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There is no correct form of the term HT ht in the literature. In Physics it is not allowed to add quantities with different units and the term HT ht is wrong. [Extracted from the article]

Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for H>12$$ \mathcal{H}>\frac{1}{2} $$ these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for H(ξ)≠12$$ \mathcal{H}\left(\xi \right)\ne \frac{1}{2} $$. Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm. [ABSTRACT FROM AUTHOR]

This paper discusses pseudospectra and stability radii for structured nonlinear matrix functions, such as Hermitian, skew‐Hermitian, H‐even, H‐odd, complex symmetric, and complex skew‐symmetric. To compute pseudospectra and stability radii, eigenvalue backward error is required. Hence, we initially present the structured eigenvalue backward error. Subsequently, we compute the structured pseudospectra using the obtained results for the eigenvalue backward error of a class of structured nonlinear matrix functions. Finally, we discuss the stability radii of the above‐structured problems arising in different applications. The paper also generalizes the results on the eigenvalue backward error of matrix polynomials in the literature for the above structures. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations ut(x,t)=Δu(x,t)+ψ(t)f(u(x,t)),(x,t)∈Ω×(0,∞),u(·,0)=u0(x),x∈Ω,u(x,t)=0,(x,t)∈∂Ω×(0,∞),$$ \left\{\begin{array}{ll}{u}_t\left(x,t\right)=\Delta u\left(x,t\right)+\psi (t)f\left(u\left(x,t\right)\right),& \left(x,t\right)\in \Omega \times \left(0,\infty \right),\\ {}u\left(\cdotp, 0\right)={u}_0(x),& x\in \Omega, \\ {}u\left(x,t\right)=0,& \left(x,t\right)\in \mathrm{\partial \Omega}\times \left(0,\infty \right),\end{array}\right. $$where Ω$$ \Omega $$ is the half‐space ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$, ψ$$ \psi $$ is a nonnegative continuous function, and f$$ f $$ is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data u0∈C0(Ω)$$ {u}_0\in {C}_0\left(\Omega \right) $$ if and only if ∫1∞ψ(t)tN+K2fϵt−N+K2dt=∞$$ {\int}_1^{\infty}\psi (t){t}^{\frac{N+K}{2}}f\left(\epsilon \kern0.1em {t}^{-\frac{N+K}{2}}\right) dt=\infty $$ for every ϵ>0$$ \epsilon >0 $$. In fact, we introduce a very special curve in ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$x^(t):=t,⋯,t⏟K‐times,xK+1,⋯,xN,t>0,$$ \hat{x}(t):= \left(\underset{K\hbox{-} \mathrm{times}}{\underbrace{\sqrt{t},\cdots, \sqrt{t}}},{x}_{K+1},\cdots, {x}_N\right),t>0, $$to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations ut(x,t)=Δu(x,t)+ψ(t)f(u(x,t)),(x,t)∈Ω×(0,∞),u(·,0)=u0(x),x∈Ω,u(x,t)=0,(x,t)∈∂Ω×(0,∞),$$ \left\{\begin{array}{ll}{u}_t\left(x,t\right)=\Delta u\left(x,t\right)+\psi (t)f\left(u\left(x,t\right)\right),& \left(x,t\right)\in \Omega \times \left(0,\infty \right),\\ {}u\left(\cdotp, 0\right)={u}_0(x),& x\in \Omega, \\ {}u\left(x,t\right)=0,& \left(x,t\right)\in \mathrm{\partial \Omega}\times \left(0,\infty \right),\end{array}\right. $$where Ω$$ \Omega $$ is the half‐space ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$, ψ$$ \psi $$ is a nonnegative continuous function, and f$$ f $$ is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data u0∈C0(Ω)$$ {u}_0\in {C}_0\left(\Omega \right) $$ if and only if ∫1∞ψ(t)tN+K2fϵt−N+K2dt=∞$$ {\int}_1^{\infty}\psi (t){t}^{\frac{N+K}{2}}f\left(\epsilon \kern0.1em {t}^{-\frac{N+K}{2}}\right) dt=\infty $$ for every ϵ>0$$ \epsilon >0 $$. In fact, we introduce a very special curve in ℝKN$$ {\mathrm{\mathbb{R}}}_K^N $$x^(t):=t,⋯,t⏟K‐times,xK+1,⋯,xN,t>0,$$ \hat{x}(t):= \left(\underset{K\hbox{-} \mathrm{times}}{\underbrace{\sqrt{t},\cdots, \sqrt{t}}},{x}_{K+1},\cdots, {x}_N\right),t>0, $$to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result. [ABSTRACT FROM AUTHOR]

