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The periodic behaviors of a linear fourth-order difference solution to the Benjamin–Bona–Mahony (BBM)-type equation with time-periodic boundaries are analyzed in this paper. Firstly, we employ a variable transformation to change the original BBM-type equation with time-periodic boundaries into a new BBM-type equation with zero boundaries. We then construct a fourth-order linear finite difference method to discrete the new BBM-type equation. The solvability, convergence, stability and accuracy of the approximating solution are discussed. The computation procedure of the present method is given in detail. Numerical results show that the proposed difference method is reliable and efficient for time-periodic simulation. [ABSTRACT FROM AUTHOR]

In this paper, we prove some existence results of a critical p -Laplacian equation involving critical and subcritical Hardy potential on compact Riemannian manifolds. [ABSTRACT FROM AUTHOR]

This paper describes two new iterative algorithms for determining absolute value equations. The algorithms are based on a splitting of the coefficient matrix. Moreover, we analyze the convergence effects of the presented algorithms via some theorems. Eventually, numerical tests are provided to confirm the credibility of our procedures. [ABSTRACT FROM AUTHOR]

This paper aims to present two hierarchies of nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy associated with a matrix Lie algebra, not being A type. The key point is to make similarity transformations for the spectral matrix, which keep the associated zero curvature equations invariant and then there follow reduced nonlocal integrable mKdV-type equations. The success lies in determining a Laurent series solution to the corresponding reduced stationary zero curvature equation, which generates temporal matrix spectral problems in the zero curvature formation. [ABSTRACT FROM AUTHOR]

In this paper, tunneling of fermions from rotating BTZ black hole is investigated using modified dispersion relation (MDR) and Rarita–Schwinger equation. The effect of MDR on the tunneling of fermions raises the Hawking temperature of rotating BTZ black hole. It is observed that the modified Hawking temperature of the black hole depends not only on the radial parameters of the black hole but also on the angular parameters of the black hole and the coupling constant σ. Further, the entropy and the heat capacity of the black hole are also studied. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlinear longitudinal wave equation (LWE) which involves mathematical physics in a magneto-electro-elastic (MEE) circular rod. By three different methods, we found that various forms for exact traveling wave solutions of longitudinal bud equation among an MEE round rod. These new exact and solitary wave solutions are derived in the form of rational, single periodic function, double periodic Jacobian elliptic functions (JEF), especially about Weierstrass elliptic function (WEF). Additionally, certain interesting (3D), (2D) figures represent the LWE provides the corporal information to explain the physically phenomena. By choosing different suitable values of the free parameters, we made some charts and we analyzed these charts to get a lot of valuable information to understand the MEE circular rod. Our work describes the dynamics of the MEE circular rod. Finally, we conclude with some perspectives for our future research work. [ABSTRACT FROM AUTHOR]

In this paper, we aim to find novel forms of wave structures by employing some innovative ideas. Various solitary wave solutions of the Zhiber–Shabat equation have been extracted using the Lie symmetry analysis and the extended direct algebraic method. In the mathematical community, the considered model has several applications, notably in integral quantum field theory, fluid dynamics, and kink dynamics. First of all, the Lie symmetry has been used to determine the corresponding similarity reductions through similarity variables and wave transformation with the help of optimal systems. Afterward, the method described has been used to create new complex, hyperbolic, rational, and trigonometric forms of solutions to the problem. Depending on the strength of the propagating pulse, these solutions reflect dark, bright, kink-type, and periodic solitary wave envelopes. Further, two-dimensional (2D), three-dimensional (3D), as well as contour 2D graphics of the results have been analyzed by giving some specific values to parameters. At last, sensitivity analysis of the evolution equation has been observed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stochastic Cahn–Hilliard–Navier–Stokes equations in two-dimensional spaces. Applying the Maslowski–Seidler method, we establish the existence of invariant measure in state space L x 2 × H 1 with the weak topology. We also prove the existence of global pathwise solutions using the stochastic compactness argument. [ABSTRACT FROM AUTHOR]

In this paper, we consider the asymptotic behavior of solutions for nonautonomous stochastic parabolic equation with nonstandard growth condition driven by nonlinear multiplicative noise for the first time. First, by making use of variational method, we prove the existence and uniqueness of solutions, and then the mean random dynamical systems generated by stochastic parabolic equations with variable exponents are obtained. Finally, due to the influence of variable indexes (dependent on space variable), we show the existence of weak mean random attractors under suitable assumptions on the variable exponents and the diffusion term. [ABSTRACT FROM AUTHOR]

