Search

Poisson equation is a partial differential equation with broad utility in theoretical physics. Dirichlet problem of Poisson Equations can be solved by Green's function. It is a very attractive problem to look for analogous results of the above problem in the fractal context. This paper studies the network set Higher-Dimensional Sierpinski Gaskets, solves Dirichlet problem of Poisson Equations on them by expressing Green's function explicitly. [ABSTRACT FROM AUTHOR]

Denote by h = h (− p) the class number of the imaginary quadratic field ℚ (− p) with p prime. It is well known that h = 1 for p ∈ { 3 , 7 , 1 1 , 1 9 , 4 3 , 6 7 , 1 6 3 }. Recently, all the solutions of the Diophantine equation x 2 + p s = 4 y n with h = 1 were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation x 2 + p s = 2 r y n in unknown integers (x , y , s , r , n) , where s ≥ 0 , r ≥ 3 , n ≥ 3 , h ∈ { 1 , 2 , 3 } and gcd (x , y) = 1. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al. [ABSTRACT FROM AUTHOR]

In recent years, SEP elements have been studied by many authors. In this paper, we obtain many new characterizations of SEP elements by using inner inverse, group inverse and Moore–Penrose inverse. Mainly, we construct a lot of equations, study the expression forms of solution to these equations in certain given set. [ABSTRACT FROM AUTHOR]

Conservation laws are well known to be a crucial part of modeling. Considering such models with the inclusion of non-local flows is becoming increasingly important in many models. On the other hand, kinetic equations provide interesting theoretical results and numerical schemes for the usual conservation laws. Therefore, studying kinetic equations associated to conservation laws for non-local flows naturally arises and is very important. The aim of this paper is to propose kinetic models associated to conservation laws with a non-local flux in dimension d and to prove the existence of solutions for these kinetic equations. This is the very first result of this kind. In order for the paper to be as general as possible, we have highlighted the properties that a kinetic model must verify in order that the present study applies. Thus, the result can be applied to various situations. We present two sets of properties on a kinetic model and two different techniques to obtain an existence result. Finally, we present two examples of kinetic model for which our results apply, one for each set of properties. [ABSTRACT FROM AUTHOR]

This paper studies the Lax pair (LP) of the (1 + 1) -dimensional Benjamin–Bona–Mahony equation (BBBE). Based on the LP, initial solution and Darboux transformation (DT), the analytic one-soliton solution will also be obtained for BBBE. This paper contains a bunch of soliton solutions together with bright, dark, periodic, kink, rational, Weierstrass elliptic and Jacobi elliptic solutions for governing model through the newly developed sub-ODE method. The BBBE has a wide range of applications in modeling long surface gravity waves of small amplitude. In addition, we will evaluate generalized breathers, Akhmediev breathers and standard rogue wave solutions for stated model via appropriate ansatz schemes. [ABSTRACT FROM AUTHOR]

This paper is concerned with complex-valued solutions of the Korteweg–de Vries equation. Interest will be focused upon the initial-value problem with initial data that is periodic in space. Derived here are results of local and global well-posedness, singularity formation in finite time and, perhaps surprisingly, results of non-existence. The overall picture is notably different from the situation that obtains for real-valued solutions. [ABSTRACT FROM AUTHOR]

The Black-Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the constructive methods of exact solutions to time fractional Black-Scholes equation. By constructing one-parameter Lie symmetry transformations and their corresponding group generators, time fractional Black-Scholes equation is reduced to a fractional ordinary differential equation and some group-invariant solutions are obtained. Using the invariant subspace method, the analytical representations of two forms of exact solutions of time fractional Black-Scholes equation are given constructively, and the characteristics and differences of the two exact solutions are compared in the sense of geometric figures. In this paper, the form of the equation is generalized, and more group invariant solutions and analytical solutions in the form of separated variables are obtained. [ABSTRACT FROM AUTHOR]