We consider the following fractional NLS with focusing inhomogeneous power‐type nonlinearity: i∂tu−(−Δ)su+|x|−b|u|p−1u=0,(t,x)∈ℝ×ℝN,$$ i{\partial}_tu-{\left(-\Delta \right)}^su+{\left|x\right|}^{-b}{\left|u\right|}^{p-1}u=0,\kern0.30em \left(t,x\right)\in \mathrm{\mathbb{R}}\times {\mathrm{\mathbb{R}}}^N, $$where N≥2$$ N\ge 2 $$, 1/2

This paper deals with the uniformly stability (US), globally uniformly asymptotically stability (GUAS) and instability of zero solution as well as integrability and boundedness of nonzero solutions of certain nonlinear integro‐differential equations (IDEs). We prove five new theorems, which include sufficient conditions related to these fundamental qualitative properties of solutions to the IDEs considered. The main tools used in the proof are two new and suitable Lyapunov–Krasovskiiˇ functionals (LKFs). In particular cases, two numerical examples are given and solved via the fourth order Runge–Kutta method (RKM) in MATLAB to illustrate the theoretical results of this paper. Compared with the existing results on the fundamental qualitative behaviors of scalar IDEs, our results are new, original, more effective and convenient for tests and applications. [ABSTRACT FROM AUTHOR]

In this paper a thorough analysis is carried out of the type of order reduction that Lawson methods exhibit when used to integrate nonlinear initial boundary value problems. In particular, we focus on nonlinear reaction‐diffusion problems, and therefore, this study is important in a large number of practical applications modeled by this type of nonlinear equations. A theoretical study of the local and global error of the total discretization of the problem is carried out, taking into account both, the error coming from the space discretization and that due to the integration in time. These results are also corroborated by the numerical experiments performed in this paper. [ABSTRACT FROM AUTHOR]

This quest paper covers the most important topic in the field of mechanics. It is the nonlinear viscoelastic wave equations with variable degenerate damping and sources acting in both equations. We seek to prove that this system has no solution with positive initial energy. So, to establish our goal, we utilize the methods and techniques used by Said‐Houari in (J. Nonlinear. Differ. Equ. Appl, vol. 18, 2011, pp. 659‐684.). [ABSTRACT FROM AUTHOR]

