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The extended Gerdjikov–Ivanov equation in dense wavelength division multiplexed system for both kerr and parabolic law nonlinearities have been studied using the exp (- ϕ (ζ)) -expansion method. Completely new parameterized travel wave solutions are extracted. When special values are assigned to the parameters, Kink soliton solutions, periodic soliton solutions, and rational function solutions are derived. Figures have been added to indicate the behavior of solutions in some selected cases. [ABSTRACT FROM AUTHOR]

We extend the results obtained in [14] by introducing a new class of boundary value problems involving non-local dynamic boundary conditions. We focus on the problem to find a solution to a local problem on a domain Ω with non-local dynamic conditions on the boundary ∂ Ω . Due to the pioneering nature of the present research, we propose here the apparently simple case of Ω = (0 , ∞) with boundary { 0 } of zero Lebesgue measure. Our results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. Moreover, in our view, we bring new light to dynamic boundary value problems and the probabilistic description of the associated models. [ABSTRACT FROM AUTHOR]

We study the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers (b n) n ≥ 1 and points of the real line (k n) n ≥ 1 , we explicitily solve the Loewner PDE ∂ f ∂ t (z , t) = - f ′ (z , t) ∑ n = 1 + ∞ 2 b n z - k n 1 - t in H × [ 0 , 1) . Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as t → 1 - . [ABSTRACT FROM AUTHOR]

In the hybrid Bayesian-frequentist approach to hypotheses tests, the power function, i.e. the probability of rejecting the null hypothesis, is a random variable and a pre-experimental evaluation of the study is commonly carried out through the so-called probability of success (PoS). PoS is usually defined as the expected value of the random power that is not necessarily a well-representative summary of the entire distribution. Here, we consider the main definitions of PoS and investigate the power related random variables that induce them. We provide general expressions for their cumulative distribution and probability density functions, as well as closed-form expressions when the test statistic is, at least asymptotically, normal. The analysis of such distributions highlights discrepancies in the main definitions of PoS, leading us to prefer the one based on the utility function of the test. We illustrate our idea through an example and an application to clinical trials, which is a framework where PoS is commonly employed. [ABSTRACT FROM AUTHOR]

This note is concerned with the linear BBM equation on the half-line. Its nonlinear counterpart originally arose as a model for surface water waves in a channel. This model was later shown to have considerable predictive power in the context of waves generated by a periodically moving wavemaker at one end of a long channel. Theoretical studies followed that dealt with qualitative properties of solutions in the idealized situation of periodic Dirichlet boundary conditions imposed at one end of an infinitely long channel. One notable outcome of these works is the property that solutions become asymptotically periodic as a function of time at any fixed point x in the channel, a property that was suggested by the experimental outcomes. The earlier theory is here generalized using complex-variable methods. The approach is based on the rigorous implementation of the Fokas unified transform method. Exact solutions of the forced linear problem are written in terms of contour integrals and analyzed for more general boundary conditions. For C ∞ -data satifisying a single compatibility condition, global solutions obtain. For Dirichlet and Neumann boundary conditions, asymptotic periodicity still holds. However, for Robin boundary conditions, we find not only that solutions lack asymptotic periodicity, but they in fact display instability, growing in amplitude exponentially in time. [ABSTRACT FROM AUTHOR]