This paper discusses the wave propagation to the nonlinear Sharma–Tasso–Olver–Burgers (STOB) equation which is used as the governing model in different fields. Natural phenomena are typically complex and nonlinear, defying simple linear superposition. Researchers have been studying a wide range of natural phenomena in depth, and nonlinear science has gradually become a part of people's consciousness. One of the most significant research questions in nonlinear science centers around the nonlinear evolution equation and its precise solution. We have secured different shapes of the solitary wave solutions including kink-type, shock-type and combined solitary wave solutions with the assistance of recently developed integration tool, namely the new extended direct algebraic method (NEDAM). Additionally, the solutions for the hyperbolic, exponential and trigonometric functions are retrieved. Moreover, based on a comparison of our results to those that are well known, the study indicates that our solutions are innovative. Using proper parameters in numerical simulations and physical explanations, it is possible to demonstrate the significance of the results. The results of this research can improve the nonlinear dynamic behavior of a system and indicate that the methodology employed is adequate. It is proposed that the offered method can be utilized to support nonlinear dynamical models applicable to a wide variety of physical situations. We hope that a wide spectrum of engineering model professionals will find this study to be beneficial. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Boltzmann equation with external force in the whole space, where the collision kernel is assumed to be hard potential and cutoff. We prove that the solutions of such Boltzmann equations are L p (1 ≤ p < ∞) stable under the perturbation of external force. Our estimate is based on the gradient estimate of the solution. The key step of this paper is to estimate the solutions of the equations propagate in different forward bi-characteristic. [ABSTRACT FROM AUTHOR]

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces L (p , q) , KBSM(L (p , q)), for q ≠ 0. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, TL 1 , n , which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, V , for knots and links in ST, via a unique Markov trace constructed on TL 1 , n . The universal invariant V is equivalent to the KBSM(ST). For passing now to the KBSM(L (p , q)), we impose on V relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in L (p , q) but not in ST, and which reflect the surgery description of L (p , q) , obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM(L (p , q)). We first present the solution for the case q = 1 , which corresponds to obtaining a new basis, ℬ p , for KBSM(L (p , 1)) with (⌊ p / 2 ⌋ + 1) elements. We note that the basis ℬ p is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case q > 1 , we first show how the new basis ℬ p of KBSM(L (p , 1)) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case q > 1. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of multiple non-radial positive and sign-changing solutions for the following Kirchhoff equation: − a + b ∫ ℝ 3 | ∇ u | 2 Δ u + (1 + λ Q (x)) u = | u | p − 2 u , in ℝ 3 , (0. 1) where a , b > 0 are constants, p ∈ (2 , 6) , λ is a parameter, and Q (x) is a potential function. Under the assumption on Q (x) with exponential decay at infinity, we construct multi-peak positive and sign-changing solutions for problem (0.1) as λ → ∞ (or 0), where the peaks concentrate at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we use new extended generalized Kudryashov and improved tan (ϑ 2) expansion approaches to investigate Kerr law nonlinearity in the extended Gerdjikov–Ivanov equation in a dense wavelength division multiplexed system. These methods rely on a traveling wave transformation and an auxiliary equation. These approaches successfully extract trigonometric, rational and hyperbolic solutions, along with some appropriate conditions imposed on parameters. To explain the dynamics of soliton profiles, a graphical description of newly discovered solutions is also presented, which exhibits distinct physical significance. The considered methods are recognized as useful and influential tools for creating solitary wave solutions to nonlinear problems in the mathematical sciences. [ABSTRACT FROM AUTHOR]

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

This paper is a key contribution with respect to the applications of solitary wave solutions to the unique solution in the presence of the auxiliary data. Hence, this study provides an insight for the unique selection of solitons for the physical problems. Additionally, the novel numerical scheme is developed to compare the result. Further, this paper deals with the stochastic Fisher-type equation numerically and analytically with a time noise process. The nonstandard finite difference scheme of stochastic Fisher-type equation is proposed. The stability analysis and consistency of this proposed scheme are constructed with the help of Von Neumann analysis and Itô integral. This model is applicable in the wave proliferation of a viral mutant in an infinitely long habitat. Additionally, for the sake of exact solutions, we used the Riccati equation mapping method. The solutions are constructed in the form of hyperbolic, trigonometric and rational forms with the help of Mathematica 11.1. Lastly, the graphical comparisons of numerical solutions with exact wave solution with the help of Neumann boundary conditions are constructed successfully in the form of 3D and line graphs by using different values of parameters. [ABSTRACT FROM AUTHOR]