In this paper, we propose integrating factor (IF)‐based methods for solving the Cahn–Hilliard (CH) equation, which was originally proposed to describe the phase separation phenomenon for binary melted alloy below the critical temperature, and since then, it has appeared in many fields. Being a nonlinear equation, it is of great importance to develop efficient numerical methods for investigating the solution behavior of the CH equation. We present a few important linear and nonlinear schemes for the CH equation and obtain error estimates for those schemes. The integrators that we present here are of first and second order in time. We show numerical results to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniqueness of nodal radial solutions to nonlinear elliptic equations in the unit ball in ℝ3$$ {\mathrm{\mathbb{R}}}^3 $$. Under suitable conditions, we prove that, for any given positive integer k$$ k $$, the problem we considered has at most one solution possessing exactly k−1$$ k-1 $$ nodes. Together with the results presented by Nagasaki [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (2): 211–232, 1989] and Tanaka [Proc. Roy. Soc. Edinburgh Sect. A. 138 (6): 1331–1343, 2008], we can prove that more types of nonlinear elliptic equations have the uniqueness of nodal radial solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the phenomenon of increasing stability in the inverse boundary value problems for the biharmonic equation. The inverse problem is to determine the potential from DtN map. It is a kind of nonlinear inverse problem. By considering a linearized form, we obtain an increasing Lipschitz‐like stability when k$$ k $$ is large. Furthermore, we extend the discussion to the linearized inverse biharmonic potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on the dimension reduction of unknown finite element (FE) solution coefficient vectors in two‐grid Crank–Nicolson FE (TGCNFE) method for the nonlinear Allen–Cahn equation. For this purpose, a new TGCNFE method for the nonlinear Allen–Cahn equation is first developed and the unconditional stability and errors of TGCNFE solutions are analyzed. Thereafter, the most important thing is to lower the dimension of the unknown FE solution coefficient vectors by a proper orthogonal decomposition, to establish a new reduced‐dimension extrapolated TGCNFE (RDETGCNFE) method, and to analyze the unconditional stability and errors of RDETGCNFE solutions. Moreover, the correctness of our theoretical results is validated by some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we shall deal with the long‐time dynamics of a nonlinear system of wave equation with a fractional damping and a small delay. Inspired by the works of Chueshov and Lasiecka on the property of quasi‐stability of dynamic systems, we show first the quasi‐stability property holds for the problem considered here and then prove the existence of global and exponential attractors. We shall also prove the upper semicontinuities of global attractors with respect to the fractional exponents. Finally, we shall show the continuities of global attractors with respect to a small parameter multiplying the delay term in a residual dense set and their upper semicontinuities in a complete metric space. [ABSTRACT FROM AUTHOR]

Mathematics and its applications try to make available a simple and accurate approach to handle nonlinear equations. The present paper investigates for the first time the application of the linearizing method to determine both the approximate frequency and displacement amplitude for the third‐order differential equations. The main merit of the linearized method is a collection of simplicity and accuracy for the solution of high‐order nonlinear conservative or non‐conservative oscillators. The current work sheds light on the approximate proposition to the damped third‐order nonlinear differential equations. The fast estimated solution is simulated to the accurate solution gained through the numerical methods. It is significant to conclude that the non‐perturbative method considered here is simpler and easier to apply than the other perturbation methods. This paper enriches some existing literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new and efficient computational method based on the hybrid of block‐pulse functions and shifted Legendre polynomials together with their exact operational vector of integration and stochastic operational matrix of integration with respect to the multifractional Brownian motion to approximate solutions of a class of nonlinear stochastic differential equations driven by multifractional Gaussian noise. The presented method transforms problems under consideration into systems of nonlinear equations which can be simply solved by the Newton method. Moreover, the convergence analysis of the presented method is investigated. Finally, the efficiency of the new method is illustrated by solving the stochastic logistic equation and three test problems. [ABSTRACT FROM AUTHOR]

This paper introduces a fractional version of reaction‐diffusion equations with non‐local boundary conditions via a non‐singular fractional derivative defined by Atangana and Baleanu. The orthonormal discrete Legendre polynomials are introduced as suitable family of basis functions to find the solution of these equations. An operational matrix is derived for fractional derivative of these polynomials. A collocation method based on the expressed polynomials and their operational matrices is developed for solving such problems. The established method transforms solving the original problem under consideration into solving a system of algebraic equations. Some numerical examples are used to investigate the validity of the presented method. [ABSTRACT FROM AUTHOR]