We study a regularized version of the Landau equation, which was recently introduced in [J. A. Carrillo, J. Hu, L. Wang and J. Wu, A particle method for the homogeneous Landau equation, J. Comput. Phys. X7 (2020) 100066, 24] to numerically approximate the Landau equation with good accuracy at reasonable computational cost. We develop the existence and uniqueness theory for weak solutions, and we reinforce the numerical findings in the above-mentioned paper by rigorously proving the validity of particle approximations to the regularized Landau equation. [ABSTRACT FROM AUTHOR]

This research paper introduces a novel fractional Caputo-type simultaneous method for finding all simple and multiple roots of polynomial equations. Without any additional polynomial and derivative evaluations using suitable correction, the order of convergence of the basic Aberth–Ehrlich simultaneous method has been increased from three to α + 3. In terms of accuracy, residual graph, computational efficiency and computation CPU time, the newly proposed families of simultaneous methods outperforms existing methods in numerical applications. [ABSTRACT FROM AUTHOR]

In this paper, two algebraic methods are applied for solving a class of conformable fractional partial differential equations (FPDEs). We use these methods for the time-fractional Radhakrishnan–Kundu–Lakshmanan equation. With these methods, further solutions can be obtained compared with other approaches and techniques. The exact particular solutions include the exponential solution, trigonometric function solution, rational solution and hyperbolic function solution. These methods are very effective to obtain exact solutions of many fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper formulates the L1 implicit difference scheme (L1IDS) for the time-fractional Kuramoto–Sivashinsky equation (TFKSE) with non-smooth solution. The TFKSE is one of useful descriptions for modeling flame-propagation, viscous flow problems, and reaction–diffusion systems. The proposed method approximates the unknown solution by using two main stages. At the first stage, the L1 method with nonuniform meshes and the general centered difference method is adopted to discretize the Caputo fractional derivative and the spatial derivative, respectively. In the second stage, the fully-discrete L1IDS is established with the help of the Galerkin scheme based on piecewise linear test functions. Meanwhile, an iterative algorithm is adopted to solve the nonlinear systems. Furthermore, the convergence and stability of the proposed method are both demonstrated and confirmed numerically. Finally, three numerical examples highlight the accuracy and efficiency of the proposed strategy. [ABSTRACT FROM AUTHOR]

In this paper, we study the continuity problem by an order of derivative for conformable parabolic equations. The problem is examined in both the linear and nonlinear cases. For the input data in suitable Hilbert scale spaces, we consider the continuity problem for the linear problem. In the nonlinear case, we prove the existence of mild solutions for a class of conformable parabolic equations once the source function is a global Lipschitz type in the L s space sense. The main results are based on semigroup theory combined with the Banach fixed point theorem and Sobolev embeddings. We also inspect the continuity problem for the nonlinear model, and prove the convergence of the mild solution to the nonlinear problem as α tends to 1 − . [ABSTRACT FROM AUTHOR]

In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of N -fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal P T -symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals. [ABSTRACT FROM AUTHOR]

In this paper, the stable analytical solutions' accuracy of the nonlinear fractional nonlinear time–space telegraph (FNLTST) equation is investigated along with applying the trigonometric-quantic-B-spline (TQBS) method. This investigation depends on using the obtained analytical solutions to get the initial and boundary conditions that allow applying the numerical scheme in an easy and smooth way. Additionally, this paper aims to investigate the accuracy of the obtained analytical solutions after checking their stable property through using the properties of the Hamiltonian system. The considered model for this study is formulated by Oliver Heaviside in 1880 to define the advanced or voltage spectrum of electrified transmission, with day-to-day distances from the electrified communication or the application of electromagnetic waves. The matching between the analytical and numerical solutions is explained by some distinct sketches such as two-dimensional, scatter matrix, distribution, spline connected, bar normal, filling with two colors plots. [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]

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 study asymptotic expansion at infinity and symmetry of zero mean curvature equations of gradient graph in dimension 2, which include the Monge–Ampère equation, inverse harmonic Hessian equation and the special Lagrangian equation. This refines the research of asymptotic behavior, gives the precise gap between exterior minimal gradient graph and the entire case, and extends the classification results of Monge–Ampère equations. [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]

