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Recently, we introduced a type of autocorrelation function (ACF) to describe a long-range dependent (LRD) process indexed with two parameters, which takes standard fractional Gaussian noise (fGn for short) as a special case. For simplicity, we call it the generalized fGn (GfGn). This short paper gives the power spectrum density function (PSD) of GfGn.

Nonlinear science is a great revolution of modern natural science. As a result of its rise, the various branches of subjects characterized by nonlinearity have been developed vigorously. In particular, more attention to acquiring the exact solutions of a wide variety of nonlinear equations has been paid by people. In this paper, three methods for solving the exact solutions of the nonlinear 2 + 1 -dimensional Jaulent-Miodek equation are introduced in detail. First of all, the exact solutions of this nonlinear equation are obtained by using the exp − ϕ z -expansion method, tanh method, and sine-cosine method. Secondly, the relevant results are verified and simulated by using Maple software. Finally, the advantages and disadvantages of the above three methods listed in the paper are analyzed, and the conclusion was drawn by us. These methods are straightforward and concise in very easier ways.

In this paper, our leading objective is to relate the fractional integral operator known as P δ -transform with the ք -extended Mathieu series. We show that the P δ -transform turns to the classical Laplace transform; then, we get the integral relating the Laplace transform stated in corollaries. As corollaries and consequences, many interesting outcomes are exposed to follow from our main results. Also, in this paper, we have converted the P δ -transform into a classical Laplace transform by changing the variable ln δ − 1 s + 1 / δ − 1 ⟶ s ; then, we get the integral involving the Laplace transform.

Transformation is an important means to study problems in analytical mechanics. It is often difficult to solve dynamic equations, and the use of variable transformation can make the equations easier to solve. The theory of canonical transformations plays an important role in solving Hamilton’s canonical equations. Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. This paper attempts to extend the canonical transformation theory of Hamilton systems to Birkhoff systems and establish the generalized canonical transformation of Birkhoff systems. First, the definition and criterion of the generalized canonical transformation for the Birkhoff system are established. Secondly, based on the criterion equation and considering the generating functions of different forms, six generalized canonical transformation formulas are derived. As special cases, the canonical transformation formulas of classical Hamilton’s equations are given. At the end of the paper, two examples are given to illustrate the application of the results.

This paper deals with a class of one element n-degree polynomial differential equations. By the fixed point theory, we obtain n periodic solutions of the equation. This paper generalizes some related conclusions of some papers.

The purpose of this paper is to establish the necessary conditions for a fuzzy optimal control problem of several variables. Also, we define fuzzy optimal control problems involving isoperimetric constraints and higher order differential equations. Then, we convert these problems to fuzzy optimal control problems of several variables in order to solve these problems using the same solution method. The main results of this paper are illustrated throughout three examples, more specifically, a discussion on the strong solutions (fuzzy solutions) of our problems.

The Laplace-Adomian Decomposition Method (LADM) and Homotopy Perturbation Method (HPM) are both utilized in this research in order to obtain an approximate analytical solution to the nonlinear Schrödinger equation with harmonic oscillator. Accordingly, nonlinear Schrödinger equation in both one and two dimensions is provided to illustrate the effects of harmonic oscillator on the behavior of the wave function. The available literature does not provide an exact solution to the problem presented in this paper. Nevertheless, approximate analytical solutions are provided in this paper using LADM and HPM methods, in addition to comparing and analyzing both solutions.

In this paper, we investigate the solutions of coupled fractional pantograph differential equations with instantaneous impulses. The work improves some existing results and contributes toward the development of the fractional differential equation theory. We first provide some definitions that will be used throughout the paper; after that, we give the existence and uniqueness results that are based on Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Two examples are given in the last part to support our study.

In this paper, a beta operator is used with Caputo Marichev-Saigo-Maeda (MSM) fractional differentiation of extended Mittag-Leffler function in terms of beta function. Further in this paper, some corollaries and consequences are shown that are the special cases of our main findings. We apply the beta operator on the right-sided MSM fractional differential operator and on the left-sided MSM fractional differential operator. We also apply beta operator on the right-sided MSM fractional differential operator with Mittag-Leffler function and the left-sided MSM fractional differential operator with Mittag-Leffler function.

In this paper, we introduce a generalization of rectangular b-metric spaces, by changing the rectangular inequality as follows: Dζa,b≤ζa,b,u,vDζa,u+Dζu,v+Dζv,b, for all distinct a,b,u,v∈X. We prove some fixed point theorems, and we use our results to present a nice application in the last section of this paper.

Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family $\{p_\nu(x)\}_{\nu=0}^\infty$ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations $\mathcal{A} p_\nu(x) =q_\nu(x) p_\nu(x)$, where $\mathcal{A}$ is a linear differential operator and each $q_\nu(x)$ is a polynomial of degree at most $n_0 \in \mathbb{N}$; $n_0$ does not depend on $\nu$. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator $\mathcal{A}$ for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials., Comment: This version contains minor improvements to the exposition to match the published version of the paper

The stochastic resonance (SR) characteristics of a generalized Langevin linear system driven by a multiplicative noise and a periodically modulated noise are studied (the two noises are correlated). In this paper, we consider a generalized Langevin equation (GLE) driven by an internal noise with long-memory and long-range dependence, such as fractional Gaussian noise (fGn) and Mittag-Leffler noise (M-Ln). Such a model is appropriate to characterize the chemical and biological solutions as well as to some nanotechnological devices. An exact analytic expression of the output amplitude is obtained. Based on it, some characteristic features of stochastic resonance phenomenon are revealed. On the other hand, by the use of the exact expression, we obtain the phase diagram for the resonant behaviors of the output amplitude versus noise intensity under different values of system parameters. These useful results presented in this paper can give the theoretical basis for practical use and control of the SR phenomenon of this mathematical model in future works.

This paper employs the computational approach known as successive linearization method (SLM) to tackle a fourth order nonlinear differential equation modelling the transient flow of an incompressible viscous fluid between two parallel plates produced by a simple wall motion. Numerical and graphical results obtained show excellent agreement with the earlier results reported in the literature. We obtain solution branches as well as a turning point in the flow field accurately. A comparison with numerical results generated using the inbuilt MATLAB boundary value solver,bvp4c, demonstrates that the SLM approach is a very efficient technique for tackling highly nonlinear differential equations of the type discussed in this paper.

In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global small H k k ⩾ 3 solution only under the condition of smallness of the H 3 norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal L p – L q 1 ⩽ p ⩽ 2 , 2 ⩽ q ⩽ ∞ -type decay rates of the solution and its higher-order derivatives.

In this paper, we define and investigate a special kind of ruled surfaces called type-2 Smarandache ruled surfaces related to the type-2 Bishop frame in E 3 . From this point and depending on the type-2 Bishop curvature, we provide the necessary and sufficient conditions that allow these surfaces to be developable in a minimal amount of time. Furthermore, an example is given to clear the results.

Some important dynamical properties of the memristor chaotic oscillator system have been studied in the paper. A novel hybrid dislocated control method and a general hybrid projective dislocated synchronization scheme have been realized for memristor chaotic oscillator system. The paper firstly presents hybrid dislocated control method for stabilizing chaos to the unstable equilibrium point. Based on the Lyapunov stability theorem, general hybrid projective dislocated synchronization has been studied for the drive memristor chaotic oscillator system and the same response memristor chaotic oscillator system. For the different dimensions, the memristor chaotic oscillator system and the other chaotic system have realized general hybrid projective dislocated synchronization. Numerical simulations are given to show the effectiveness of these methods.

The prime numbers have attracted mathematicians and other researchers to study their interesting qualitative properties as it opens the door to some interesting questions to be answered. In this paper, the Random Matrix Theory (RMT) within superstatistics and the method of the Nearest Neighbor Spacing Distribution (NNSD) are used to investigate the statistical proprieties of the spacings between adjacent prime numbers. We used the inverse χ 2 distribution and the Brody distribution for investigating the regular-chaos mixed systems. The distributions are made up of sequences of prime numbers from one hundred to three hundred and fifty million prime numbers. The prime numbers are treated as eigenvalues of a quantum physical system. We found that the system of prime numbers may be considered regular-chaos mixed system and it becomes more regular as the value of the prime numbers largely increases with periodic behavior at logarithmic scale.

In the present paper, by considering the Gauss equation in place of the Codazzi equation, we derive new optimal inequality for the second fundamental form of CR-warped product submanifolds into a generalized Sasakian space form. Moreover, the inequality generalizes some inequalities for various ambient space forms.

In this paper, we define the notation of admissible hybrid Z -contractions in the setting of extended b -metric spaces, which unifies and generalizes previously existing results in literature. Furthermore, as an application, we discuss Ulam-Hyers stability and well-posedness of a fixed point problem.