The paper focuses on a problem of nonlinear electromagnetic wave propagation in a plane shielded waveguide filled with nonlinear medium. We look for nonlinear eigenvalues that correspond to propagation constants of the waveguide. One of the main feature of the problem is that it has nonperturbative solutions and therefore the corresponding linearized problem cannot be used in any perturbation approach to find such eigenvalues (moreover, it turns out that the linearized problem is solvable in exceptional cases only). To solve the main problem, we develop a nonlinear perturbation approach based on usage of a certain simpler nonlinear problem that also has nonlinearizable solutions. Using this approach, we prove existence of nonlinearizable solutions to the main problem. Numerical results are also presented. Briefly, we present electromagnetic background for the main problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iterative scheme with memory for solving nonlinear equations numerically in order to achieve higher order of convergence in comparison to the cubically convergent Chebyshev–Halley‐type method. Several modifications on Chebyshev–Halley‐type methods without memory have been considered in order to extend it to the scheme with memory. We have used self accelerating parameter in order to attain acceleration of convergence speed which is estimated from the current and previous iterations using divided differences. Therefore, the order of convergence increases from 3 to 3.30 without any further functional evaluation. We study the complex and real dynamics of the proposed family. The parameter spaces and dynamical planes are presented. From the parameter spaces, we can detect different members of the proposed family that have good and bad convergence properties. From the dynamical planes, we can analyze the stability of the proposed family in terms of different values of the parameter involved. This study aids in determining the family members with stable behavior which in turn are suitable for practical problems. Numerical examples and comparisons with some of the existing methods are included to confirm the theoretical results. Furthermore, basins of attraction are included to describe a clear picture of the convergence of the proposed as well as some of the existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of a Nash‐type equilibrium for a non‐potential nonlinear system by combining variational methods with the monotonicity approach. The advance over existing research is that we can consider systems of Dirichlet problems in which the operator is not necessarily linear. [ABSTRACT FROM AUTHOR]

Heat transfer can be enhanced in various equipment by utilizing fins that have pores in their media. Efforts have been made to solve the porous fins problem by numerical, semianalytical, and analytical methods. In this paper, after the extraction of equations and the expression of boundary conditions, the Sinc collocation method is employed for problem solving the first time. By using the properties of the Sinc collocation method, the nonlinear problem is discritized into a nonlinear system of equations. The results support the accuracy and speed of the solution. The effects of effective physical parameters such as fins porosity, thermal conductivity of fin, Raley's number, and Darcy parameter have been considered in solving the problem. Also, the variations in temperature distribution, efficiency, and performance of fins are investigated. Therefore, changes in the parameters of the solving method indicate the convergence of the problem in the domain of many issues. In general, the results show that the proposed method can be considered as a powerful method for solving different types of fines. [ABSTRACT FROM AUTHOR]

In this paper, we study two inverse problems for the nonlinear Boussinesq system for incompressible viscoelastic nonisothermal Kelvin–Voigt fluids. The studying inverse problems consist of determining an intensities of density of external forces and heat source under given integral overdetermination conditions. Two types of boundary conditions for the velocity v$$ \mathbf{v} $$ are considered: sticking and sliding conditions on boundary. In both cases of these boundary conditions, the local and global in time existence and uniqueness of weak and strong solutions are established under suitable assumptions on the data. The large time behavior of weak solutions is also studied. [ABSTRACT FROM AUTHOR]

In the present paper, we study a class of nonlinear time‐fractional pseudo‐parabolic equations with bounded time‐delay. Firstly, we analyze the well‐posedness of the problem by the well‐known Galerkin's method and some compactness criteria. Next, the existence of the stationary solutions is established by utilizing a lemma of ours, which is a corollary of the Brouwer fixed point theorem. Finally, with regard to the stability analysis of the stationary solutions, we prove the local stability via the direct method of energy estimations and show the asymptotic stability by the construction of appropriate Lyapunov functionals. [ABSTRACT FROM AUTHOR]

Split variational inclusion problems are core and useful technique that connects many important problems in nonlinear analysis. This paper proposes a resolvent‐based superiorization method for solving split variational inclusion problems, and some strong convergence theorems are proved under some mild conditions. Superiorization iterative procedures are simple yet powerful method that interlaces two algorithmic procedures under the assumption that one of the algorithmic procedure called the basic algorithm is resilient to the disturbances of its iterates caused by the activities of the other algorithm called perturbations. The numerical results in this paper show that our results are improvements and extensions of the corresponding results announced by many others. [ABSTRACT FROM AUTHOR]

