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This research paper investigates the behavior of optical fiber pulses by studying the solitary wave solutions of the perturbed Chen–Lee–Liu (CLL) equation. The perturbed CLL equation is derived from the perturbed Schrödinger equation. Two analytical approaches for obtaining the innovative soliton-wave solutions are presented in this paper, and their reliability is examined by using a well-established numerical method. The accuracy of pulse wave analysis in an optical cable is demonstrated through a series of graphical representations. Our study introduces scientific novelty, as evidenced by the comparative analysis of our data with those of previous research papers. This research contributes to enhancing the understanding of the kinetics and physical behavior of pulses in optical fibers, which holds implications for the advancement of optical communication technologies. [ABSTRACT FROM AUTHOR]

In this paper, we establish a generalized version of Gibbons’ conjecture in the context of the master equation (∂t − Δ)su(x,t) = f(t,u(x,t))in ℝn × ℝ. We show that, for each t ∈ ℝ, the bounded entire solution u(x,t) must be monotone increasing in one direction, and furthermore it is one-dimensional symmetric under certain uniform convergence assumption on u and an appropriate decreasing condition on f. These conditions are slightly weaker than their counter parts proposed in the original Gibbons’ conjecture. To overcome the difficulties in proving the Gibbons’ conjecture and the impediments caused by the strong correlation between space and time of fully fractional heat operator (∂t − Δ)s, we introduce some new ideas and provide several new insights. More precisely, we first derive a weighted average inequality, which not only provides a straightforward proof for the maximum principle in bounded domains, but also plays a crucial role in further deducing the maximum principle in unbounded domains. Such average inequality and maximum principles are essential ingredients to carry out the sliding method, and then we apply this direct method to prove the Gibbons’ conjecture in the setting of the master equation. It is important to note that the holistic approach developed in this paper is highly versatile, and will become useful tools in investigating various qualitative properties of solutions as well as in establishing the Gibbons’ conjecture for a broad range of fractional elliptic and parabolic equations and systems. [ABSTRACT FROM AUTHOR]

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures. [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]

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]

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 take variable separation method to study Klein–Kramers (KK) equation. By choosing different eigenvalues and noise functions, we can get different probability density functions (PDFs) of KK equation. These PDFs contain not only normal distributions but also other distributions that correspond to anomalous diffusion phenomena. For example, power-law distribution, truncated Cauchy–Lorentz distribution, Weibull distribution, log-logistic distribution, Gamma distribution. We also show the 3D and 2D profiles of these PDFs to analyze the corresponding dynamic properties and illustrate the possible practical applications of these results. In addition, we also find some exact solutions that are not PDFs. They are also listed to ensure the completeness of the results and to illustrate the potential applications of these exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak strong uniqueness of the Dirichlet type problems of fractional Laplace (Poisson) equations. We construct the Green’s function and the Poisson kernel. We then provide a somewhat sharp condition for the solution to be unique. We also show that the solution under such condition exists and must be given by our Green’s function and Poisson kernel. In doing these, we establish several basic and useful properties of the Green’s function and Poisson kernel. Based on these, we obtain some further a priori estimates of the solutions. Surprisingly those estimates are quite different from the ones for the local type elliptic equations such as Laplace equations. These are basic properties to the fractional Laplace equations and can be useful in the study of related problems. [ABSTRACT FROM AUTHOR]

The center of this paper is to establish the generalized variational structure (GVS) of the (3 + 1) -dimensional Zakharov–Kuznetsov–Burgers equation (ZKBe) by taking advantage of the Semi-inverse method (SIM). Two different GVSs are extracted and the derivation process is presented in detail. The extracted GVSs reveal the energy conservation law and can offer some new insights on the study of the variational method. [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 attains cubic-quartic optical solitons of the Biswas–Milovic equation with dual power law and Kerr law nonlinearities. Dark, bright, singular and combo-singular solitons solutions are obtained with the F-expansion method. The solutions gained in our this work are newer than the obtained solutions using different method before this work in the literature. In order to better convey the physical significance of the solutions, we also include 2D and 3D graphics. The computational software MATHEMATICA is the helping tool for visualizing and analyzing the derived solutions. [ABSTRACT FROM AUTHOR]

