Solution Of Equation Research Articles

Formulation of Special Pythagorean Triangles through Integer Solutions of the Hyperbola 𝑦𝟐 = (𝑘𝟐 + 𝟐𝑘)𝑥𝟐 + 𝟏

Objectives: The objective of this research paper is to formulate Pythagorean triangles, each of which satisfies the relation: Hypotenuse = (𝑘 + 1) times the leg with even values added with unity, through employing the integer solutions to the Hyperbola 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1. Methods: The sides of the Pythagorean triangle satisfying the requirement are obtained by suitably choosing its generators consisting of the integer solutions to the considered hyperbola. Findings : There are plenty of Pythagorean triangles satisfying the given characterization for each value of k > 0 in the binary quadratic equation given by 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1.The characterizations of special pattern of the Pythagorean triangle for 𝑘 = 2is illustrated. Novelty: From the integer solutions of a binary quadratic equation, namely, a hyperbola (Two-dimensional geometrical representation), it is possible to obtain integer solutions to a ternary quadratic equation, namely, Pythagorean triangle (Threedimensional geometrical representation). Keywords: Binary quadratic equation; Ternary quadratic equation; Hyperbola; Pythagorean triangle; Integer solutions

Solving Fuzzy Delay Differential Equations using Fuzzy Elzaki Transform and Fuzzy Elzaki Decomposition Method

Objective: In this work, the fuzzy Elzaki transform and fuzzy Elzaki decomposition method are used to solve fuzzy delay differential equations. The fuzzy Elzaki decomposition method is a combination of the fuzzy Elzaki transform and Adomian decomposition method in a fuzzy environment. The use of transforms in solving differential equations makes the solution process simpler. Methodology: The fuzzy Elzaki transform is applied in the fuzzy delay differential equation. The nonlinear terms are decomposed using Adomian polynomials in a nonlinear fuzzy delay differential equation. In the transformed domain, the resulting algebraic equation is solved. The solution in the original time domain is obtained by applying the inverse Elzaki transform. Findings: The fuzzy Elzaki transform and fuzzy Elzaki decomposition method are useful for handling the complexities associated with fuzzy functions and delays in differential equations. The technique used to solve the fuzzy delay differential equation gives results with better accuracy compared to the exact solution. The applicability of the method is illustrated with numerical examples. Novelty: The Elzaki transform is a helpful tool for solving delay differential equations with discontinuities. For the resilient solution of complicated differential equations, especially those including nonlinearity and fuzzy logic, the fuzzy Elzaki decomposition approach is useful. It gives an organized way to find the solution to fuzzy delay differential equation by utilizing the advantages of both the Elzaki transform and the Adomian decomposition method in a fuzzy situation. Keywords: Triangular Fuzzy Number, Fuzzy Elzaki Transform, Adomian decomposition method, Fuzzy Elzaki decomposition method, Fuzzy Delay Differential Equations (FDDE)

Mountain-Pass Solution for a Kirchhoff Type Elliptic Equation

Mountain-Pass Solution for a Kirchhoff Type Elliptic Equation

Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Existence of solutions for time fractional semilinear parabolic equations in Besov–Morrey spaces

AbstractWe consider the Cauchy problem for a time fractional semilinear heat equation "Equation missing"where $$0<\alpha <1,\, \gamma >1,\, N\in \mathbb {Z}_{\geqslant 1}$$ 0 < α < 1 , γ > 1 , N ∈ Z ⩾ 1 and $$\mu (x)$$ μ ( x ) belongs to inhomogeneous/homogeneous Besov–Morrey spaces. The fractional derivative $$^{C}\partial ^{\alpha }_{t}$$ C ∂ t α is interpreted in the Caputo sense. We present sufficient conditions for the existence of local/global-in-time solutions to problem (P). Our results cover all existing results in the literature and can be applied to a large class of initial data.

The Impact of Noise on the Solution of the Stochastic Fractional Chiral Schrodinger Equation Forced by Multiplicative Noise in Ito Sense

The Impact of Noise on the Solution of the Stochastic Fractional Chiral Schrodinger Equation Forced by Multiplicative Noise in Ito Sense

On the absence of weak solutions for a sequential time-fractional Klein-Gordon equation with quadratic nonlinearity

On the absence of weak solutions for a sequential time-fractional Klein-Gordon equation with quadratic nonlinearity

Analytical solutions of the Caudrey-Dodd-Gibbon equation using Khater II and variational iteration methods.

This study focuses on solving the Caudrey-Dodd-Gibbon ( ) equation using the Khater II ( ) method and the Variational Iteration ( ) method. The equation is a pivotal mathematical model in nonlinear wave dynamics, essential for understanding the evolution, interaction, and preservation of wave forms in dispersive media. Its applications span various fields, including fluid dynamics, nonlinear optics, and plasma physics, where it plays a crucial role in analyzing solitons and complex wave interactions. In this research, we meticulously implement the and methods to derive solutions for this nonlinear partial differential equation. Our findings reveal new aspects of the equation's behavior, offering deeper insights into nonlinear wave phenomena. The significance of this study lies in its contribution to advancing the understanding of these phenomena and their practical applications in the academic realm. The results underscore the effectiveness of the employed methods, their innovative contributions, and their relevance to applied mathematics.