In this paper, we report on the discovery of a previously-unknown type of long-range instability phenomenon for the one-dimensional linear Schrödinger (LS) equation on the vacuum spacetime quarter-plane. More specifically, the inhomogeneous LS on the half-line, with generic initial data, boundary conditions and forcing term, is addressed, as an illustrative paradigm of our techniques, in a classical, smooth context via the formula proposed by the linear Fokas' unified transform method. We, first, present a new and suitable decomposition of that formula in the complex plane in order to appropriately interpret various terms appearing in the formula, thus securing convergence in a strictly defined sense. We also write the solution in a form consistent with the fundamental principle of Ehrenpreis and Palamodov. This novel analysis then allows for the necessary rigorous a posteriori verification of the full initial-boundary-value problem, for the first time. This is followed by a thorough investigation of the behavior of the solution near the boundaries of the spatiotemporal domain. We prove that the integrals in this representation converge uniformly to 'prescribed' values and the solution admits a smooth extension up to the boundary only if certain compatibility conditions are satisfied by the data (with direct implications for efficient numerics, well-posedness and control). Importantly, moreover, based on our analysis, we perform an effective asymptotic study of far-field dynamics. This leads to new explicit asymptotic formulae which characterize the properties of the solution in terms of (in)compatibilities of the data at the 'corner' of the quadrant. In particular, we found out that the asymptotic behavior of the solution is sensitive to perturbations of the data at the corner. In all cases, even assuming the initial data to belong to the Schwartz class, the solution loses this property as soon as time becomes positive (implying an infinite-speed type of singularity propagation). Hereby, the recent discovery of a novel type of a long-range instability effect for the Stokes equation is further corroborated and elucidated by revisiting a celebrated lower-order linear evolution partial differential equation (PDE). It thence transpires that our rigorous analytical approach is straightforwardly extendable to other Schrödinger-like evolution equations as well as more general problems with dispersion formulated on domains with a quasi-infinite boundary. Finally, although occurrence of the new instability is most stunning in the case discussed herein, it is naturally conjectured that analogous phenomena shall too appear in variable-coefficient and nonlinear settings which remain to be accordingly investigated. [ABSTRACT FROM AUTHOR]

We consider the 3D Mikhalev system, u t = w x , u y = w t - u w x + w u x , which has first appeared in the context of KdV-type hierarchies. Under the reduction w = f (u) , one obtains a pair of commuting first-order equations, u t = f ′ u x , u y = (f ′ 2 - u f ′ + f) u x , which govern simple wave solutions of the Mikhalev system. In this paper we study higher-order reductions of the form w = f (u) + ϵ a (u) u x + ϵ 2 [ b 1 (u) u xx + b 2 (u) u x 2 ] + ⋯ , which turn the Mikhalev system into a pair of commuting higher-order equations. Here the terms at ϵ n are assumed to be differential polynomials of degree n in the x-derivatives of u. We will view w as an (infinite) formal series in the deformation parameter ϵ . It turns out that for such a reduction to be non-trivial, the function f(u) must be quadratic, f (u) = λ u 2 , furthermore, the value of the parameter λ (which has a natural interpretation as an eigenvalue of a certain second-order operator acting on an infinite jet space), is quantised. There are only two positive allowed eigenvalues, λ = 1 and λ = 3 / 2 , as well as infinitely many negative rational eigenvalues. Two-component reductions of the Mikhalev system are also discussed. We emphasise that the existence of higher-order reductions of this kind is a reflection of linear degeneracy of the Mikhalev system, in particular, such reductions do not exist for most of the known 3D dispersionless integrable systems such as the dispersionless KP and Toda equations. [ABSTRACT FROM AUTHOR]