This paper presents a numerical technique to solve the generalized fractional Lane–Emden–Fowler equation. We use the Haar wavelet technique to approximate the solution of the generalized fractional Lane–Emden–Fowler equation. In this method, a differential equation transforms into a system of nonlinear equations, and this system is further solved to obtain Haar coefficients. We also showed the convergence of this method by establishing an upper bound. The rate of convergence is also discussed in our algorithm. Many numerical examples have been taken, and the results of those numerical experiments have been written in tabular form. We also documented the experiments graphically to show the efficiency and accuracy of this method. [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, we have conducted and investigated the existence of solutions for a fourth-order quasilinear elliptic equation. This equation incorporates a perturbed 1-biharmonic problem, represented as Δ 1 2 v = g (x , v) + h (x). To achieve this, we established two distinct sets of assumptions for the function g and demonstrated that each set of conditions yields solutions with unique characteristics. Our approach is based on variational method. [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]

By means of He's fractal derivative, a new fractal (2 + 1)-dimensional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation is extracted in this paper. The semi-inverse method is employed to establish the generalized fractal variational principle. The generalized fractal variational principle can show the conservation laws through the energy form in the fractal space. Moreover, some semi-domain solutions are also explored by applying the variational approach and the one-step method namely Wang's direct mapping method-II. The dynamics of the extracted solutions on the Cantor set are unveiled graphically. The findings of this study are expected to provide some new insights into the exploration of the fractal PDEs. [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]

In this paper, a reliable and effective local discontinuous Galerkin (LDG) scheme for numerically solving the classical Rosenau–Hyman equation with non-periodic boundary conditions has been proposed. This study employs the third-order nonlinearly stable total variation diminishing Runge–Kutta method and the LDG method, respectively, to discretize the temporal and spatial derivatives. Finally, numerical simulations are performed on various test problems and compared with the exact results as well as results produced by a few other numerical methods, to analyze the reliability and efficiency of the proposed method. The results generated, which validate the expected order of accuracy, are presented through multiple tables. In addition, several graphical representations of the problem are presented to depict the behavior of the solution. [ABSTRACT FROM AUTHOR]

The aim of this review paper is to revisit the geometric framework of the Teukolsky equation in a form that is suitable for analysts working on this equation. We introduce spinor bundles, the Newman–Penrose formalism and the Geroch–Held–Penrose (GHP) formalism. In particular, we develop the case of Kerr spacetimes, for which we provide detailed computations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the monotonicity and one-dimensional symmetry of solutions for parabolic equations related to the weighted fractional Laplacian on the whole space. We first establish a generalized weighted average inequality and the maximum principle in unbounded domains, and then carry out the sliding method to obtain monotonicity and one-dimensional symmetry of solutions. [ABSTRACT FROM AUTHOR]

This paper addresses the new (3 + 1) -dimensional Mikhailov–Novikov–Wang (MNW) equation with arbitrary order derivative and presents novel exact solutions of it by implementing exp (− φ (ξ)) -expansion, modified Kudryashov, generalized (G ′ / G) -expansion, and modified extended tanh-function methods. This equation emphasizes significant connection between the integrability and water waves' phenomena. Employing the conformable derivative definition, a variety of soliton (bright, dark, anti-kink) solutions of the model are obtained. Therefore, it would appear that these approaches might yield noteworthy results in producing the exact solutions to the fractional differential equations in a wide range. In addition, 2D, 3D, and contour plots of the solutions are drawn for specific values to demonstrate the physical behaviors of the solutions. [ABSTRACT FROM AUTHOR]

Let (M n , F , m) be a compact Finsler manifold. This paper will obtain a global estimate and a differential Harnack inequality for positive solutions to the Newell–Whitehead–Segal equation ∂ t u (x , t) = Δ m u (x , t) + a u − b u 3 ,   (x , t) ∈ M × [ 0 , T ] , on compact Finsler manifolds, where a , b > 0 are real constants. [ABSTRACT FROM AUTHOR]

The Novikov equation is a generalization of the Camassa–Holm one. It models the propagation of unidirectional shallow water waves on a flat bottom. In this paper, we prove the well-posedness of H 2 and H 3 solutions of the Cauchy problem, associated with this equation. [ABSTRACT FROM AUTHOR]

