturbations of a Hamiltonian system with a hyper-elliptic Hamiltonian of degree six, we prove the existence of isolated periodic wave solutions with certain wave speed in an open interval. It is also shown that isolated periodic wave solutions persist for any energy parameter h in an open interval under small perturbation. Furthermore, we prove that the wave speed c (h) of periodic wave is strictly monotonically increasing with respect to h by analyzing Abelian integral having three generating elements. Moreover, the upper and lower bounds of the limiting wave speed are obtained. Our analysis is mainly based on Melnikov theory, Chebyshev criteria, and symbolic computation, which may be useful for other problems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a reaction–diffusion mathematical model augmented with self/cross-diffusion in 2D domain which describes the oncolytic virotherapy treatment of a tumor with its growth following the logistic law. The tumor cells are divided into uninfected and infected cells and the virus transmission is supposed to be in a direct mode (from cell to cell). In the absence of cross-diffusion, we establish well posedness of the problem, non-negativity and boundedness of solutions, nonexistence of positive solutions, local and global stability of the nontrivial steady-state and the nonoccurrence of Turing instability. In the presence of cross-diffusion, we prove the occurrence of Turing instability by using the cross-diffusion coefficient of infected cells as a parameter. To have an idea about different patterns, we derive the corresponding amplitude equation by using the nonlinear analysis theory. In the end, we perform some numerical simulations to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

Algebraic operations on generalized hesitant fuzzy numbers are key tools to address the problems with decision uncertainty. In this paper, by studying the arithmetic operations on generalized trapezoidal hesitant fuzzy numbers, modified arithmetic operations are introduced for this class of numbers so that, using these arithmetic operations, the multiplication and division of two generalized trapezoidal hesitant fuzzy numbers are always generalized trapezoidal hesitant fuzzy numbers. Furthermore, a generalized trapezoidal hesitant fuzzy number raised to the power of a real number is a generalized trapezoidal hesitant fuzzy number, and in the defined division, the case where the denominator becomes zero is not considered. Numerical examples are used to show the shortcomings of the previous arithmetic operations as well as the efficiencies of the arithmetic operations proposed in this research for generalized trapezoidal hesitant fuzzy numbers. Finally, the application of the proposed new arithmetic operations to generalized trapezoidal hesitant fuzzy numbers in solving the generalized trapezoidal hesitant fully fuzzy equation is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the number of nontrivial limit cycles in a kind of piecewise smooth generalized Abel equation d x d t = x p p − 1 + ∑ i = 1 2 i (A i − (t) x p + B i − (t) x 2 p − 1) , − 1 ≤ t ≤ 0 , x ∈ R , − x p p − 1 + ∑ i = 1 2 i (A i + (t) x p + B i + (t) x 2 p − 1) , 0 ≤ t ≤ 1 , x ∈ R , where p ∈ Z + ∖ { 1 } , A 1 ± = a 1 0 ± + a 1 1 ± t , B 1 ± = b 1 0 ± + b 1 1 ± t , A 2 ± = a 2 0 ± + a 2 1 ± t , B 2 ± = b 2 0 ± + b 2 1 ± t and | | ≪ 1. Under the condition b 1 1 + ≠ 0 , employing Melnikov functions of any order and using properties of Chebyshev systems, we prove that if p is odd, then the maximum number of nontrivial limit cycles bifurcating from the periodic annulus of the unperturbed system is 6 and it is attainable, and if p is even, then the maximum number is 3, and it can be attained too. [ABSTRACT FROM AUTHOR]

In this paper, the methodologies of dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] are applied to find the solutions in the form of u (x , t) = ϕ (x − c t) e i (κ x − ω t) for the Hirota-type peakon equation posed by [Anco & Mobasheramini, 2017; Anco et al., 2021]. For the function ϕ (ξ) therein, under the parameter conditions of κ = 5 3 , c = 8 0 7 ω 1 0 0 or c ≠ 8 0 7 ω 1 0 0 , the existence of some possible bounded solutions (solitary wave solutions, periodic wave solutions and periodic peakons, as well as compactons) is proved, with exact explicit parametric representations obtained, except for the peakon solution. [ABSTRACT FROM AUTHOR]