In this paper, we study the smoothness properties of solutions to a onedimensional coupled nonlinear Schrödinger system equations that describe some physical phenomena such as propagation of polarized laser beams in birefringent Kerr medium in nonlinear optics. We show that the equations dispersive nature leads to a gain in regularity for the solution. In particular, if the initial datum (u0, v0) possesses certain regularity and sufficient decay as |x| → ∞, then the solution (u(t), v(t)) will be smoother than (u0, v0) for 0 < t ≤ T, where T is the existence time of the solution. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analytical study of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion. Utilizing the bifurcation method of dynamical systems, the existence of compacton and peakon solutions are established. Under certain parameter conditions, the compacton and peakon solutions can be bifurcated by smooth periodic wave solutions, smooth solitary wave solutions, and singular cusp solutions. Numerical simulations are supplied to corroborate the analytical results. Some previous results are extended. [ABSTRACT FROM AUTHOR]

In this work, we extend previous research on exponential integrators for stiff semilinear delay differential equations to the nonlinear case. In addition to the stiffness, there are two new issues that should be handled properly: nonlinear term and delay term. For nonlinear problems, a badly chosen linearization can cause a severe step size restriction. In this work, we linearize the equation along the numerical solution in each step. For the delay term, the interpolation based on the numerical values at the mesh points rather than inner stage values is adopted to significantly reduce the number of stiff order conditions. We focus on the construction and convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations. The main result of this paper is that under the framework of strongly continuous semigroup, the explicit exponential Rosenbrock method is proved to be stiffly convergent of order p$$ p $$ even if the order conditions of order p$$ p $$ hold in a weak form. Moreover, by pointing that there does not exist fifth‐order method with less than or equal to four stages, we present the construction of a fifth‐order method with five stages. Finally, numerical tests are carried out to validate the theoretical results and to demonstrate the superiority of high‐order methods. [ABSTRACT FROM AUTHOR]

The (3 + 1)‐dimensional Vakhnenko–Parkes mathematical model has a wide range of applications in science and engineering. In this paper, the model studied is investigated and analyzed by using two effective schemes such as sine‐Gordon expansion method and its newly developed version, rational SGEM. Moreover, many novel properties of model studied are extracted in detail. Furthermore, their wave distributions properties and graphs are also plotted under the strain conditions. Interactions of the gravitational potential and high‐frequency wave perturbation properties are also reported in a detailed manner. [ABSTRACT FROM AUTHOR]

In this paper, a semi‐discrete rotational velocity‐correction projection method is proposed to solve the Kelvin–Voigt viscoelastic fluid equations. In this method, the rotational velocity‐correction projection method can preserve the divergence free of the velocity u$$ u $$, and the nonlinear equation was linearized. Then, the unconditional stability and optimal convergence order will be provided by the numerical analysis. Finally, our analysis are confirmed by some numerical results, and the algorithm is effective. [ABSTRACT FROM AUTHOR]