The authors of this paper solve the fractional space-time advection-dispersion equation (ADE). In the advection-dispersion process, the solute movement being nonlocal in nature and the velocity of fluid flow being nonuniform, it leads to form a heterogeneous system which approaches to model the same by means of a fractional ADE which generalizes the classical ADE, where the time derivative is substituted through the Caputo fractional derivative. For the study of such fractional models, various numerical techniques are used by the researchers but the nonlocality of the fractional derivative causes high computational expenses and complex calculations so the challenge is to use an efficient method which involves less computation and high accuracy in solving such models numerically. Here, in order to get the FADE solved in the form of convergent infinite series, a novel method NHPM (natural homotopy perturbation method) is applied which couples Natural transform along with the homotopy perturbation method. The homotopy peturbation method has been applied in mathematical physics to solve many initial value problems expressed in the form of PDEs. Also, the HPM has an advantage over the other methods that it does not require any discretization of the domains, is independent of any physical parameters, and only uses an embedding parameter p ∈ 0 , 1 . The HPM combined with the Natural transform leads to rapidly convergent series solutions with less computation. The efficacy of the used method is shown by working out some examples for time-fractional ADE with various initial conditions using the NHPM. The Mittag-Leffler function is used to solve the fractional space-time advection-dispersion problem, and the impact of changing the fractional parameter α on the solute concentration is shown for all the cases.

In this paper, we study a stochastic Holling-type II predator-prey model with stage structure and refuge for prey. Firstly, the existence and uniqueness of the global positive solution of the system are proved. Secondly, the stochastically ultimate boundedness of the solution is discussed. Next, sufficient conditions for the existence and uniqueness of ergodic stationary distribution of the positive solution are established by constructing a suitable stochastic Lyapunov function. Then, sufficient conditions for the extinction of predator population in two cases and that of prey population in one case are obtained. Finally, some numerical simulations are presented to verify our results.

The aim of this paper is to study representations of 3-dimensional simple multiplicative Hom-Lie algebras (whose structure is of A1-type). In this paper we can see that a finite dimensional representation of is not always completely reducible, and a representation of is irreducible if and only if it is a regular Lie-type representation.

Fractional diffusion on the sphere plays a large role in the study of physical phenomena customs and meteorology and geophysics. In this paper, we examine two types of the sphere problem: the initial value problem and the end value problem. We are interested in focus on the solution existence in a local or global form. In order to overcome difficult evaluations when evaluating, we need some new techniques. The main analytical tool is the use of the Banach fixed point theorem.

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

In this paper, we classified the paracontact metric κ , μ -manifold satisfying the Miao-Tam critical equation with κ > − 1 . We proved that it is locally isometric to the product of a flat n + 1 -dimensional manifold and an n -dimensional manifold of negative constant curvature − 4 .

In this paper, the Kudryashov method to construct the new exact solitary wave solutions for the newly developed ( 2 + 1 )-dimensional Benjamin-Ono equation is successfully employed. In the same vein, also the new ( 2 + 1 )-dimensional Benjamin-Ono equation to ( 3 + 1 )-dimensional spaces is extended and then analyzed and investigated. Different forms of exact solitary wave solutions to this new equation were also determined. Graphical illustrations for certain solutions in both equations are provided. We alternatively offer that the determining method is general, impressive, outspoken, and powerful and can be exerted to create exact solutions of various kinds of nonlinear models originated in mathematical physics and engineering.

Nowadays it is practically forgotten that for observables with degenerate spectra the original von Neumann projection postulate differs crucially from the version of the projection postulate which was later formalized by Lüders. The latter (and not that due to von Neumann) plays the crucial role in the basic constructions of quantum information theory. We start this paper with the presentation of the notions related to the projection postulate. Then we remind that the argument of Einstein-Podolsky-Rosen against completeness of QM was based on the version of the projection postulate which is nowadays called Lüders postulate. Then we recall that all basic measurements on composite systems are represented by observables with degenerate spectra. This implies that the difference in the formulation of the projection postulate (due to von Neumann and Lüders) should be taken into account seriously in the analysis of the basic constructions of quantum information theory. This paper is a review devoted to such an analysis.

This paper discusses a boundary value problem of nonlinear fractional integrodifferential equations of order 1 < α ≤ 2 and 1 < β ≤ 2 and boundary conditions of the form x 0 = x 1 = D c β x 1 = D c β x 0 = 0 . Some new existence and uniqueness results are proposed by using the fixed point theory. In particular, we make use of the Banach contraction mapping principle and Krasnoselskii’s fixed point theorem under some weak conditions. Moreover, two illustrative examples are studied to support the results.