In switching algebra, there exist standard forms of Boolean functions such as the disjunctive or conjunctive form. This paper discusses the theory of the extended standard forms of Boolean functions. In addition to the four existing standard forms, two further forms are introduced and thus expanded to six basic forms. On the one hand, the existence of the extended forms is presented and on the other hand new formulas and equations are illustrated. Equations relating to the resolution and/or solution of conjunction/disjunctions are detailed and proven to be valid. In addition, equations for conversion between forms are exemplified. Finally, certain formal relations between the basic forms that are valid under certain conditions are featured. [ABSTRACT FROM AUTHOR]

State-based peridynamics (SPD) is effective for simulating fracture and damage in different materials. However, the solutions may suffer from numerical instabilities, particularly for strong nonlinearity cases or large displacement cases, leading to inaccurate predictions or oscillation in responses. This paper proposed a novel practical method to solve the instability problem of SPD based on stress equilibrium equation, referred to herein as the stress correction method (SCM). A correction force is applied on each SPD point surrounding the loading points. The correction force is defined as the difference between an internal force obtained by the stress equilibrium equation and that obtained by the force states of the SPD. Four examples are presented herein to verify the accuracy and stability of the proposed method in various conditions, e.g. static and dynamic analyses of elastic and plastic models subjected to force and displacement boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the bifurcations and exact traveling wave solutions of the Lakshmanan–Porsezian–Daniel model with parabolic law nonlinearity. Employing the bifurcation theory of planar dynamical system, we obtain the phase portraits of the corresponding traveling wave system. The existence of the two singular straight lines leads to the dynamical behavior of solutions with two scales. Corresponding to some special level curves, we give the explicit exact solutions under different parameter conditions, containing peakon solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic wave solutions and periodic peakon solutions. Moreover, we collect all these traveling solutions into a theorem. Finally, we discuss a small perturbation of the LPD equation and persistence of the traveling waves. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to form a complete group classification for the Gilson–Pickering equation. Exact traveling wave solutions and bifurcations of the Gilson–Pickering equation are studied by the method of dynamical systems. The study on the wave solutions of the model derives a planar Hamiltonian system. Based on phase portraits, many exact explicit parametric representations of wave solutions are obtained under different parametric conditions. [ABSTRACT FROM AUTHOR]

This paper studies traveling wave solutions of a generalized Sasa–Satsuma equation introduced in [Adem et al., 2020]. For the cases of n = 0 , 1 , 2 , under given parameter conditions, the bifurcations of traveling wave solutions in the parameter space are investigated for the corresponding traveling systems. All possible explicit exact parametric representations of various solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we develop an efficient wavelet method based on the Fibonacci polynomials for solving the Pennes bioheat transfer equation. The formulation of the proposed technique is started with the construction of Fibonacci wavelets by using Fibonacci polynomials and then applying spectral collocation technique to transform the given problem into a system of an algebraic equation, that can be solved using the Newton method. Some results related to error estimate and convergence analysis of the proposed scheme are also investigated. The applicability and accuracy of the present technique are elucidated by a comparison with the exact solution and those of the other methods found in the recent literature. The obtained results show that the proposed technique is an effective tool for solving Pennes bioheat transfer equations and can also be used for solving similar types of various partial differential equations numerically. [ABSTRACT FROM AUTHOR]

A modified generalized KdV equation is considered in this paper. Under the given parameter conditions, the corresponding traveling wave system is a singular planar dynamical system with three singular straight lines. The bifurcations and traveling wave solutions of the system are investigated in the parameter space from the perspective of dynamical systems. The existence of solitary wave solutions, periodic peakon solutions, pseudo-peakon solutions, kink and anti-kink wave solutions and compactons is proved. Furthermore, possible exact explicit parametric representations of various solutions are given. Particularly, the model has uncountably infinite many solitary wave and pseudo-peakon solutions. [ABSTRACT FROM AUTHOR]

In this paper, based on the Lagrange Multiplier approach in time and the Fourier-spectral scheme for space, we propose efficient numerical algorithms to solve the phase field crystal equation. The numerical schemes are unconditionally energy stable based on the original energy and do not need the lower bound hypothesis of the nonlinear free energy potential. The unconditional energy stability of the three semi-discrete schemes is proven. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes. [ABSTRACT FROM AUTHOR]

A second-order time-stepping method for numerically solving the linearized time fractional KdV equation (TFKDVE) whose solution has initial singularity is investigated. Alikhanov's scheme is used to discretize the Caputo fractional derivative on temporal graded mesh, and the space is approximated by a Petrov–Galerkin spectral method. Detailed stability and error estimate of the full-discrete scheme are given, and the final error bound is α -robust. Numerical examples are given to illustrate the sharpness of the error analysis. [ABSTRACT FROM AUTHOR]