This paper investigates numerical solutions for the unperturbed and general perturbed Newell–Whitehead–Segel-type equations with the help of the local discontinuous Galerkin method. The stability analysis and error estimations of the proposed local discontinuous Galerkin algorithm are extensively examined. First, the spatial variables are discretized to provide a semidiscrete method of lines scheme. This generates an ordinary differential equation system in the temporal variable, which is subsequently solved using the total variation diminishing Runge–Kutta method of higher order. The generated numerical results are compared to the exact results and a few other existing numerical methods via various tables and figures to illustrate the efficiency and accuracy of the proposed method. The numerical results show that the proposed method is an effective numerical scheme for solving the Newell–Whitehead–Segel equation since the solutions obtained using the local discontinuous Galerkin method are highly close to the exact solutions with significantly less error. [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]

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 initiate the study of critical point equation and *-critical point equation within the substructure of Sasakian manifolds. We prove that if a compact Sasakian manifold admits CPE, then either the manifold is Einstein or the potential function is harmonic in an open subset. Later, we demonstrate that if the manifold satisfies *-CPE, it is η -Einstein and obtain an expression for the *-Ricci tensor. Finally, we construct an example to illustrate the existence of CPE and *-CPE on the Sasakian manifold. [ABSTRACT FROM AUTHOR]

In this paper, a (3 + 1) -dimensional generalized KP-Boussinesq equation is introduced and its associate Hirota bilinear form is also given. Based on finding the positive quadratic function solutions of the associate Hirota bilinear equation, the lump solutions of the proposed (3 + 1) -dimensional generalized KP-Boussinesq equation and its corresponding reduced equations in (2 + 1) dimensions are obtained. Furthermore, the sufficient and necessary conditions for guaranteeing the analyticity and rational localization of lump solutions are derived and expressed in the form of free parameters, which are involved in lump solutions and play a key role in controlling the dynamic properties of lump solutions. The localized properties are also analyzed and shown graphically. [ABSTRACT FROM AUTHOR]

The coupled-Higgs equation, which depicts a system of preserved scalar nucleons cooperating with unbiased scalar mesons, is investigated using the ϕ 6 -model expansion approach in this paper. We want to look at the various particles that are created when scalar nucleons collide with unbiased scalar mesons. The modulus of the elliptic function m → 0 and m → 1 is used to obtain the trigonometric and hyperbolic functions. Under the stated limited constraints, all found solutions are used to validate the partial differential equation. We displayed 3D surfaces and 2D visuals of selected solutions produced with the assistance of a computer program to shed more light on the dynamic behavior of the acquired solutions. [ABSTRACT FROM AUTHOR]

In this paper, the Allen–Cahn equation with time noise is under consideration. The extended fan-sub technique is used to find the exact solutions. The solutions are successfully extracted in the form of hyperbolic, trigonometric and mixed forms of solitons. Importantly, the physical unique value problems of the solutions are discussed using the different values of parameters. The 2D, 3D, and their corresponding contour behaviors of these solutions are depicted by choosing the different values of parameters. The stability is controlled through the Borel coefficient in the noise term. [ABSTRACT FROM AUTHOR]

In this paper, we study the blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations of radiation hydrodynamics. First, we improve the blowup result in [P. Jiang and Y. G. Wang, Initial-boundary value problems and formation of singularities for one-dimensional non-relativistic radiation hydrodynamic equations, J. Hyperbolic Differential Equations 9 (2012) 711–738] on the half line [ 0 , + ∞) for large initial data by removing a restrict condition. Next, we obtain a new blowup result on the half line [ 0 , + ∞) by introducing a new momentum weight. Finally, we present two non-global existence results for the smooth solutions to the one-dimensional Euler–Boltzmann equations with vacuum on the interval [ 0 , 1 ] by introducing some new average quantities. [ABSTRACT FROM AUTHOR]