This study uses a delayed matrix exponential to illustrate the solution of both linear and nonlinear delay problems using permutable matrices. We outline several prerequisites that are necessary to ensure the exponential stability of the trivial solution of delay systems. The conditions that are necessary and sufficient for the linear system to possess relative controllability are established by introducing a delayed Grammian matrix‐like criterion. A sufficient condition is obtained by Krasnoselskii's fixed point theorem for the relative controllability of nonlinear systems. Numerical examples provided at the end of the paper demonstrate the validity of our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a derivative‐free iterative method for finding the solutions of convex constrained nonlinear equations (CCNE) using the projection strategy. The new approach is free from gradient evaluations at each iteration. Also, the search direction generated by the proposed method satisfies the sufficient descent property, which is independent of the line search. Compared with traditional methods for solving CCNE that assumes Lipschitz continuity and monotonicity to establish the global convergence result, an advantage of our proposed method is that the global convergence result does not require the assumption of Lipschitz continuity. Moreover, the underlying operator is assumed to be pseudomonotone, which is a milder condition than monotonicity. As an applications, we solve the LASSO problem in compressed sensing. Numerical experiments illustrate the performances of our proposed algorithm and provide a comparison with related algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we address a Von Karman equation with nonlinear dissipation, logarithmic source effect, and acoustic boundary conditions. We devote to investigate a blow‐up result of the solutions corresponding to the equation. Owing to the presence of nonlinear dissipation instead of linear dissipation, we have some difficulty in use of the concavity method. Thus, using the energy method, we establish a blow‐up result of the solutions with some positive initial energy. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and the nonexistence of some positive solutions for nonlinear equations involving variable exponent Laplace operator and concave–convex second term: −Δp(x)v=λk(x)vq(x)±h(x)vr(x),$$ -{\Delta}_{p(x)}v&#x0003D;\lambda k(x){v}&#x0005E;{q(x)}\pm h(x){v}&#x0005E;{r(x)}, $$under Robin boundary condition in a regular open‐bounded domain Ω$$ \Omega $$ of ℝN,N≥2$$ {\mathbb{R}}&#x0005E;N,N\ge 2 $$. The p(x)$$ p(x) $$‐Laplacian is given by Δp(x)v=div|∇v|p(x)−2∇v,$$ {\Delta}_{p(x)}v&#x0003D;\operatorname{div}\left({\left&#x0007C;\nabla v\right&#x0007C;}&#x0005E;{p(x)-2}\nabla v\right), $$ where p∈C1(Ω‾)$$ p\in {C}&#x0005E;1\left(\overline{\Omega}\right) $$ and p≥1$$ p\ge 1 $$. Our proofs are based on the subsupersolutions method mixed with variational arguments. [ABSTRACT FROM AUTHOR]

Nonlinear Blasius equation of boundary layer plays a crucial role in different physics and engineering fields. For nearly a century, researchers have been trying to find a convergent and closed analytical solution of the Blasius equation. In this paper, an analytical self‐consistent method (ASCM) is presented to solve this equation. Then, the method was implemented in other forms of Blasius equation, that is, Sakiadis flow and shrinking sheet problem. The analytical solutions obtained show the success of ASCM and agree closely with the corresponding numerical results. The reliability and efficiency of the proposed method are demonstrated on the numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is interested in the decay of solutions to 3D Navier–Stokes equations with a nonlinear damping term α|u|β−1u$$ \alpha {\left|u\right|}^{\beta -1}u $$. We discuss the bounds of the global strong solutions in the negative Sobolev space H˙−s$$ {\dot{H}}^{-s} $$ and establish the optimal decay rates in L2$$ {L}^2 $$ of the global strong solutions. Moreover, the upper bound of the derivative is also obtained. Finally, we investigate the asymptotic stability of the solutions to the system. [ABSTRACT FROM AUTHOR]

In this paper, we study different ways for introducing memory to a parametric family of optimal two‐step iterative methods. We study the convergence and the stability, by means of real dynamics, of the methods obtained by introducing memory in order to compare them. We also perform several numerical experiments to see how the methods behave. [ABSTRACT FROM AUTHOR]

In this paper, we study a one‐dimension nonlinear problem for thermoelastic coupled beam equations with memory term. Using the Faedo–Galerkin method and the linearization method for nonlinear terms, we first prove the local existence and the uniqueness of a weak solution. Next, by establishing assumptions and constructing energy functionals suitably, we consider the global existence and general decay behavior of the solution. Finally, the blow‐up property in the special case of this problem is also given. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the existence and bifurcation of positive solutions for Kirchhoff‐type eigenvalue problems involving the fractional p$$ p $$‐Laplacian. First, we investigate the properties of the first eigenvalue for fractional p$$ p $$‐Laplacian equations with weighted functions. Furthermore, by using fixed‐point argument and modified global bifurcation theorem of Rabinowitz, together with topological degree theory, we obtain the existence of unbounded continuum of positive weak solutions to Kirchhoff‐type equations with subcritical and critical nonlinearities, where the bifurcation emanates from (0,0)$$ \left(0,0\right) $$. It is worth mentioning that our main results fill in some gaps of the available results. [ABSTRACT FROM AUTHOR]