A novel method is presented for explicitly solving inhomogeneous initial-boundary-value problems (IBVPs) on the half-line for a well-known coupled system of evolution partial differential equations. The so-called double-diffusion model, which is based on a simple, yet general, inhomogeneous diffusion configuration, describes accurately several important physical and mechanical processes and thus emerges in miscellaneous applications, ranging from materials science, heat-mass transport and solid–fluid dynamics, to petroleum and chemical engineering. For instance, it appears in nanotechnology and its inhomogeneous version has recently appeared in the area of lithium-ion rechargeable batteries. Our approach is based on the extension of the unified transform (also called the Fokas method), so that it can be applied to systems of coupled equations. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. The novel solution formulae are also utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. In particular, we prove uniform convergence of the solution to the data, its rapid decay at infinity as well as its smoothness up to (and beyond) the boundary axes, provided certain data compatibility conditions at the quarter-plane corner are satisfied. As a sample of important applications of our analysis and investigation of the boundary behavior of the solution and its derivatives, we both prove a novel uniqueness theorem and construct a 'non-uniqueness counterexample'. These supplement the preceding 'constructive existence' result, within the framework of well-posedness. Moreover, one of the advantages of the unified transform is that it yields representations which are defined on contours in the complex Fourier λ -plane, which exhibit exponential decay for large values of λ . This important characteristic of the solutions is expected to allow for an efficient numerical evaluation; this is envisaged in future numerical-analytic investigations. The new formulae and the findings reported herein are also expected to find utility in the study of questions pertaining to well-posedness for nonlinear counterparts too. In addition, our rigorous approach can be extended to IBVPs for other significant models of mathematical physics and potentially also to higher-dimensional and variable-coefficient cases. [ABSTRACT FROM AUTHOR]

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of τ -functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff–Capel totally discrete Toda equations. The hypergeometric τ -functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre–Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases. [ABSTRACT FROM AUTHOR]

In this article, we propose a new method to construct many new exact solutions with parameters for the generalized KdV–mKdV (GKdV–mKdV) equation with higher-order nonlinear terms. The proposed method is a generalization of the well-known G ′ G , 1 G -expansion method in the case of the hyperbolic function solutions. Also, we use a direct algebraic method based on the generalized Liénard equation to find other diffrent new exact solutions of the above GKdV–mKdV equation. Soliton solutions, periodic solutions, rational functions solutions, hyperbolic functions solutions and symmetrical hyperbolic Lucas functions solutions are obtained. Comparing our new results obtained in this article with the well-known results are given. The generalized G ′ G , 1 G -expansion method presented in this article is straightforward, concise and it can also be applied to other nonlinear partial differential equations in mathematical physics. [ABSTRACT FROM AUTHOR]

Using a version of Hironaka’s resolution of singularities for real-analytic functions, any elliptic multiplier Op (p) of order d > 0 , real-analytic near p - 1 (0) , has a fundamental solution μ 0 . We give an integral representation of μ 0 in terms of the resolutions supplied by Hironaka’s theorem. This μ 0 is weakly approximated in H loc t (R n) for t < d - n 2 by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols. [ABSTRACT FROM AUTHOR]

In this paper, we study the general rogue wave solutions and their patterns in the vector (or M-component) nonlinear Schrödinger (NLS) equation. By applying the Kadomtsev–Petviashvili reduction method, we derive an explicit solution for the rogue wave expressed by τ functions that are determinants of K × K block matrices ( 1 ≤ K ≤ M ) with an index jump of M + 1 . Patterns of the rogue waves for M = 3 , 4 and K = 1 are thoroughly investigated. It is found that when one of the internal parameters is large enough, the wave pattern is linked to the root structure of a generalized Wronskian–Hermite polynomial hierarchy in contrast with rogue wave patterns of the scalar NLS equation, the Manakov system, and many others. Moreover, the generalized Wronskian–Hermite polynomial hierarchy includes the Yablonskii–Vorob’ev polynomial and Okamoto polynomial hierarchies as special cases, which have been used to describe the rogue wave patterns of the scalar NLS equation and the Manakov system, respectively. As a result, we extend the most recent results by Yang et al. for the scalar NLS equation and the Manakov system. It is noted that the case M = 3 displays a new feature different from the previous results. The predicted rogue wave patterns are compared with the ones of the true solutions for both cases of M = 3 , 4 . An excellent agreement is achieved. [ABSTRACT FROM AUTHOR]

In this work we consider a nonlinear parabolic higher order partial differential equation that has been proposed as a model for epitaxial growth. This equation possesses both global-in-time solutions and solutions that blow up in finite time, for which this blow-up is mediated by its Hessian nonlinearity. Herein, we further analyze its blow-up behaviour by means of the construction of explicit solutions in the square, the disc, and the plane. Some of these solutions show complete blow-up in either finite or infinite time. Finally, we refine a blow-up criterium that was proved for this evolution equation. Still, existent blow-up criteria based on a priori estimates do not completely reflect the singular character of these explicit blowing up solutions. [ABSTRACT FROM AUTHOR]