In this paper, the boundary value inverse problem related to the generalized Burgers–Fisher and generalized Burgers–Huxley equations is solved numerically based on a spline approximation tool. B-splines with quasilinearization and Tikhonov regularization methods are used to obtain new numerical solutions to this problem. First, a quasilinearization method is used to linearize the equation in a specific time step. Then, a linear combination of B-splines is used to approximate the largest order of derivatives in the equation. By integrating from this linear combination, some approximations have been obtained for each of the functions and derivatives with respect to time and space. The boundary and additional conditions of the problem are also applied in these approximations. The Tikhonov regularization method is used to solve the system of linear equations using noisy data. Several numerical examples are provided to illustrate the accuracy and efficiency of the method.

This paper is devoted to the initial and boundary value problems for a class of nonlinear metaparabolic equations u t − β u x x − k u x x t + γ u x x x x = f u x x . At low initial energy level ( J u 0 < d ), we not only prove the existence of global weak solutions for these problems by the combination of the Galerkin approximation and potential well methods but also obtain the finite time blow-up result by adopting the potential well and improved concavity skills. Finally, we also discussed the finite time blow-up phenomenon for certain solutions of these problems with high initial energy.

This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε , taking arbitrary value in the interval 0 , 1 . For small values of ε , solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε .

In this paper, we study the Kirchhoff-type equation: − a + b ∫ ℝ 3 ∇ u 2 d x Δ u + V x u = Q x f u , in ℝ 3 , where a , b > 0 , f ∈ C 1 ℝ 3 , ℝ , and V , Q ∈ C 1 ℝ 3 , ℝ + . V x and Q x are vanishing at infinity. With the aid of the quantitative deformation lemma and constraint variational method, we prove the existence of a sign-changing solution u to the above equation. Moreover, we obtain that the sign-changing solution u has exactly two nodal domains. Our results can be seen as an improvement of the previous literature.

Let E be a separable symmetric space on 0 , ∞ and E M the corresponding noncommutative space. In this paper, we introduce a kind of quasimartingale spaces which is like but bigger than E M and obtain the following interpolation result: let E ^ M be the space of all bounded E M -quasimartingales and 1 < p < p E < q E < q < ∞ . Then, there exists a symmetric space F on 0 , ∞ with nontrivial Boyd indices such that E ^ M = L ^ p M , L ^ q M F , K .

In this paper, applying the weak maximum principle, we obtain the uniqueness results for the hypersurfaces under suitable geometric restrictions on the weighted mean curvature immersed in a weighted Riemannian warped product I × ρ M f n whose fiber M has f -parabolic universal covering. Furthermore, applications to the weighted hyperbolic space are given. In particular, we also study the special case when the ambient space is weighted product space and provide some results by Bochner’s formula. As a consequence of this parametric study, we also establish Bernstein-type properties of the entire graphs in weighted Riemannian warped products.

In this paper, we consider the problem of the rotational motion of a rigid body with an irrational value of the frequency ω . The equations of motion are derived and reduced to a quasilinear autonomous system. Such system is reduced to a generating one. We assume a large parameter μ proportional inversely with a sufficiently small component r o of the angular velocity which is assumed around the major or the minor axis of the ellipsoid of inertia. Then, the large parameter technique is used to construct the periodic solutions for such cases. The geometric interpretation of the motion is obtained to describe the orientation of the body in terms of Euler’s angles. Using the digital fourth-order Runge-Kutta method, we determine the digital solutions of the obtained system. The phase diagram procedure is applied to study the stability of the attained solutions. A comparison between the considered numerical and analytical solutions is introduced to show the validity of the presented techniques and solutions.

The aim of this paper is to introduce a notion of ϕ , F -contraction defined on a metric space with w -distance. Moreover, fixed-point theorems are given in this framework. As an application, we prove the existence and uniqueness of a solution for the nonlinear Fredholm integral equations. Some illustrative examples are provided to advocate the usability of our results.