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for one nonclassical, third‐order, partial differential equation with gradient nonlinearity |∇u(x,t)|q$$ {\left&#x0007C;\nabla u\left(x,t\right)\right&#x0007C;}&#x0005E;q $$. The solution to this problem is understood in a weak sense. We show that for 13/2$$ q>3/2 $$, such a solution exists. For 3/2

This paper presents and studies three two‐grid stabilized quadratic equal‐order finite element algorithms based on two local Gauss integrations for the steady Navier–Stokes equations with damping. In these algorithms, we first solve a stabilized nonlinear problem on a coarse grid, and then pass the coarse grid solution to a fine grid and solve a stabilized linear problem. Using some nonlinear analysis techniques, we analyze stability of the algorithms and derive optimal order error estimates of the approximate solutions. Theoretical and numerical results show that, when the algorithmic parameters are chosen appropriately, the accuracy of the approximate solutions computed by our two‐grid stabilized algorithms is comparable to that of solving a fully stabilized nonlinear problem on the same fine grid; however, our two‐grid algorithms save a large amount of CPU time than the one‐grid stabilized algorithm. [ABSTRACT FROM AUTHOR]

In the present paper, a nonlinear fractional Schrödinger integro‐differential equation is considered in a Hilbert space. Operator approach is applied on multidimensional problems with nonlinearity that deserve a studious treatment. In this paper, theorems on existence and uniqueness of a bounded solution for the abstract problem are achieved. Additionally, existence theorems are obtained for first and second orders of accuracy difference schemes of the abstract problem. Furthermore, theorems are applied on a one‐dimensional problem with nonlocal condition and a multidimensional problem with Dirichlet boundary condition. Numerical results and illustrations are presented to show the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the well‐posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained in previous results to allow weak internal dampings and removing the dimensional restriction 1≤n≤4$$ 1\le n\le 4 $$. The method we use is based on a finite‐dimensional approach by combining the Faedo‐Galerkin method with suitable energy estimates and multiplier techniques. [ABSTRACT FROM AUTHOR]

This research paper develops an efficient Gegenbauer wavelet‐based approximation method to solve nonlinear fractional‐time regularized long wave (RLW) equations arising in ocean engineering. The operational matrices of derivatives, as well as the fractional order integration, are engaged in the proposed method. Using the operational matrix method and collocation point, the nonlinear RLW equations are converted into a system of algebraic equations, and these equations are further solved. Some results regarding the error‐bound estimation of this method have been developed. The method's accuracy and efficiency are confirmed by error estimation. The obtained results are compared with other available results and exact solutions. Moreover, the use of Gegenbauer wavelets is found to be a simple, straightforward forward and efficient tool for solving nonlinear PDEs arising in ocean engineering. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence of extreme solutions and their properties for a general σ$$ \sigma $$‐Hessian equation involving a nonlinear operator. By introducing a suitable growth condition and developing a iterative technique, some new results on existence and asymptotic estimates of minimum and maximum solutions are derived. Moreover, we also establish the iterative sequences that converge uniformly to the extreme solutions. [ABSTRACT FROM AUTHOR]

In this paper, we apply the basic one‐way propagator in order to introduce complex regularized long wave (RLW) equation. The complex RLW can be transformed into a system of nonlinear equations. By adopting a consequence nonlinear system, we can derive an energy conservation law of a complex regularized long wave equation. We then investigate how the Ansatz method may be applied to find a class of solitary wave solutions. Simultaneously, a numerical scheme for solving the model is implemented using a finite difference method based on the energy‐preserving Crank–Nicolson/Adams–Bashforth technique. It is worth mentioning that the obtained system is nonlinear. However, by using the present algorithm, we are able to linearize this system and solve it because of the implicit nature of the system of equations. An a priori estimate of the numerical solutions is derived to obtain a convergence and stability analysis; this yields second‐order accuracy in both time and space. Additionally, some numerical experiments verify computational efficiency. The results indicate that this method is an excellent way to preserve energy conservation, providing second‐order accuracy both in time and space with a maximum norm. In addition, we use the proposed scheme to study the effects of dispersive parameters when proceeding with an initial complex Gaussian condition. [ABSTRACT FROM AUTHOR]