In this work, the generalized scale-invariant analog of the Korteweg–de Vries equation is studied. For the first time, the tanh–coth methodology is used to find traveling wave solutions for this nonlinear equation. The considered generalized equation is a connection between the well-known Korteweg–de Vries (KdV) equation and the recently investigated scale-invariant of the dependent variable (SIdV) equation. The obtained results show many families of solutions for the model, indicating that this equation also shares bell-shaped solutions with KdV and SIdV, as previously documented by other researchers. Finally, by executing the symbolic computation, we demonstrate that the used technique is a valuable and effective mathematical tool that can be used to solve problems that arise in the cross-disciplinary nonlinear sciences.

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces B p , r s with s > 1 + 1 p and s = 1 + 1 p , r = 1 , p ∈ [ 1 , ∞). Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces. [ABSTRACT FROM AUTHOR]

We consider uniform plane Hall plates with an arbitrary number of holes exposed to a uniform perpendicular magnetic field of arbitrary strength. The plates have extended contacts on the outer boundary and on the boundaries of the holes. No symmetry is presumed. Pairs of complementary Hall plates are defined, where contacts and insulating segments on the boundaries are swapped. A unique stream function exists in these multiply-connected domains if the total current through each boundary vanishes. Then the voltages between neighbouring contacts and the currents into contacts of pairs of complementary Hall plates are linked in a peculiar way, which results in identical power density. Relations between the impedances of complementary multiply-connected Hall plates are derived. The prominent role of complementary symmetric Hall plates—with or without holes—is revealed. For arbitrary magnetic field, their resistance equals the one of a square without holes and with contacts fully covering two opposite edges. [ABSTRACT FROM AUTHOR]

This work is devoted to get a new family of analytical solutions of the (2+1)-coupled dispersive long wave equations propagating in an infinitely long channel with constant depth, and can be observed in an open sea or in wide channels. The solutions are obtained by using the invariance property of the similarity transformations method via one-parameter Lie group theory. The repeated use of the similarity transformations method can transform the system of PDEs into system of ODEs. Under adequate restrictions, the reduced system of ODEs is solved. Numerical simulation is performed to describe the solutions in a physically meaningful way. The profiles of the solutions are simulated by taking an appropriate choice of functions and constants involved therein. In each animation, a frame for dominated behavior is captured. They exhibit elastic multisolitons, single soliton, doubly solitons, stationary, kink and parabolic nature. The results are significant since these have confirmed some of the established results of S. Kumar et al. (2020) and K. Sharma et al. (2020). Some of their solutions can be deduced from the results derived in this work. Other results in the existing literature are different from those in this work.

The generalized Hirota-Satsuma coupled KdV system with fractional-order derivative plays a significant role to simulate the interaction of nearby identical-weight particles in a crystal lattice structure, two long waves interaction with different dispersion relationships, the description of shallow-water wave propagation, ion-acoustic waves, plasma physics and some other fields. In this study, diverse analytical wave solutions in general and standard structures are established of the stated equation by introducing two viable techniques, namely, the generalized Kudryashov scheme and the two variables G'/G,1/G-expansion approach. The solutions are established in terms of elementary functions, specifically trigonometric functions, rational functions, exponential functions, and hyperbolic functions. For definite values of free parameters, the obtained analytical wave solutions transform into solitary wave solutions. The graphical patterns of the wave solutions with there-, two-, and contour plots are depicted magnificently to elucidate the internal structure of the phenomenon. The methods contribute as powerful mathematical tools and appear to be further effective, computerized, and user-friendly to investigate nonlinear fractional equations as well as comprehensive analytical solutions of nonlinear evolution equations in engineering, technology, and mathematical physics.

