Search Results (329)

Fractional Langevin models are found to be useful in the study of physical phenomena such as diffusion processes, gait variability, etc. Langevin equations involving different fractional–order operators and boundary conditions have been addressed by many researchers. In this article, we formulate a new Langevin model consisting of a coupled system of Riemann–Liouville and Hadamard–type nonlinear fractional differential equations and coupled multipoint–integral boundary conditions. We present the existence and Ulam–Hyers stability criteria for solutions of the given model problem. Our study is based on the tools of the fixed–point theory. Numerical examples with graphical representations of solutions are offered to demonstrate the application of the obtained results. Our work is novel and useful in the given configuration, and specializes to some new results. Full article

While the standard Schwarzschild metric is overwhelmingly employed in general relativity (GR) as the starting point for various spherical spacetime metric calculations, its isotropic (ISO) form is mentioned in more specialized contexts and its derivation is barely discussed in published GR literature. In this work, we review the isotropic metric, stressing that it stands out as a useful spherically symmetric metric to be employed also in traditional GR problems. We start by deriving the ISO metric through solving the vacuum field equations in Cartesian coordinates, thereby obtaining the Ricci tensor also in spherical coordinates. We then analytically calculate the Riemann tensor in Cartesian coordinates, proving its consistency with the Ricci tensor calculation for pedagogical reasons. Finally, from the Riemann tensor we exactly evaluate the Kretschmann scalar, which lacks metric singularities, a result consistent with the known singular behavior of the standard Schwarzschild metric. We conclude that the isotropic metric naturally emerges as a suitable candidate for modeling static neutron stars and regular black holes, thereby complementing the present attempts to understand these rapidly evolving research fields. Full article

This work presents the first systematic development of the inverse scattering transform for matrix nonlinear Schrödinger equations in the case where the discrete spectrum has triple poles, under nonzero boundary conditions at infinity. These systems arise physically as reductions modeling spinor Bose-Einstein condensates with hyperfine spin $F=1$ and find applications in nonlinear optics. A uniformization variable is employed to map the underlying Riemann surface to the complex plane, enabling a complete characterization of the analyticity, symmetries, and asymptotic behaviors of the Jost functions and scattering data. Extending the established framework for simple and double poles, we show that $\mathrm{rank} P(x,t,z_n)=3$ requires a third-order zero of $\det a\left(z\right)$ at $z=z_n$, while $\mathrm{rank} P(x,t,z_n)=2$ necessitates a fourth-order zero—a nontrivial feature absent in lower-order cases. The discrete spectrum for both rank configurations is fully characterized, and the full singular behavior near a triple pole is derived, respecting the quartet symmetry $z_n$, $z_n^{*}$, $v{k}_0^2/z_n$, $v{k}_0^2/z_n^{*}$ imposed by the nonzero boundary conditions. Solving the resulting matrix Riemann-Hilbert problem with triple poles yields the potential reconstruction formula and, in the reflectionless case, explicit expressions for general N-triple-pole soliton solutions, with a detailed example for $N=1$ presented to illustrate the construction. Full article

The aim of the present work is to study a class of nabla fractional problems with two different nabla Riemann–Liouville operators and three-point parameter-dependent boundary conditions. First, we derive the expression of the Green’s function; then, we deduce a few useful inequalities with it, and we establish an interval for the parameter in which the Green’s function is always positive. Using these properties, we manage to prove some non-existence, existence and multiplicity results using different fixed-point theorems. At the end, we give a few examples that verify and clarify the applications of our results. Full article

In this paper we compare the rotation of a rigid body in the real three-dimensional Euclidean space E³ and its representation in the complex plane (Klein space), on one hand, with the transformation of polarization states of light (SOPs) by the phase-shifters figured in the complex plane and on the Poincaré sphere, on the other hand. Both the Klein space, in classical mechanics, and the Poincaré sphere, in polarization theory, are abstract spaces, whose construction is based on the classical stereographical projection between Riemann sphere and the simple complex plane C¹. They are classical abstract spaces, even if they have been largely used for representing quantum spinorial physical realities too. At the interface of classical/quantum physics persist some misaperceptions about what is intrinsically of quantum origin and nature, and what is imported from the classical domain. In this context we examine some misunderstandings that take place in the field of these spaces. I shall focus on the double angle relationship between the rotation of representative points of the SOPs on the Poincaré sphere with respect to the corresponding rotations of the azimuthal and ellipticity angles of the “form of the SOPs”, at a transformation of state given by a phase shifter. This is a classical result, that is transferred on the sphere from the complex plane, on the basis of the stereographic bijective connection between the points on the sphere and those in the complex plane. In any textbook of quantum mechanics “the double angle/half angle problem” is presented as a pure quantum spinorial one, avoiding its classical face and origin. A quantum spinorial approach, obviously, recovers the classical results, together with the specific spinorial ones, but with regards to the double angle/half angle issue it is superfluous. We shall also briefly examine the classical and quantum spinorial content of what we know today under the global name of Pauli spin matrices. Often in papers or textbooks of physics the results are presented in a mélange in which it is difficult to establish from which point on one needs to appeal to spinorial or quantum aspects. Full article