Considering the importance of using nonlinear evolution equations in the investigation of many natural phenomena, in this paper, we consider the (2 + 1) -dimensional Date–Jimbo–Kashiwara–Miwa ((2 + 1) -dimensional DJKM) equation, we will investigate the solutions for this equation. Using the multiple exp functions method, we obtain analytical solutions for this equation, which are one-soliton, two-soliton and three-soliton solutions and these solutions include three categories of soliton wave solutions, i.e., one-wave solutions, two-wave solutions and three wave solutions. We have performed all calculations with a computer algebra system such as Maple and have also provided a graphical representation of the obtained solutions. [ABSTRACT FROM AUTHOR]

This paper tackles the recently introduced (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation (4D-BLMPE) utilized to model wave phenomena in incompressible fluid and fluid mechanics. Modified extended tanh expansion method (METEM) and the new Kudryashov scheme are implemented to produce analytical soliton solutions for the presented equation. The traveling wave transformation is constructed, and the homogeneous balance principle is utilized to apply the two proposed techniques. Furthermore, the flat-kink, smooth-kink, singular, and periodic singular solutions are successfully extracted. Some produced solutions are illustrated graphically to understand the physical meaning of the presented model. Moreover, for the first time in this study, the effect of model parameters on kink soliton dynamics is examined, and graphical representations are depicted and interpreted. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial-boundary value (IBV) problem for a generalized Sasa–Satsuma equation with 3 × 3 Lax pair by Fokas unified method on the half line. Based on the analyticity and asymptotics of the eigenfunctions, the IBV problem is formulated as a Riemann–Hilbert (RH) problem. Further, the global relation among IBVs is established and the map from the Dirichlet boundary value to Neumann boundary value is obtained. [ABSTRACT FROM AUTHOR]

A new generalized KdV equation, describing the motions of long waves in shallow water under the gravity field, is considered in this paper. By adopting a series of well-organized methods, the Bäcklund transformation, the bilinear form and diverse wave structures of the governing model are formally extracted. The exact solutions listed in this paper are categorized as lump-type, complexiton, and soliton solutions. To exhibit the physical mechanism of the obtained solutions, several graphical illustrations are given for particular choices of the involved parameters. As a direct consequence, diverse wave structures given in this paper enrich the studies on the KdV-type equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the question of the existence and nonexistence of solutions for some fractional equations with variable exponents. This paper generalizes some analog results in the classical fractional one. As we know, there are no previous results on the nonexistence of solutions for nonlinear equations with fractional p (⋅ , ⋅) -Laplacian. [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]