This paper shows the impact of randomness on solutions to the Heisenberg ferromagnetic spin chain equation utilizing the consolidated solver approach. Namely, we construct solitary waves in forms of rational, trigonometric and hyperbolic solutions via arbitrary parameters. The unified approach presents the closed form of solutions. The reported solutions are very significant for explaining different complex phenomena in plasma physics, chemical engineering, optical fiber, super fluids, quantum mechanics. The proposed study depicts that the proposed technique is efficacious and sturdy. Beta and exponential statistical distributions are selected to represent the neighboring interaction along the diagonal random input. To describe the attitude of the stochastic wave solutions, statistical properties such as the expectation and the probability density function are depicted graphically via suitable parameters. Finally, we study the effect of the randomness on the solutions via beta and exponential statistical distributions. [ABSTRACT FROM AUTHOR]

Nonlinear evolution equations (NLEEs) are frequently employed to determine the fundamental principles of natural phenomena. Nonlinear equations are studied extensively in nonlinear sciences, ocean physics, fluid dynamics, plasma physics, scientific applications, and marine engineering. The generalized exponential rational function (GERF) technique is used in this article to seek several closed-form wave solutions and the evolving dynamics of different wave profiles to the generalized nonlinear wave equation in (3+1) dimensions, which explains several more nonlinear phenomena in liquids, including gas bubbles. A large number of closed-form wave solutions are generated, including trigonometric function solutions, hyperbolic trigonometric function solutions, and exponential rational functional solutions. In the dynamics of distinct solitary waves, a variety of soliton solutions are obtained, including single soliton, multi-wave structure soliton, kink-type soliton, combo singular soliton, and singularity-form wave profiles. These determined solutions have never previously been published. The dynamical wave structures of some analytical solutions are graphically demonstrated using three-dimensional graphics by providing suitable values to free parameters. This technique can also be used to obtain the soliton solutions of other well-known equations in engineering physics, fluid dynamics, and other fields of nonlinear sciences.

The generalized exponential rational function (GERF) method is used in this work to obtain analytic wave solutions to the Kudryashov-Sinelshchikov (KS) equation. The KS equation depicts the occurrence of pressure waves in mixtures of liquid-gas bubbles while accounting for thermal expansion and viscosity. By applying the GERF method to the KS equation, we obtain analytic solutions in terms of trigonometric, hyperbolic, and exponential functions, among others. These solutions include solitary wave solutions, dark-bright soliton solutions, singular soliton solutions, singular bell-shaped solutions, traveling wave solutions, rational form solutions, and periodic wave solutions. We discuss the two-dimensional and three-dimensional graphics of some obtained solutions under the accurate range space by selecting appropriate values for the involved arbitrary parameters to make this research more praiseworthy. The obtained analytic wave solutions specify the GERF method’s dependability, capability, trustworthiness, and efficiency.

This paper systematically investigates the exact solutions to an extended (2+1)-dimensional Boussinesq equation, which arises in several physical applications, including the propagation of shallow-water waves, with the help of the Lie symmetry analysis method. We acquired the vector fields, commutation relations, optimal systems, two stages of reductions, and exact solutions to the given equation by taking advantage of the Lie group method. The method plays a crucial role to reduce the number of independent variables by one in each stage and finally forms an ODE which is solved by taking relevant suppositions and choosing the arbitrary constants that appear therein. Furthermore, Lie symmetry analysis (LSA) is implemented for perceiving the symmetries of the Boussinesq equation and then culminating the solitary wave solutions. The behavior of the obtained results for multiple cases of symmetries is obtained in the present framework and demonstrated through three-and two-dimensional dynamical wave profiles. These solutions show single soliton, multiple solitons, elastic behavior of combo soliton profiles, and stationary waves, as can be seen from the graphics. The outcomes of the present investigation manifest that the considered scheme is systematic and significant to solve nonlinear evolution equations.