Biaxial Gaussian Beams, Hermite–Gaussian Beams, and Laguerre–Gaussian Vortex Beams in Isotropy-Broken Materials

We have developed the paraxial approximation for electromagnetic fields in arbitrary isotropy-broken media in terms of the ray–wave tilt and the curvature of materials’ Fresnel wave surfaces. We have obtained solutions of the paraxial equation in the form of biaxial Gaussian beams, which is a novel class of electromagnetic field distributions in generic isotropy-broken materials. Such beams have been previously observed experimentally and numerically in hyperbolic metamaterials but have evaded theoretical analysis in the literature up to now. Biaxial Gaussian beams have two axes: one in the direction of the Abraham momentum, corresponding to the ray propagation, and another in the direction of the Minkowski momentum, corresponding to the wave propagation, in agreement with the recent theory of refraction, ray–wave tilt, and hidden momentum [Durach, 2024]. We show that the curvature of the wavefronts in the biaxial Gaussian beams correspond to the curvature of the Fresnel wave surface at the central wave vector of the beam. We obtain the higher-order modes of the biaxial beams, including the biaxial Hermite–Gaussian and Laguerre–Gaussian vortex beams, which opens avenues toward studies of the optical angular momentum (OAM) in isotropy-broken media, including generic anisotropic and bianisotropic materials.

On the large time behaviour of the solutions of an evolutionary-epidemic system with spatial dispersal.

This paper investigates some properties of the large time behaviour of the solutions of a spatially distributed system of equations modelling the evolutionary epidemiology of a plant-pathogen system. The model takes into account the phenotypic trait and the mutation of the pathogen, which is described by a non-local operator. We roughly speaking prove that the solutions separate the phenotype trait from the spatio-temporal evolution in the large time asymptotic. This feature is obtained by investigating the positive and bounded entire solutions of the problem, that are shown to exhibit such a separation of the variables property, by reformulating them as the positive solutions of suitable integral equations in some ordered Banach space. In addition, some numerical simulations are performed to support our theoretical results.

Closed causal curves in f(ℛ,𝒜,AμνAμν) gravity

In this paper, we investigate a [Formula: see text] solution of Einstein field equations, an extension of [Formula: see text] Misner space in the context of Ricci-inverse ([Formula: see text])-gravity theory. For that, we consider two different class, namely, Class-II model which is defined by the function [Formula: see text] and Class-III model defined by the function [Formula: see text]. Here, [Formula: see text] is the anti-curvature tensor with [Formula: see text] as its scalar, and [Formula: see text] is the Ricci scalar. Using this [Formula: see text] metric as background, the modified field equations for Ricci-inverse gravity were derived and obtain the results. These derived field equations are then solvable by choosing the pure radiation field as the energy-momentum tensor, including a cosmological constant as does in the general relativity. This implies that the chosen [Formula: see text] space-time is a valid solution in this novel gravity theory, and thereby violate the causality condition, analogous to general relativity.

On Solutions of Some Functional Equations of Radical Type

Many authors have studied various functional equations of forms patterned on the equation expressed as ga2+b2=g(a)+g(b), which can be considered, e.g., for real functions. Such equations are usually referred to as radical functional equations or of the radical type. Authors mainly study the so-called Ulam stability of such equations, i.e., they investigate how much the mappings satisfying the equations approximately (in a sense) differ from the exact solutions of these equations. Quite often, information about the solutions of these equations is also provided, but unfortunately, sometimes, such information is given in a misleading or incomplete way. It seems, therefore, that there is a need for a publication containing simple descriptions of such solutions (with appropriate examples), which would help in easy correction of such information and avoidance of similar problems for future authors. This is the main motivation for this expository paper. We present a general approach to the topic and consider two general forms of such equations. Moreover, the results presented in this paper show significant symmetry between the solutions of numerous functional equations and the solutions of equations of the radical type that correspond to them. To make this publication accessible to a wider audience, we omit various related information, avoid advanced generalizations, and present several simple examples.

On the positive solutions for some diffusive logistic equations

In this paper, we study the positive solutions of the diffusive logistic equations. It is shown that the asymptotic profiles of the positive solutions display differences between nonlocal dispersal equation and semilinear elliptic equation. When the degeneracy domain becomes small, we find that the positive solution of semilinear elliptic equation admits a blow-up profile. However, when the non-degeneracy domain becomes large, the positive solution of nonlocal equation exhibits a blow-up profile.