In this paper, we are interested in the Einstein vacuum equations on a Lorentzian manifold displaying (1) symmetry. We identify some freely prescribable initial data, solve the constraint equations and prove the existence of a unique and local in time solution at the H 3 level. In addition, we prove a blow-up criterium at the H 2 level. By doing so, we improve a result of Huneau and Luk in [Einstein equations under polarized (1) symmetry in an elliptic gauge, Commun. Math. Phys. 361(3) (2018) 873–949] on a similar system, and our main motivation is to provide a framework adapted to the study of high-frequency solutions to the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As a consequence we work in an elliptic gauge, particularly adapted to the handling of high-frequency solutions, which have large high-order norms. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solutions are proposed for the 1D Benjamin–Bona–Mahony (BBM) equation and 2D coupled BBM system by using Galerkin finite element technique. In this regard, the cubic B-splines and linear triangular elements are used, respectively. In 1D space, a proposed numerical scheme is implemented to a test problem including the motion of a single solitary solution. To verify practicality and robustness of our new procedure, the error norms L 2 , L ∞ and three constants I 1 , I 2 and I 3 are evaluated. Stability analysis of the linearized technique indicates that it is unconditionally stable. Moreover, a tsunami wave in 2D space is used to investigate the efficiency of the considered method. Also, the improved tanh (Γ (ϖ)) - coth (Γ (ϖ)) function technique (IThChT) and the combined tan (Γ (ϖ)) - cot (Γ (ϖ)) function technique (ITCT) are obtained in the mentioned BBM equation. The presented methods are seen to be robust, impressive and economical to employment as compared to the existing finite difference techniques and other earlier papers for discovering the numerical solutions for numerous types of linear and nonlinear PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we study the function Δ ∗ (k , n) , which we define as the smallest number s of variables needed to guarantee that the equation ∑ i = 1 s a i x i k + ∑ i = 1 s b i y i n = 0 has nontrivial solutions in each of the p -adic fields ℚ p , regardless of the rational integer coefficients. This generalizes the Γ ∗ (k) function of Davenport and Lewis. In this paper, we give a sharp upper bound for Δ ∗ (k , n) and compute its value for various choices of the degrees. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a new class of nonlinear multi-term impulsive anti-periodic boundary value problems involving Caputo type fractional q -derivative operators of different orders and the Riemann–Liouville fractional q -integral operator. The uniqueness of solutions to the given problem is proved with the aid of Banach's fixed point theorem. Applying a Shaefer-like fixed point theorem, we also obtain an existence result for the problem at hand. Examples are constructed for illustrating the obtained results. The paper concludes with certain interesting observations concerning the reduction of the results proven in the paper to some new results under an appropriate choice of the parameters involved in the governing equation. [ABSTRACT FROM AUTHOR]

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

In this paper, new solutions of the time-fractional Hirota–Satsuma coupled KdV equation model the intercommunication between two long waves that have well-defined dispersion connection received successfully by the unified method, the improved F -expansion method and the homogeneous balance method. In contrast, these methods are simple and efficient, and can obtain different exact solutions to this equation. By symbolic calculation, polynomial solutions, hyperbolic function solutions, trigonometric function solutions, rational function solutions, etc. are acquired. Furthermore, we plot and analyze some solutions. [ABSTRACT FROM AUTHOR]

This paper investigates the (3 + 1) -dimensional extended Sakovich equation, which represents an essential nonlinear scientific model in the field of ocean physics. The Lie symmetry analysis has been utilized for extracting the non-traveling wave solutions of the (3 + 1) -dimensional extended Sakovich equation. These solutions are investigated through infinitesimal generators, which are obtained from Lie's continuous group of transformations. As there are infinite possibilities for the linear combination of infinitesimal generators, so a one-dimensional optimal system of subalgebra has been established using Olver's standard approach. Moreover, by considering the optimal system of subalgebra, the extended Sakovich equation is converted into a solvable nonlinear PDE through symmetry reductions. Finally, the conservation laws for the governing equation have been derived using Ibragimov's generalized theorem and quasi-self-adjointness condition. [ABSTRACT FROM AUTHOR]

This paper adopts the rational extended sine-cosine and cosh-sinh methods to construct the Bogoyavlenskii equation's exact solutions. To the best of our knowledge, the Bogoyavlenskii equation has not been investigated by aforementioned techniques. In this paper, we find the precise traveling wave solutions of the Bogoyavlenskii equation. Finally, 3D and 2D graphics of the obtained solutions are illustrated for the applicability and reliability of the proposed strategy for various special values. [ABSTRACT FROM AUTHOR]