This paper is devoted to the global conservative solutions of a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solutions of the semilinear system. Returning to the original variables, we get the global conservative solution of the equation. Finally, we show that the peakon solutions of the equation still conserve in H 1 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the characteristic Goursat problem with regular singular data for a class of linear partial differential equations of second order in the complex domain. We give an explicit representation of the solutions. Such representations permit to see the structure of the solutions and to show that they are singular on the union of the characteristic surfaces issued from data. [ABSTRACT FROM AUTHOR]

The (1 + 1) -and (1 + 2) -dimensional inviscid Rossby wave equations are analysed using Lie symmetry techniques. The travelling-wave reductions for the 1 D equation lead to a third-order ordinary differential equation from which the propagation properties are derived. It is observed that the wave has easterly phase velocity and westerly group velocity. Also, the wave propagates slightly faster in the f-plane than the β-plane. The dispersion relation derived from the third-order equation shows that the 2 D Rossby equation transports energy both eastward and westward and its speed is reduced successively and the propagation remains eastward. As for the two-dimensional Rossby wave equation, certain solutions which behave like solitary waves after a certain time are plotted. Interestingly, for certain symmetries the reductions lead to the Riccati's, Abel's, Euler's type and to some linearized equations. Moreover, certain reductions lead to highly nonlinear equations which are analysed by the singularity analysis method. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the local well-posednesss for the Cauchy problem of a generalized Camassa–Holm (gCH) equation in Besov spaces B p , 1 1 + 1 p with 1 ≤ p < + ∞. Then we gain two blow-up criterions, and present two new blow-up results. Finally, we prove the ill-posedness of the gCH equation in critical Besov spaces B 2 , r 3 2 , r ∈ (1 , + ∞ ]. [ABSTRACT FROM AUTHOR]

In this article, we present a new form of solutions to the Chiral/perturbed Chiral nonlinear Schrödinger equation, using the unified solver method. This solver introduces the closed formula in an explicit way. In each problem, different families of elliptic functions solutions are found under suitable conditions. Moreover, some cases of hyperbolic solutions are obtained. The acquired solutions may be applicable for explaining various complex phenomena in the quantum Hall effect. The presented study displays that the proposed solver is simple, powerful and sturdy. For more details about the physical dynamical representation of the presented solutions, we introduce some graphs of some selected solutions using Matlab software. Finally, the presented solver can be used to solve many other models in real life problems. [ABSTRACT FROM AUTHOR]

In this work, meticulous solutions of the three-dimensional Zakharov-Kuznetsov equation are gained. Lie group analysis is utilized to convey the integration of the underlying equation. We also employ the ansatz method to obtain further exact solutions. The results obtained are solitary waves. Moreover, conservation laws are derived by using the characteristic method. [ABSTRACT FROM AUTHOR]

The whole world is suffering from a pandemic virus known as COVID-19 nowadays. This virus is also known as novel corona virus. There are various statements coming out regarding its generation but till now one thing is coming is that its origin place seems to be a lab somewhere in Wuhan city of China. It is also said off and on the record that covid-19 virus came into existence near November 2019 but on record, the WHO office (China) reported on 31st December 2019 that an unknown virus is detected in Wuhan city. On 7th January 2020, Chinese authorities announced about the existence of this virus. At the initial stage, it was supposed that this virus might be bat-origin but after that several other statements are coming day by day. Authors are not going to discuss about the causes of its origin. But since this virus has become the pandemic around the world so it is necessary to study this virus and its nature to predict the future outbreaks and also control strategies by using mathematical modeling. In this paper, we are going to study the model by using Atangana-Baleanu fractional operator. Also we calculated the numeric results with graphical representation. [ABSTRACT FROM AUTHOR]