In this paper, we continue the rigorous study of classical effective field theories (EFTs) that was recently initiated in the work of Reall and Warnick. We study a system with one light and one heavy field with cubic coupling and prove global existence [of the ultraviolet (UV) solution] under an effective norm in the high mass limit. Furthermore, we prove that the global solution linearly scatters (in Lax–Philips sense) and that this final state has an expansion in inverse powers of the mass, and can be recovered from the EFT equation alone. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of bi-fractional Black–Scholes equation derived by the fractional G-Brownian motion. Then some exact solutions are obtained and the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, the new conservation theorem and the generalization of the Noether operators are developed to construct the conservation laws for bi-fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

In this paper, we conduct a comprehensive exploration of the relativistic quantum dynamics of spin-0 scalar particles, as described by the Duffin–Kemmer–Petiau (DKP) equation, within the framework of a magnetic space-time. Our focus is on the Bonnor–Melvin–Lambda (BML) solution, a four-dimensional magnetic universe characterized by a magnetic field that varies with axial distance. To initiate this investigation, we derive the radial equation using a suitable wave function ansatz and subsequently employ special functions to solve it. Furthermore, we extend our analysis to include Duffin–Kemmer–Petiau oscillator fields within the same BML space-time background. We derive the corresponding radial equation and solve it using special functions. Significantly, our results show that the geometry's topology and the cosmological constant (both are related to the magnetic field strength) influence the eigenvalue solution of spin-0 DKP fields and DKP-oscillator fields, leading to substantial modifications in the overall outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we study the decaying dynamics in the mirror-field interaction by means of the intrinsic decoherence scheme. Factorization of the mirror-field Hamiltonian with the use of displacement operators allows us to calculate the explicit solution to Milburn's equation for arbitrary initial conditions. We show expectation values, correlations, and Husimi functions for the solutions obtained. [ABSTRACT FROM AUTHOR]

In this paper, the Titchmarsh–Weyl theory of the impulsive q -Dirac equation is studied. [ABSTRACT FROM AUTHOR]

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

The Fokker–Planck equation (FPE) is an important deterministic tool for investigating stochastic dynamical systems. In this paper, we consider the space-time fractional FPE driven by multiplicative Marcus Lévy noises. Efficient numerical schemes are presented to solve the equations. Stability and convergence of the methods are also discussed. We give some numerical experiments to validate our schemes, and examine the effects of parameters on solutions. Additionally, we analyze the maximal likely trajectories and the critical time for the change of the most probability location. [ABSTRACT FROM AUTHOR]

On pp. 243 and 244 of his second notebook, Ramanujan recorded seven modular equations of degree 11 followed by two more which he had crossed out. The first modular equation in the list was proved by Berndt using theta function identities. The same was also proved by Venkatachaliengar in a different way. To the best of our knowledge, the only available proofs to the other six modular equations in the list are by Berndt. These proofs employ the theory of modular forms. Interestingly these are the first set of modular equations, the proofs to which by Berndt take a shift from the elementary algebraic methods to the theory of modular forms. In this paper, we reprove all these modular equations of degree 11 (except the first one in the list) by elementary algebraic techniques. First, we use two series identities due to Ye and Liu to prove one of the modular equations and its reciprocal. The remaining modular equations are proved by the method of parametrization. [ABSTRACT FROM AUTHOR]

The fractional derivatives have been widely applied in many fields and has attracted widespread attention. This paper extracts a new fractional (3+1)-dimensional modified Zakharov–Kuznetsov equation (MZKe) with the local fractional derivative (LFD) for the first time. Two special functions, namely, the LT δ (Ξ δ) and LC δ (Ξ δ) functions that are derived on the basis of the Mittag-Leffler function (MLF) defined on the Cantor set (CS), are employed to construct the auxiliary trial function to look into the exact solutions (ESs). Aided by Yang's non-differentiable (ND) transformation, six groups of the ND ESs are found. The ND ESs on the CS for δ = ln 2 / ln 3 are depicted graphically. Additionally, as a comparison, the ESs of the classic (3+1)-dimensional MZKe for δ = 1 are also illustrated. The outcomes reveal that the derived method is powerful and effective, and can be used to deal with the other local fractional PDEs. [ABSTRACT FROM AUTHOR]