On the number of solutions of some Frobenius type equations in finite groups

Let G be a finite group with automorphism group Aut ( G ) and g ∈ G . For any subgroup H of Aut ( G ) , define a map ξ H : G → C by ξ H ( g ) = | { ( x , α ) ∈ G × H : [ x , α ] = g } | , where [ x , α ] : = x − 1 α ( x ) is called the autocommutator of x and α in G. In this paper, we show that ξ H is a character of G when H is any subgroup of Aut ( G ) containing Inn ( G ) and H = Autcent ( G ) , where Inn ( G ) is the inner automorphism group of G and Autcent ( G ) is the group of all the central automorphisms of G. In these cases expressions for ξ H are also obtained in terms of irreducible characters of G. Our results generalize a classical result of Frobenius regarding the number of solutions of the commutator equation in finite groups. As applications of our results, we compute g-autocommuting probabilities of some finite groups and obtained certain bounds for g-autocommuting probability of G in terms of commuting and autocommuting probabilities.

Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Delay

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the (μ1,μ2)-exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the (μ1,μ2)-exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research.

Computational analysis of the conformable nonlinear Schrödinger equation with Kudryashov’s refractive index model and generalized non-local nonlinearity

This paper investigates the nonlinear conformable Schrödinger equation with nonlocal nonlinearities and Kudryashov’s arbitrary refractive index. The study employs the Uniform Method (UM) and the Extended Hyperbolic Function Method (EHFM) to derive various optical soliton solutions for this present conformable equation. To demonstrate the importance of the novel solutions, the paper presents two-dimensional, three-dimensional, and contour plots, illustrating kink-type, wave, bright, bell-shaped, and mixed dark–bright soliton solutions. Further, the impact of the fractional order parameter, temporal parameter, and nonlinearity coefficient on these optical solutions is analyzed, providing valuable insights into the conformable nonlinear Schrödinger model. The behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the conformable order derivative, temporal parameter, and nonlinearity coefficient. The methodologies applied in this research show potential for broader application to various nonlinear Schrödinger equations in fields such as applied mathematics and nonlinear optics. The research suggests that these methods could be applied in the future to investigate other differential equations with fractional and integer orders across different fields of applied sciences. This study expands our understanding of nonlinear optics and has potential practical implications in various fields like optical signal processing, laser technology, and telecommunications.

A Generalised Difference Equation and Its Dynamics and Solutions

Rational difference equations have a wide range of applications in various fields of science. To illustrate, the equation xn+1=a+bxnc+dxn,n=0,1,..., known as the Riccati difference equation, has been applied in the field of optics. In this study, the global asymptotic stability of the difference equation xn+1=Axn−2k+j+1B+Cxn−(k+j)xn−2k+j+1,n=0,1,..., is proved. The solutions of this difference equation are obtained by applying the standard iteration method, and the periodicity of these solutions is determined. Furthermore, this difference equation represents a generalisation of the results obtained in previous studies.

Long-time behavior of solutions for time-periodic reaction-diffusion equations and applications

This article concerns the asymptotic behavior of solutions for time-periodic reaction-diffusion equations with a drift term in one dimensional space. Assuming that drift term is decaying, we analyze the effect of the decaying rate of the drift term on the propagation speed of solutions. Also we show some applications of our results in high-dimensional domains. For more information see https://ejde.math.txstate.edu/Volumes/2024/69/abstr.html

Propagation of traveling wave solution of the strain wave equation in microcrystalline materials

Abstract This study focuses on the propagation behavior of traveling wave solution in microcrystalline materials using the polynomial complete discriminant system method. By establishing a complete discriminant system, we systematically analyze the formation and evolution process of traveling wave solution in microcrystalline materials. Specifically, we apply the cubic polynomial extension to the strain wave equation to obtain more accurate analytical solutions. Additionally, two-dimensional, three-dimensional, and contour plots are generated to visually illustrate the characteristics of the obtained solutions, facilitating a more intuitive understanding of their physical significance. These findings not only help reveal the propagation mechanism of traveling wave solution but also provide a theoretical foundation for the application of microcrystalline materials.

Nonequilibrium Molecular Dynamics Method to Generate Poiseuille-Like Flow between Lipid Bilayers.

There are various flows inside and outside cells in vivo. Nonequilibrium molecular dynamics (NEMD) simulation is a useful tool for understanding the effects of these flows on the dynamics of biomolecules. We propose an NEMD method to generate a Poiseuille-like flow between lipid bilayers. We extended the conventional equilibrium MD method to produce a flow by adding constant external force terms to the water molecules. Using the Lagrange multiplier method, the center of mass of the lipid bilayer is constrained so that the flow does not sweep away the lipid bilayer, but the individual lipid molecules fluctuate. The temperature of the system is controlled properly in the solution and membrane by using the Nosé-Hoover thermostat. We found that the flow velocity increases linearly as the applied external force term increases. It is possible to estimate the appropriate value of acceleration to generate a flow with an arbitrary velocity using this proportional relation once a single short MD simulation is performed. We also found that the flow between two lipid bilayers is slower than the analytical solution of the Navier-Stokes equations between rigid parallel plates due to the interactions between water molecules and the membrane. This method can be applied not only to a flow on lipid membranes but also to a flow on soft surfaces generally.