In this work, we emphasize the use of singularity analysis in obtaining analytic solutions for equations for which standard Lie point symmetry analysis fails to make any lucid decision. We study the higher-dimensional Kadomtsev–Petviashvili, Boussinesq, and Kaup–Kupershmidt equations in a more general sense. With higher-order equations, there can be a commensurate number of resonances and when consistency for the full equation is examined at each resonance the constant of integration is supposed to vanish from the expression so that it remains arbitrary, but if there is an instance of this not happening, the consistency can be partially established by giving the offending constant the value from the defining equation. If consistency is otherwise not compromised, the equation can be said to be partially integrable, i.e., integrable on a surface of the complex space. Furthermore, we propose an approach that is meant to magnify the scope of singularity analysis for equations admitting higher values for resonances or positive leading-order exponent. [ABSTRACT FROM AUTHOR]

This work deals with applying a rapidly convergent approximation method to obtain some (weak) nonsmooth solitary or periodic solutions of a generic version of the Korteweg-de Vries and the Benjamin-Bona-Mahony equations, the generalized Gardner equation with dual power-law nonlinearity. A novel approach has been adopted to derive conditions among parameters for which the obtained solution may be bounded. Explicit parameter dependence of few constants of the motion corresponding to the nonsmooth solitary or periodic solutions obtained here has been presented. Results derived here may be helpful in interpretations of large-amplitude internal waves in the ocean and for irregularly large-amplitude waves in other fields of nonlinear physics, e.g., optics and dusty- or magneto-plasmas. [ABSTRACT FROM AUTHOR]

The physical principles of natural occurrences are frequently examined using nonlinear evolution equations (NLEEs). Nonlinear equations are intensively investigated in mathematical physics, ocean physics, scientific applications, and marine engineering. This paper investigates the Boiti-Leon-Manna-Pempinelli (BLMP) equation in (3+1)-dimensions, which describes fluid propagation and can be considered as a nonlinear complex physical model for incompressible fluids in plasma physics. This four-dimensional BLMP equation is certainly a dynamical nonlinear evolution equation in real-world applications. Here, we implement the generalized exponential rational function (GERF) method and the generalized Kudryashov method to obtain the exact closed-form solutions of the considered BLMP equation and construct novel solitary wave solutions, including hyperbolic and trigonometric functions, and exponential rational functions with arbitrary constant parameters. These two efficient methods are applied to extracting solitary wave solutions, dark-bright solitons, singular solitons, combo singular solitons, periodic wave solutions, singular bell-shaped solitons, kink-shaped solitons, and rational form solutions. Some three-dimensional graphics of obtained exact analytic solutions are presented by considering the suitable choice of involved free parameters. Eventually, the established results verify the capability, efficiency, and trustworthiness of the implemented methods. The techniques are effective, authentic, and straightforward mathematical tools for obtaining closed-form solutions to nonlinear partial differential equations (NLPDEs) arising in nonlinear sciences, plasma physics, and fluid dynamics.

A method is developed to determine the Hall voltage for the case of a Hall plate exposed to a piecewise constant magnetic field. The electromagnetic field is discontinuous at the interface between any two subdomains. The conditions satisfied by the electric field on a discontinuity curve (interface) inside a Hall plate with four contacts are obtained as: (a) the continuity of the potential function along the whole plate and (b) a discontinuity relationship for the flux function across the interface in terms of the potential function on the interface. An application of the method is given for the case of four point contacts where the voltage along the interface is computed using a complex variable interface element method. This voltage is used to calculate the Hall voltage of the entire plate structure. [ABSTRACT FROM AUTHOR]