This paper focuses on the numerical investigation of the fractal modification of the (3+1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation. A variational approach based on the two-scale fractal complex transformation and the variational principle is presented for solving this fractal equation. The fractal potential-YTSF equation can be transformed as the original potential-YTSF equation by means of the fractal complex transformation. Some fractal soliton-type solutions and fractal periodic wave solutions are provided by using the variational principle proposed by He, which are not touched in the existing literature. Some remarks about the variational formulation and the wave solutions for the original potential-YTSF equation by Manafian et al. [East Asian J. Appl. Math. 10(3) (2020) 549–565] are also given. Numerical results of the fractal wave solutions with different fractal dimensions and amplitudes are presented to show the propagation behavior. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the Zakharov–Kuznetsov (ZK) equation in the n -dimensional setting with n ≥ 2 and investigate its smoothness properties. We extend the well-known regularity propagation phenomenon observed in the 2D and 3D cases, where the regularity of the initial data on certain half-spaces propagates with infinite speed, to the case where the regularity of the initial data is measured on a fractional scale. To achieve this, we introduce new localization formulas that enable us to describe the regularity of the solution on a specific class of subsets in Euclidean space. This work provides insights into the regularity behavior of solutions of the ZK equation in higher dimensions and with more general initial data. [ABSTRACT FROM AUTHOR]

This research uses the Sinc Collocation method to numerically examine the Degasperis–Procesi and Benjamin–Bona–Mahony equations, achieving a high level of precision and accuracy on computational grounds with a variety of mesh points. The proposed technique involves global collocation using Sinc bases function (SBF) as an activation function. Initially, the time derivative discretization has been performed through finite difference scheme and the remaining spatial derivatives are approximated by θ -weighted scheme to minimize the complexity in numerical computations. The main contribution of research is in modifying the proposed scheme to overcome the complexity involved in the whole process of finding solutions with a cardinal expansion of Sinc functions in nonlinear and singular systems. The solutions of Degasperis–Procesi and Benjamin–Bona–Mahony stiff nonlinear systems are illustrated numerically and graphically and the validation and verification of the obtained results are examined through stability analysis by calculating the spectral radius by bounding the value of θ. A comparative study is presented with some of the well-known numerical techniques, which assured the accuracy and reliability of our technique. [ABSTRACT FROM AUTHOR]

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

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety X Σ . The algorithm lends its name from a construction, described by Cox, of X Σ as a GIT quotient X Σ = (ℂ k ∖ Z) / / G of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space ℂ k of X Σ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of X Σ . It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the G -orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of X Σ . In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound. [ABSTRACT FROM AUTHOR]

We define a conformally invariant action on gauge connections on a closed pseudo-Riemannian manifold M of dimension 6. At leading order this is quadratic in the gauge connection. The Euler–Lagrange equations of , with respect to variation of the gauge connection, provide a higher-order conformally invariant analogue of the (source-free) Yang–Mills equations. For any gauge connection A on M , we define (A) by first defining a Lagrangian density associated to A. This is not conformally invariant but has a conformal transformation analogous to a Q -curvature. Integrating this density provides the conformally invariant action. In the special case that we apply to the conformal Cartan-tractor connection, the functional gradient recovers the natural conformal curvature invariant called the Fefferman–Graham obstruction tensor. So in this case, the Euler–Lagrange equations are exactly the "obstruction-flat" condition for 6-manifolds. This extends known results for 4-dimensional pseudo-Riemannian manifolds where the Bach tensor is recovered in the Yang–Mills equations of the Cartan-tractor connection. [ABSTRACT FROM AUTHOR]

It is well-known that solutions to deterministic nonlocal aggregation-diffusion models may blow up in two or higher dimensions. Various mechanisms hence have been proposed to "regularize" the deterministic aggregation-diffusion equations in a manner that allows pattern formation without blow-up. However, stochastic effect has not been ever considered among other things. In this work, we consider a nonlocal aggregation-diffusion model with multiplicative noise and establish the local existence and uniqueness of strong solutions on ℝ d (d ≥ 2). If the noise is non-autonomous and linear, we establish the global existence and large-time behavior of strong solutions with decay properties by combining the Moser-Alikakos iteration technique and some decay estimates of Girsanov type processes. If the noise is nonlinear and strong enough, we show that blow-up can be prevented. As such, our results assert that certain multiplicative noise can also regularize the aggregation-diffusion model. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with generating solutions of a class of Convection–Diffusion–Reaction (CDR) equation from the solutions of another CDR equation through the Darboux transformations. The method is elucidated by cases with certain types of the reaction coefficients. We have also discussed briefly supersymmetric pairs of CDR equations connected through similarity solutions. [ABSTRACT FROM AUTHOR]