Quenching characteristics based on the two-dimensional (2D) nonlinear unsteady convection-reaction-diffusion equation are creatively researched. The study develops a 2D compact finite difference scheme constructed by using the first and the second central difference operator to approximate the first-order and the second-order spatial derivative, Taylor series expansion rule, and the reminder-correction method to approximate the three-order and the four-order spatial derivative, respectively, and the forward difference scheme to discretize temporal derivative, which brings the accuracy resulted meanwhile. Influences of degenerate parameter, convection parameter, and the length of the rectangle definition domain on quenching behaviors and performances of special quenching cases are discussed and evaluated by using the proposed scheme on the adaptive grid. It is feasible for the paper to offer potential support for further research on quenching problem.

In this paper, we focus on high-order approximate solutions to two-level systems with quasi-resonant control. Firstly, we develop a high-order renormalization group (RG) method for Schrödinger equations. By this method, we get the high-order RG approximate solution in both resonance case and out of resonance case directly. Secondly, we introduce a time transformation to avoid the invalid expansion and get the high-order RG approximate solution in near resonance case. Finally, some numerical simulations are presented to illustrate the effectiveness of our RG method. We aim to provide a mathematically rigorous framework for mathematicians and physicists to analyze the high-order approximate solutions of quasi-resonant control problems.

In this paper, we present a common fixed point result for a pair of mappings defined on a b-metric space, which satisfies quasi-contractive inequality with nonlinear comparison functions. An application in solving a class of integral equations will support our results.

The aim of this paper is to give existence results for a class of coupled systems of fractional integrodifferential equations with Hilfer fractional derivative in Banach spaces. We first give some definitions, namely the Hilfer fractional derivative and the Hausdorff’s measure of noncompactness and the Sadovskii’s fixed point theorem.

This paper is devoted to identify a space-dependent source function in a multiterm time-fractional diffusion equation with nonhomogeneous boundary condition from a part of noisy boundary data. The well-posedness of a weak solution for the corresponding direct problem is proved by the variational method. We firstly investigate the uniqueness of an inverse initial problem by the analytic continuation technique and the Laplace transformation. Then, the uniqueness of the inverse source problem is derived by employing the fractional Duhamel principle. The inverse problem is solved by the Levenberg-Marquardt regularization method, and an approximate source function is found. Numerical examples are provided to show the effectiveness of the proposed method in one- and two-dimensional cases.

The aim of this paper is to establish the existence of solutions for singular double-phase problems depending on one parameter. This work improves and complements the existing ones in the literature. There seems to be no results on the existence of solutions for singular double-phase problems.

In this paper, based on a bilinear differential equation, we study the breather wave solutions by employing the extended homoclinic test method. By constructing the different forms, we also consider the interaction solutions. Furthermore, it is natural to analyse dynamic behaviors of three-dimensional plots.

In this paper, we study a class of the Kirchhoff-Schrödinger-Poisson system. By using the quantitative deformation lemma and degree theory, the existence result of the least energy sign-changing solution u0 is obtained. Meanwhile, the energy doubling property is proved, that is, we prove that the energy of any sign-changing solution is strictly larger than twice that of the least energy. Moreover, we also get the convergence properties of u0 as the parameters b↘0 and λ↘0.

In this paper, we study the following fractional Schrödinger equation in ℝ3−Δσu−λu=up−2u, in ℝ3 with σ∈0,1,λ∈ℝ and p∈2+σ,2+4/3σ. By using the constrained variational method, we show the existence of solutions with prescribed L2 norm for this problem.

Type-II censored data is an important scheme of data in lifetime studies. The purpose of this paper is to obtain E-Bayesian predictive functions which are based on observed order statistics with two samples from two parameter Burr XII model. Predictive functions are developed to derive both point prediction and interval prediction based on type-II censored data, where the median Bayesian estimation is a novel formulation to get Bayesian sample prediction, as the integral for calculating the Bayesian prediction directly does not exist. All kinds of predictions are obtained with symmetric and asymmetric loss functions. Two sample techniques are considered, and gamma conjugate prior density is assumed. Illustrative examples are provided for all the scenarios considered in this article. Both illustrative examples with real data and the Monte Carlo simulation are carried out to show the new method is acceptable. The results show that Bayesian and E-Bayesian predictions with the two kinds of loss functions have little difference for the point prediction, and E-Bayesian confidence interval (CI) with the two kinds of loss functions are almost similar and they are more accurate for the interval prediction.

In this paper, we consider a class of nonlinear Caputo fractional differential equations with impulsive effect under multiple band-like integral boundary conditions. By constructing an available completely continuous operator, we establish some criteria for judging the existence and uniqueness of solutions. Finally, an example is presented to demonstrate our main results.