The present study investigates the influence of uniform magnetic field on the flow of a Newtonian fluid sandwiched between two micropolar fluid layers through a rectangular (horizontal) porous channel. Fluid flow in the every region is steady, incompressible and the fluids are immiscible. Uniform magnetic field is applied in a direction perpendicular to the direction of fluid motion. The governing equations of micropolar fluid are expressed in Eringen's approach and further modified by Nowacki's approach. For respective porous channels, expressions for linear velocity, microrotations, stresses (shear and couple) are obtained analytically. Continuity of velocities, continuity of microrotations and continuity of stresses are employed at the porous interfaces; conditions of no slip and no spin are applied at the impervious boundaries of the composite channel. Numerical values of flow rate, wall shear stresses and couple stresses at the porous interfaces are evaluated by MATHEMATICA and listed in tables. Graphs of flow rate and fluid velocity are plotted and their behaviors discussed. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equation without any additional assumptions. Firstly, introducing a new set of variables, we transform the single-cycle pulse equation into an equivalent semilinear system. Using the standard ordinary differential equation theory, the global solution of the semilinear system is studied. Secondly, returning to the original coordinates, we get a global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, choosing some vital test functions which are different from [Bressan (Discrete Contin. Dyn. Syst 35:25-42, 2015), Brunelli (Phys. Lett. A 353:475-478, 2006)], we find a equation to single out a unique characteristic curve through each initial point. Moreover, the uniqueness of global admissible conservative weak solution is obtained. [ABSTRACT FROM AUTHOR]

We present a method to obtain explicit solutions of the complex eikonal equation in the plane. This equation arises in the approximation of the Helmholtz equation by the WKBJ or Evanescent Wave Tracking methods. We obtain the complex-valued solutions (called eikonals) as parametrizations in a complex variable. We consider both the cases of constant and non-constant index of refraction. In both cases, the relevant parametrizations depend on some holomorphic function. In the case of a non-constant index of refraction, the parametrization also depends on some extra exponential complex-valued function and on a quasi-conformal homeomorphism. This is due to the use of the theory of pseudo-analytic functions and the related similarity principle. The parametrizations give information about the formation of caustics and the light and shadow regions for the relevant eikonals. [ABSTRACT FROM AUTHOR]

In this paper we like to explore the full power of Adomian decomposition method (ADM), specially its symbolic capability. We will demonstrate the standard ADM and ADM with integration factor to compute explicit closed form solutions of first order scalar partial differential equations with unprescribed initial conditions, and even with parameters. These features are those numerical methods fail to do. Our examples include linear/nonlinear, constant/variable coefficients and homogeneous/nonhomogeneous equations. The method of characteristics is also tested and compared with these two ADM methods. We conclude that ADM is excellent among all existing methods.

In this paper we investigate solutions to the characteristic Goursat problem with analytic data for a class of linear operators of second order without the Levi's generalized condition. We give an explicit representation of the solutions of the problem involving Kummer functions and show that they are generally singular on the surface K tangent to the characteristic surface S. [ABSTRACT FROM AUTHOR]

In the paper, similarity solutions are constructed for a model equation with multiple characteristics of an arbitrary integer order. It is shown that the structure of similarity solutions depends on the mutual simplicity of the orders of derivatives with respect to the variable x and y, respectively. Frequent cases are considered in which they are shown as fundamental solutions of well-known equations, expressed in a linear way through the constructed similarity solutions. [ABSTRACT FROM AUTHOR]

In this article, we established an application of homotopy perturbation method using Laplace transform (LT-HPM) to elaborate the analytical solution of heat conduction equation in the heterogeneous casting-mould system. The solution of the problem is provided with our supposition of an ideal contact between the cast and the mould. In the proposed method, we have chosen initial approximations of unknown constants which can be implemented by imposing the boundary and initial conditions. Examples have been discussed and confirmed the usefulness of this method. [ABSTRACT FROM AUTHOR]

We study some functionals associated with a process driven by a fractional boundary value problem (FBVP for short). By FBVP we mean a Cauchy problem with boundary condition written in terms of a fractional equation, that is an equation involving time-fractional derivative in the sense of Caputo. We focus on lifetimes and additive functionals characterizing the boundary conditions. We show that the corresponding additive functionals are related to the fractional telegraph equations. Moreover, the fractional order of the derivative gives a unified condition including the elastic and the sticky cases among the others. [ABSTRACT FROM AUTHOR]