This paper derives a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation based on the conformable fractional derivative for the first time. Some new types of the fractal traveling wave solutions are successfully constructed by applying a novel approach which is called the fractal semi-inverse variational method. To our knowledge, the obtained results are all new and have not reported in the other literature. In addition, the dynamic characteristics of the different solutions on the fractal space are discussed and presented via the 3D plots, 2D contour and 2D curves. It can be found that: (1) The fractal order can not only affect the peak value of the fractal traveling waves, but also affect the wave structures, that is, the smaller the fractional order value is, the more curved the waveform is, and the slower waveform changes. (2) In the fractal space, the fractal wave keeps its shape unchanged in the process of the propagation and still meets the energy conservation. The methods in this paper can be used to study the other fractal PDEs in the physics, and the findings are expected to bring some new thinking and inspiration toward the fractal theory in physics. [ABSTRACT FROM AUTHOR]

The variational theory has triggered skyrocketing interest in the solitary theory, and the semi-inverse method has laid the foundation for the search for a variational formulation for a nonlinear system. This paper gives a brief review of the last development of the fractal soliton theory and discusses the variational principle for fractal Boussinesq-like B (m , n) equation in the literature. The paper establishes a variational formulation for B (m , 1) equation to show the effectiveness of the semi-inverse method, and a general trial-Lagrange function with two free parameters is established for B (m , n) equation, the identification of the unknown parameters and the unknown function involved in the trial-Lagrange function is shown step by step. This paper opens a new path for the fractal variational theory. [ABSTRACT FROM AUTHOR]

In this paper, we obtain novel solutions of a coupled ϕ 4 , a coupled nonlinear Schrödinger equation and a coupled modified Korteweg de Vries equation which can be re-expressed as a linear superposition of either the sum or the difference of two hyperbolic pulse solutions or the sum of either a two-kink or a kink and an antikink solution. These results demonstrate that the notion of superposed solutions extends to coupled nonlinear equations as well. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Kirchhoff-type parabolic equation with logarithmic nonlinearity. By making a more general assumption about the Kirchhoff function, we establish a new finite time blow-up criterion. In particular, the blow-up rate and the upper and lower bounds of the blow-up time are also derived. These results generalize some recent ones in which the blow-up results were obtained when the Kirchhoff function was assumed to be a very special form. [ABSTRACT FROM AUTHOR]

Finding the solution of the absolute value equations (AVEs) has attracted much attention in recent years. In this paper, we propose and analyze two generalized accelerated overrelaxation (AOR) methods for solving AVEs A x − | x | = b , where A ∈ R n × n is an M -matrix. Furthermore, we discuss the convergence of the methods under some suitable assumptions. Numerical results are given to verify the effectiveness of our methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the q -Shehu transform. Further, we define the generalized hyper-Bessel function. Also, we state the q -Shehu transform for some elementary functions. The present aim in this paper is to obtain the solutions of the q -fractional kinetic equations in terms of the established generalized hyper-Bessel function by applying the established q -Shehu transform. Also, we give some special cases of our main results. At the end of this paper, we give the numerical values and the graphical representations of these solutions by using the software MATLAB. [ABSTRACT FROM AUTHOR]

In this paper, we explore how to generate solitary, peakon, periodic, cuspon and kink wave solution of the well-known partial differential equation Korteweg–de Vries (KdV) by using exp-function and modified exp-function methods. The presented methods construct more efficiently almost all types of soliton solution of KdV equation that can be rarely seen in the history. These methods appear to be straightforward and symbolic calculations are used to solve the problem. All resulting answers are verified for accuracy using the symbolic computation program with M a p l e. To show the physical appearance of the model, 3D plots of all the generated solutions are then displayed. The obtained solutions revealed the compatibility of the proposed techniques which provide the general solution with some free parameters. This is the key benefit of these methods over the other methods. [ABSTRACT FROM AUTHOR]

This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of O (h k) in the energy norm, where k represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach. [ABSTRACT FROM AUTHOR]