A set { a 1 , a 2 , ... , a m } of positive integers is called a Diophantine m -tuple if a i a j + 1 is a perfect square for all 1 ≤ i < j ≤ m. Let { a , b , c } be the Diophantine triple with c > max { a , b }. In this paper, we find the condition for the reduction of third element c , and using this result, we prove the extendibility of Diophantine pair { F k − 1 F k + 1 , F k − 2 F k + 2 } , where F n is the n -th Fibonacci number. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new kind of super warped product spaces M ̄ (I) = I 1 | 0 × f M m | n , M ̄ (I I) = I 0 | 1 × f M m | n , and M ̄ (I I I) = I 1 | 1 × f M m | n , where M m | n is a supermanifold of dimension m | n , I δ | δ ′ is a superdomain with I = (0 , 1) and δ , δ ′ ∈ { 0 , 1 } , subject to the warp functions f (t) , f (t ̄) , and f (t , t ̄) , respectively. In each super warped product space, M ̄ (I) , M ̄ (I I) , and M ̄ (I I I) , it is shown that Einstein equations Ḡ A B = − Λ ̄ ḡ A B , with cosmological term Λ ̄ are reducible to the Einstein equations G α β = − Λ g α β on the super space M m | n with cosmological term Λ , where Λ ̄ and Λ are functions of f(t), f (t ̄) , and f (t , t ̄) , as well as (m, n). This dependence points to the origin of cosmological terms which turn out to be within the warped structure of the super spacetime. By using the generalized Robertson–Walker spacetime, as a super spacetime, and demanding for constancy of Λ ̄ , we can determine the warp functions and Λ which result in finding the solutions for Einstein equations Ḡ A B = − Λ ̄ ḡ A B and G α β = − Λ g α β . We have discussed the cosmological solutions, for each kind of super warped product space, in the special case of M 3 | 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time asymptotic behavior of the Cauchy problem for the complex nonlinear transverse oscillation equation. Based on the corresponding Lax pair, the original Riemann–Hilbert problem is constructed by introducing some spectral function transformations and variable transformations, and the solution of the complex nonlinear transverse oscillation equation is transformed into the solution of the resulted Riemann–Hilbert problem. Various Deift–Zhou contour deformations and the motivation behind them are given, from which the original Riemann–Hilbert problem is further transformed into a solvable model problem. The long-time asymptotic behavior of the Cauchy problem for the complex nonlinear transverse oscillation equation is obtained by using the nonlinear steepest decent method. [ABSTRACT FROM AUTHOR]

This paper obtains soliton solutions to Sasa–Satsuma equation, which is studied in birefringent fibers with multiplicative noise. The extended auxiliary equation approach reveals solutions in terms of Jacobi's elliptic functions from which bright, dark and singular solitons emerge when the limiting operation is applied to the modulus of ellipticity, with the limit approaching zero or unity. [ABSTRACT FROM AUTHOR]

This paper aims to develop numerical approximations of the Keller–Segel equations that mimic at the discrete level the lower bounds and the energy law of the continuous problem. We solve these equations for two unknowns: the organism (or cell) density, which is a positive variable, and the chemoattractant density, which is a non-negative variable. We propose two algorithms, which combine a stabilized finite element method and a semi-implicit time integration. The stabilization consists of a nonlinear artificial diffusion that employs a graph-Laplacian operator and a shock detector that localizes local extrema. As a result, both algorithms turn out to be nonlinear and can generate cell and chemoattractant numerical densities fulfilling lower bounds. However, the first algorithm requires a suitable constraint between the space and time discrete parameters, whereas the second one does not. We design the latter to attain a discrete energy law on acute meshes. We report some numerical experiments to validate the theoretical results on blowup and nonblowup phenomena. In the blowup setting, we identify a locking phenomenon that relates the L ∞ (Ω) -norm to the L 1 (Ω) -norm limiting the growth of the singularity when supported on a macroelement. [ABSTRACT FROM AUTHOR]

In this paper, by Symmetric Mountain Pass Lemma, we study the existence and multiplicity of solutions to the following nonlocal ψ -Hilfer fractional p -Laplasian equation: H D T α , β ; ψ (| H D 0 + α , β ; ψ ξ (x) | p − 2 H D 0 + α , β ; ψ ξ (x)) − γ (x) | ξ (x) | r − 2 ξ (x) − g (x , ξ (x)) = 0 , I 0 + β (β − 1) ; ψ ξ (0) = I T β (β − 1) ; ψ ξ (T) = 0 , where H D 0 + α , β ; ψ ξ (x) and H D T α , β ; ψ are ψ -Hilfer fractional derivatives left-sided and right-sided of order 1 p < α < 1 , 0 ≤ β ≤ 1 , r > p and I 0 + β (β − 1) ; ψ (.) and I T β (β − 1) ; ψ (.) are ψ -Riemann–Liouville fractional integrals left-sided and right-sided, γ : [ 0 , T ] → ℝ and g : [ 0. T ] × ℝ → ℝ are continuous functions. Finally, we give some examples to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal J generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal J, and describe the minimal graded free resolution of the symmetric algebra of J. Finally, we provide a method to compute the defining equations of the Rees algebra of J using three moving planes that follow the parametrization. [ABSTRACT FROM AUTHOR]