The improved modified extended tanh‐function approach was used to study optical stochastic soliton solutions and other exact stochastic solutions for the nonlinear Schrödinger‐Hirota equation with multiplicative white noise. The derived solutions include stochastic bright solitons, stochastic singular solitons, stochastic periodic solutions, stochastic singular periodic solutions, stochastic exponential solutions, stochastic rational solutions, and stochastic Jacobi elliptic doubly periodic solutions. Constraints on the parameters were taken into account to ensure the existence of the obtained stochastic soliton solutions. Additionally, selected solutions were presented graphically to illustrate the physical characteristics of the stochastic solutions. In this paper, we used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stability problem of a kind of boundary value problem for quasilinear hyperbolic systems. For small boundary data, we prove that the C1$$ {C}&amp;amp;#x0005E;1 $$ solution exists globally in time when the system is weakly linearly degenerate. In the special case of linear degeneracy, the smallness assumption on the boundary data is weakened. The error estimate in LT∞L1$$ {L}_T&amp;amp;#x0005E;{\infty }{L}&amp;amp;#x0005E;1 $$ space between two different solutions with different boundary data is also obtained. In our proof, an important estimate which describes the interaction of different waves is established by constructing a continuous Glimm function. This estimate together with uniform a prior estimates with respect to the time leads to the global‐in‐time existence of smooth solutions. Finally, we apply the stability results to the isentropic Euler equations and the system of the motion of an elastic string. [ABSTRACT FROM AUTHOR]

This paper investigates a class of higher order neutral functional dynamic equations with Laplacians. By using the generalized Riccati technique, integral averaging technique, and inequality technique, several new oscillation criteria are established. Our findings build on many previous results (Hassan and Kong, doi:10.1002/mma.4283; Wu and Sun, https://doi.org/10.1515/ms‐2015‐0166) and address cases that are not covered by existing criteria in the literature. Finally, we present some interesting examples to illustrate the versatility of new theorems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotical stability of fractional‐order Hopfield neural networks with multiple delays. The problem is actually a generalization of stability for linear fractional‐order delayed differential equations: 0CDtαX(t)=MX(t)+CX(t−τ)$$ {}_0^C{\mathrm{D}}_t^{\alpha }X(t)= MX(t)+ CX\left(t-\tau \right) $$, which is widely studied when |Arg(λM)|>π2$$ \mid \mathrm{Arg}\left({\lambda}_M\right)\mid >\frac{\pi }{2} $$. However, the stability is rarely known when απ2<|Arg(λM)|≤π2$$ \frac{\alpha \pi}{2}<\mid \mathrm{Arg}\left({\lambda}_M\right)\mid \le \frac{\pi }{2} $$. Hence, this work is mainly devoted to the stability analysis for απ2<|Arg(λM)|≤π2$$ \frac{\alpha \pi}{2}<\mid \mathrm{Arg}\left({\lambda}_M\right)\mid \le \frac{\pi }{2} $$. By virtue of the Laplace transform method and a decoupling technique for the characteristic equation, we propose a necessary and sufficient condition to ensure the stability, which improves the existing stability results for |Arg(λM)|>π2$$ \mid \mathrm{Arg}\left({\lambda}_M\right)\mid >\frac{\pi }{2} $$. Afterward, by a linearization technique, a necessary and sufficient stability condition is also presented for fractional‐order Hopfield neural networks with multiple delays. The conditions are established by delay‐independent coefficient‐type criteria. Finally, several numerical simulations are given to show the effectiveness of our results. [ABSTRACT FROM AUTHOR]