For the generalized Camassa–Holm–Degasperis–Procosi (CH–DP) type equation, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to analyze its corresponding traveling wave systems, which depend on four parameters, it is found that under different parameter conditions its bifurcation portraits exhibit all possible exact explicit bounded solutions, such as solitary wave solutions, periodic wave solutions, peakon as well as periodic peakons. A total of 30 explicit exact parametric representations of the traveling wave system of the CH–DP type equation are presented. [ABSTRACT FROM AUTHOR]

For the cubic Camassa–Holm type equation, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to analyze its corresponding traveling wave system, it was found that under different parameter conditions, its bifurcation portraits exhibit all possible exact explicit bounded solutions (solitary wave solutions, periodic wave solutions, peakon as well as periodic peakons). A total of 19 explicit exact parametric representations of the traveling wave system of the Camassa–Holm type equation are presented. [ABSTRACT FROM AUTHOR]

Accurate predictions for the spread and evolution of epidemics have significant societal and economic impacts. The temporal evolution of infected (or dead) persons has been described as an epidemic wave with an isolated peak and tails. Epidemic waves have been simulated and studied using the classical SIR model that describes the evolution of susceptible (S), infected (I), and recovered (R) individuals. To illustrate the fundamental dynamics of an epidemic wave, the dependence of solutions on parameters, and the dependence of predictability horizons on various types of solutions, we propose a Korteweg–de Vries (KdV)–SIR equation and obtain its analytical solutions. Among classical and simplified SIR models, our KdV–SIR equation represents the simplest system that produces a solution with both exponential and oscillatory components. The KdV–SIR model is mathematically identical to the nondissipative Lorenz 1963 model and the KdV equation in a traveling-wave coordinate. As a result, the dynamics of an epidemic wave and its predictability can be understood by applying approaches used in nonlinear dynamics, and by comparing the aforementioned systems. For example, a typical solitary wave solution is a homoclinic orbit that connects a stable and an unstable manifold at the saddle point within the I – I ′ space. The KdV–SIR equation additionally produces two other types of solutions, including oscillatory and unbounded solutions. The analysis of two critical points makes it possible to reveal the features of solutions near a turning point. Using analytical solutions and hypothetical observed data, we derive a simple formula for determining predictability horizons, and propose a method for predicting timing for the peak of an epidemic wave. [ABSTRACT FROM AUTHOR]

This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration. Moreover, with the help of fixed point theory, we study the existence and uniqueness conditions for the positive solution and prove some new results. Also, obtain the Ulam–Hyers stabilities for the proposed problem. Two generalized examples are considered to show the method's applicability and compared with other existing numerical methods. The present method performs exceptionally well in terms of efficiency and simplicity. Further, we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution. [ABSTRACT FROM AUTHOR]

Using bifurcation analytic method of dynamical systems, we investigate the nonlinear waves and their bifurcations of the generalized KdV–mKdV-like equation. We obtain the following results : (i) Three types of new explicit expressions of nonlinear waves are obtained. They are trigonometric expressions, exp-function expressions, and hyperbolic expressions. (ii) Under different parameteric conditions, these expressions represent different waves, such as solitary waves, kink waves, 1-blow-up waves, 2-blow-up waves, smooth periodic waves and periodic blow-up waves. (iii) Two kinds of new interesting bifurcation phenomena are revealed. The first phenomenon is that the single-sided periodic blow-up waves can bifurcate from double-sided periodic blow-up waves. The second phenomenon is that the double-sided 1-blow-up waves can bifurcate from 2-blow-up waves. Furthermore, we show that the new expressions encompass many existing results. [ABSTRACT FROM AUTHOR]

A new spectral three-term conjugate gradient algorithm in virtue of the Quasi-Newton equation is developed for solving large-scale unconstrained optimization problems. It is proved that the search directions in this algorithm always satisfy a sufficiently descent condition independent of any line search. Global convergence is established for general objective functions if the strong Wolfe line search is used. Numerical experiments are employed to show its high numerical performance in solving large-scale optimization problems. Particularly, the developed algorithm is implemented to solve the 100 benchmark test problems from CUTE with different sizes from 1000 to 10,000, in comparison with some similar ones in the literature. The numerical results demonstrate that our algorithm outperforms the state-of-the-art ones in terms of less CPU time, less number of iteration or less number of function evaluation. [ABSTRACT FROM AUTHOR]