In this paper, we study the Dunkl–Killingbeck problem in two dimensions. We apply the Lie algebraic approach within the framework of quasi-exact solvability to the radial part of the Dunkl–Killingbeck problem to find the general exact expressions for the energies and corresponding wave functions. The allowed values of the potential parameters are the representation space of sl(2) Lie algebra. In addition, we discuss that the effective potential of the Dunkl–Killingbeck is the same as the obtained from the bi-confluent Heun equation by a suitable variable transformation. Following earlier results, we follow the explicit solutions of this differential equation expressed as a series expansion of Hermite functions and obtain the expansion coefficients from a three-term recurrence relation. In the sequel, we present that this construction leads to the known quasi-exactly solvable (QES) form of the Dunkl–Killingbeck problem. Therefore, we find that the expressions for the energy eigenvalues and wave functions of the corresponding potential term are in agreement with those from the QES formalism. Then, we derive the ladder operators for the Dunkl–Killingbeck problem within the algebraic approach. It seems that this method is the Dunkl–Killingbeck rotation problem solved by operators of the su (1 , 1) Lie algebra in a specific way. [ABSTRACT FROM AUTHOR]

In this paper, we study the following mean field type equation: (M F) − Δ g u = ϱ K e u ∫ Σ K e u d V g − 1 in  Σ , where (Σ , g) is a closed oriented surface of unit volume V o l g (Σ) = 1, K positive smooth function and ϱ = 8 π m , m ∈ ℕ. Building on the critical points at infinity approach initiated in [M. Ahmedou, M. Ben Ayed and M. Lucia, On a resonant mean field type equation: A "critical point at infinity" approach, Discrete Contin. Dyn. Syst. 37(4) (2017) 1789–1818] we develop, under generic condition on the function K and the metric g, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters B m (Σ). We derive from these Morse inequalities at infinity various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. ϱ ∈ 8 π ℕ. [ABSTRACT FROM AUTHOR]

In this paper, we establish a global C 2 estimates for a Hessian type equation with homogeneous Dirichlet boundary. By the method of sub and sup solution, we get an existence and uniqueness result for the eigenvalue problem of a Hessian type operator. [ABSTRACT FROM AUTHOR]

In this paper, the local discontinuous Galerkin method is used to analyze numerical solutions for nonlinear Allen–Cahn equations with nonperiodic boundary conditions. To begin with, the spatial variables are discretized to generate a semidiscrete method of lines scheme. This yields an ordinary differential equation system in the temporal variable, which is then solved using the higher-order total variation diminishing Runge–Kutta method. A comparison of the generated numerical results to the exact results for various test problems using different tables and figures provides insight into the effectiveness and accuracy of the proposed method. The numerical results confirm that the proposed method is an effective numerical scheme for solving the Allen–Cahn equation since the obtained solutions are extremely close to the exact solutions while exhibiting substantially less error. [ABSTRACT FROM AUTHOR]

This paper establishes the exponential behavior and stability of the stochastic three-dimensional primitive equations driven by Lévy noise via Burkholder–Davis–Gundy inequality and Itô formula. In particular, we prove that under some conditions on the forcing terms, the weak solution converges exponentially in the mean square and almost surely exponentially to the stationary solution. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the convergence problem of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with rough data and the stochastic continuity of free reduced Ostrovsky equation in Fourier–Lebesgue spaces with random data. On the one hand, we establish the pointwise convergence related to the free reduced Ostrovsky equation in Fourier–Lebesgue spaces Ĥ 1 p , p 2 (R) (4 ≤ p < ∞) with rough data. In particular, we show that s ≥ 1 p is the necessary condition for the maximal function estimate in Ĥ s , p 2 (R) , which means that s = 1 p is optimal for rough data. On the other hand, we present the stochastic continuity of free reduced Ostrovsky equation at t = 0 in Fourier–Lebesgue spaces L ̂ r (R) (2 ≤ r < ∞) with random data. [ABSTRACT FROM AUTHOR]

In this paper, microtubules nonlinear dynamics has been investigated through two different approaches — the first integral method and the G ′ G expansion method — with an objective of deriving traveling wave solutions to the considered model governed by nonlinear partial differential equation by first reducing it to nonlinear ordinary differential equation. The classes of solutions, furnished by these two methods, form a completely new contribution. In some cases, we also demonstrate that various solutions reported earlier in literature can also be recovered as special cases from the solutions constructed in this study. [ABSTRACT FROM AUTHOR]