In this paper, under the observation of extended modified rational expansion method based on symbolic computation, the multiple solitary wave solutions for nonlinear two-dimensional Jaulent–Miodek Hierarchy (JMH) equation are constructed. In this investigation, we use the computer software Mathematica for the construction of multiple solitary wave solutions. The interested and important things in this work are the multiple solitary wave solutions which have various kinds of physical structures such as kink soliton, periodic traveling wave, bright soliton, anti-kink soliton, dark soliton, combined bright and dark solitons, topological soliton and peakon soliton. We are sure that the various kinds of soliton solutions are found first time by using one method in the existing literature works. On the basis of this research, we can say that the applied technique is very efficient, reliable, fruitful and powerful. The constructed soliton solutions for nonlinear JMH equation will play an important role in the investigation of different physical phenomena in nonlinear sciences. [ABSTRACT FROM AUTHOR]

In this paper, fractal space–time, the Hubble horizon and the energy–momentum tensor are examined in relation to the FLRW metric. It offers a Fractal Friedman equation along with its answer. Also included is the scale factor, which includes fractal structures for closed, flat and open universes. They offer fresh insights into the behavior and evolution of the universe through detailed plots that vividly illustrate their potential cosmological implications. [ABSTRACT FROM AUTHOR]

In general, a perfect fluid spacetime is not a generalized Robertson–Walker spacetime and the converse is also not true. In this paper, it is shown that if a perfect fluid spacetime satisfies the critical point equation, then either the spacetime becomes a generalized Robertson–Walker spacetime and represents dark era or the vorticity of the fluid vanishes as well as the spacetime is expansion free. Besides, we prove that if a generalized Robertson–Walker spacetime with constant scalar curvature satisfies the critical point equation, then the spacetime becomes a perfect fluid spacetime. Next, the existence of critical point equation is established by a non-trivial example. Finally, we discuss the critical point equation in f (r) -gravity. For the model f (r) = r − α (1 − e − r α ) (α = constant and r is the scalar curvature of the spacetime), various energy conditions in terms of the scalar curvature are examined and state that the Universe is in an accelerating phase and satisfies the weak, null, dominant, and strong energy conditions. [ABSTRACT FROM AUTHOR]

In this paper, we first study the dual fractional parabolic equation ∂ t α u (x , t) + (− Δ) s u (x , t) = f (u (x , t)) in  B 1 (0) × ℝ , subjected to the vanishing exterior condition. We show that for each t ∈ ℝ , the positive bounded solution u (⋅ , t) must be radially symmetric and strictly decreasing about the origin in the unit ball in ℝ n . To overcome the challenges caused by the dual nonlocality of the operator ∂ t α + (− Δ) s , some novel techniques were introduced. Then we establish the Liouville theorem for the homogeneous equation in the whole space ∂ t α u (x , t) + (− Δ) s u (x , t) = 0 in  ℝ n × ℝ. We first prove a maximum principle in unbounded domains for antisymmetric functions to deduce that u (x , t) must be constant with respect to x. Then it suffices for us to establish the Liouville theorem for the Marchaud fractional equation ∂ t α u (t) = 0 in  ℝ. To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator ∂ t α , we bring in some new ideas and simpler approaches. Instead of disturbing the antisymmetric function, we employ a perturbation technique directly on the solution u (t) itself. This method provides a more concise and intuitive way to establish the Liouville theorem for one-sided operators ∂ t α , including even more general Marchaud time derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we seek to find solutions of the local time fractional Telegraph equation (LTFTE) by employing the local time fractional reduced differential transform method (LTFRDTM). This method produces a numerical approximate solution having the form of an infinite series that converges to a closed form solution in many cases. We apply LTFRDTM on four different LTFTEs to examine the efficiency of the proposed method. The yielded results established the effectiveness of LTFRDTM as a reliable and solid approach for obtaining solutions of LTFTEs. The solutions coincided with the exact solution in the ordinary case when μ = 1. It also required minimal amount of computational work and saved a lot of time. [ABSTRACT FROM AUTHOR]

This paper deals with the Wong–Zakai approximations and random attractors for stochastic Ginzburg–Landau equations with a white noise. We first prove the existence of a pullback random attractor for the approximate equation under much weaker conditions than the original stochastic equation. In addition, when the stochastic Ginzburg–Landau equation is driven by an additive white noise, we establish the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation tends to zero. [ABSTRACT FROM AUTHOR]