This paper addresses the numerical study of variable-order fractional differential equation based on finite-difference method. We utilize the implicit numerical scheme to find out the solution of two-dimensional variable-order fractional modified sub-diffusion equation. The discretized form of the variable-order Riemann–Liouville differential operator is used for the fractional variable-order differential operator. The theoretical analysis including for stability and convergence is made by the von Neumann method. The analysis confirmed that the proposed scheme is unconditionally stable and convergent. Numerical simulation results are given to validate the theoretical analysis as well as demonstrate the accuracy and efficiency of the implicit scheme. [ABSTRACT FROM AUTHOR]

This paper develops optimal family of fourth-order iterative techniques in order to find a single root and to generalize them for simultaneous finding of all roots of polynomial equation. Convergence study reveals that for single root finding methods, its optimal convergence order is 4, while for simultaneous methods, it is 12. Computational cost and numerical illustrations demonstrate that the newly developed family of methods outperformed the previous methods available in the literature. [ABSTRACT FROM AUTHOR]

This research paper aims to introduce an extension of the k -Hurwitz–Lerch ζ -function of matrix arguments and interpret its several properties, like generating matrix relations, derivative formulae, Mellin transforms and integral representations. Further, we discuss the solution of the fractional kinetic equations (FKEs) involving an extended k -Hurwitz–Lerch ζ -function of matrix arguments. In addition, the results investigated in view of an extended k -Hurwitz–Lerch ζ -matrix function are rather general in nature and can easily give various known and novel FKEs. [ABSTRACT FROM AUTHOR]

A fractal modification of the combined KdV–mKdV equation which plays a key role in various fields of physics is presented in this work for the first time. Aided by the fractal two-scale transform, the homogeneous balance method is employed to construct the fractal Bäcklund transformation. By means of the Bäcklund transformation, some new exact explicit solutions such as the algebraic solitary wave solution of rational function, single-soliton solution, double-soliton solutions, N-soliton solutions, singular traveling solutions and the periodic wave solutions of trigonometric function are obtained. Finally, some solutions are illustrated with different fractal orders in the form of the 3D plot, 3D density and 2D curves by assigning reasonable parameters with the help of Mathematica. The findings in this paper are expected to present some new insights into the fractal theory of the fractal PDEs. [ABSTRACT FROM AUTHOR]

The Boussinesq–Kadomtsev–Petviashvili-like model is a famous wave equation which is used to describe the shallow water waves in ocean beaches and lakes. When shallow water waves propagate in microgravity or with unsmooth boundaries, the Boussinesq–Kadomtsev–Petviashvili-like model is modified into its fractal model by the local fractional derivative (LFD). In this paper, we mainly study the fractal Boussinesq–Kadomtsev–Petviashvili-like model (FBKPLM) based on the LFD on Cantor sets. Two efficient and reliable mathematical approaches are successfully implemented to obtain the different types of fractal traveling wave solutions of the FBKPLM, which are fractal variational method (FVM) and fractal Yang wave method (FYWM). Finally, some three-dimensional (3D) simulation graphs are employed to elaborate the properties of the fractal traveling wave solutions. [ABSTRACT FROM AUTHOR]

This paper encompassed various physically realistic models of static and spherically symmetric wormholes in extended f (R) gravitational framework by incorporating the features of non-compact matter density distribution of non-commutative geometry. For this assessment, we formulated the f (R) field equations and obtained their exact analytical solutions in presence of cosmologically well-consistent power law type f (R) = a R n corresponding to distinct values of the parameter n. We conducted the physical adequacy of such proposed solutions through the graphical manifestation for the necessary existence of wormhole structures. Moreover, we revisited our field equations with a feasible choice of shape function together with reconstructed form of F (R) to provide an again authentication to the existence of wormholes structures. These all implications vigorously depicted that our discussed models are able to expect the required wormhole structures. [ABSTRACT FROM AUTHOR]

This paper is concerned with the mean asymptotic behavior of the Kuramoto–Sivashinshy equation with stochastic perturbation. We define the mean random dynamical systems for the stochastic Kuramoto–Sivashinshy equation in Bochner spaces. Then we obtain the so-called weak pullback mean random attractor for the stochastic Kuramoto–Sivashinshy equation with odd initial conditions. [ABSTRACT FROM AUTHOR]

In this short paper, we focus on the blowup phenomenon of stochastic parabolic equations. We first discuss the probability of the event that the solutions keep positive. Then, the blowup phenomenon in the whole space is considered. The probability of the event that the solutions blow up in finite time is given. Lastly, we obtain the probability of the event that blowup time of stochastic parabolic equations larger than or less than the deterministic case. [ABSTRACT FROM AUTHOR]