Let B be an one-point extension of a finite-dimensional k -algebra A by a simple A -module at a source point i. In this paper, we classify the τ -tilting modules over B. Moreover, it is shown that there are equations | τ − tilt B | = | τ − tilt A | + | τ − tilt A / 〈 e i 〉 | and | s τ − tilt B | = 2 | s τ − tilt A | + | s τ − tilt A / 〈 e i 〉 |. As a consequence, we can calculate the numbers of τ -tilting modules and support τ -tilting modules over linearly Dynkin type algebras whose square radical are zero. [ABSTRACT FROM AUTHOR]

Hirota's bilinear method (HBM) has been successfully applied to the (2 + 1) -dimensional Pavlov equation to analyze the different wave structures in this paper. The (2 + 1) -dimensional Pavlov equation is used for the study of integrated hydrodynamic chains and Einstein–Weyl manifolds. In our research, we find new solutions in the forms of lump solutions, breather waves, and two-wave solutions. The modulation instability (MI) of the governing model is also discussed. Moreover, a variety of 3D, 2D, and contour profiles are used to illustrate the physical behavior of the reported results. Acquired findings are useful in understanding nonlinear science and its related nonlinear higher-dimensional wave fields. Through the use of Mathematica, the obtained results are verified by inserting them into the governing equation. The strengthening of representative calculations we've made gives us a strong and effective mathematical framework for dealing with the most difficult nonlinear wave problems. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of the time-fractional Gilson–Pickering equation using the Kansa method, in which the multiquadrics were utilized as the radial basis function. To achieve this, a meshless numerical scheme based on the finite difference along with the Kansa method has been presented. First, the finite difference approach has been utilized to discretize the temporal derivative, and subsequently, the Kansa method is employed to discretize the spatial derivatives. The stability and convergence analysis of the numerical scheme are also elucidated in this paper. Furthermore, the soliton solutions have been acquired by implementing the Kudryashov method for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we use the genus properties in critical point theory to prove the existence of infinitely many solutions for fractional p -Laplacian equations of Schrödinger-Kirchhoff type. [ABSTRACT FROM AUTHOR]

In this paper, the analytical solution that describes the evolution in time of the fractal damped Duffing equation subjected to external forces of elliptic type is derived using He's two-scale fractal transform and the elliptic balance method (EBM). This solution predicts the evolution in time of the Duffing equation and unveils qualitative and quantitative system behavior when the values of the fractal parameter varies, and how these affect the frequency, the wavelength, and the oscillation amplitude from the start of the motion. Comparison of the amplitude–time response curves over the selected time-interval with those obtained from numerical simulations confirms the accuracy of the derived analytical solution. [ABSTRACT FROM AUTHOR]

The governing equations for fluid flows, i.e. Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) model equations represent a water wave model. These model equations describe the bidirectional propagating water wave surface. In this paper, an auto-Bäcklund transformation is being generated by utilizing truncated Painlevé expansion method for the considered equation. This paper determines the new bright soliton solutions for (2 + 1) and (3 + 1) -dimensional nonlinear KP-BBM equations. The simplified version of Hirota's technique is utilized to infer new bright soliton solutions. The results are plotted graphically to understand the physical behavior of solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlocal Cahn–Hilliard equation (CHE) in the framework of random walk spaces, which includes as particular cases, the CHE on locally finite weighted connected graphs, the CHE determined by finite Markov chains or the Cahn–Hilliard Equations driven by convolution integrable kernels. We consider different transitions for the phase and the chemical potential, and a large class of potentials including obstacle ones. We prove existence and uniqueness of solutions in L 1 of the Cahn–Hilliard Equation. We also show that the Cahn–Hilliard equation is the gradient flow of the Ginzburg–Landau free energy functional on an appropriate Hilbert space. We finally study the asymptotic behavior of the solutions. [ABSTRACT FROM AUTHOR]