This investigation provides a comprehensive analytical framework for the topological morphology and global convergence dynamics governing a specific family of sixth-order iterative schemes designed for nonlinear equations with multiple roots. By invoking a Möbius conjugacy transformation upon the specialized polynomial class $f\left(z\right)={\left((z-p)(z-q)\right)}^m$, we project the iterative sequence onto the Riemann sphere $\widehat{\mathbb{C}}$, effectively recasting the algorithm as a discrete complex dynamic system. The core of this study lies in the bifurcation analysis of the associated parameter space. We meticulously chart the stability manifolds, tracing the evolution of critical orbits to distinguish between regions of predictable convergence and those characterized by chaotic instability. By examining the iterative methods generated by these rational endomorphisms, the research unveils the intricate fractal boundaries that delineate the basin of attraction, offering a profound insight into the structural robustness of higher-order methods. In the dynamical plane, the geometry of the basins of attraction is scrutinized to evaluate the robustness of the numerical flow and its sensitivity to the configuration of weight functions. By analyzing the fractal complexity of the boundaries within these basins, we provide a detailed characterization of the iterative morphology and its global reliability. The analytical findings are supported by high-resolution graphical representations and comparative numerical data, illustrating the superior performance and structural integrity of the proposed methods in solving nonlinear problems. Full article

This paper investigates the equivalent integral equations (EIEs) of two impulsive right-sided Riemann–Liouville fractional-order systems (IRRFOSs). The limit properties of one IRRFOS are employed to establish the linear additivity of impulsive effects. A computational approach based on fractional calculus for piecewise functions is then employed to construct the EIE corresponding to a single impulse. With the aid of this linear additivity, the EIE of the considered IRRFOS is obtained, and through the relationship between the two IRRFOSs, the EIE of the other IRRFOS is further derived. The results indicate that the solutions of both EIEs consist of linear combinations of $\varphi \left(t\right)$ and ${\mathrm{\Phi }}_j\left(t\right)$$(j=1,2,\dots ,N)$ containing an arbitrary constant, which implies the non-uniqueness of solutions to the two IRRFOSs. Finally, the computational procedure for deriving the EIEs of the two IRRFOSs is presented, and the non-uniqueness of solutions is illustrated through two numerical examples. Full article

This article presents an algorithm for solving the direct and inverse problem for a model consisting of a fractional differential equation with non-integer order derivatives with respect to time and space. The Caputo derivative was taken as the fractional derivative with respect to time, and the Riemann–Liouville derivative in the case of space. On one of the boundaries of the considered domain, a fractional boundary condition of the third kind was adopted. In the case of the direct problem, a differential scheme was presented, and a metaheuristic optimization algorithm, namely the Group Teaching Optimization Algorithm (GTOA), was used to solve the inverse problem. The article presents numerical examples illustrating the operation of the proposed methods. In the case of inverse problem, a function occurring in the fractional boundary condition was identified. The presented approach can be an effective tool for modeling the anomalous diffusion phenomenon. Full article

This article studies the abstract discrete-time Cauchy problem involving the Riemann–Liouville type difference operator. Sufficient conditions for the existence of unique solution to the semilinear Cauchy problem in Lebesgue and weighted Lebesgue vector-valued spaces are shown. Finally, some examples are presented to illustrate the main results. Full article

In this paper, we investigate the existence of positive solutions of the eigenvalue problem for a singular tempered fractional equation with a Riemann–Stieltjes integral boundary condition and signed measures. By establishing the Green function and its properties, an eigenvalue interval for the existence of positive solutions is outlined based on Schauder’s fixed-point theorem and the upper and lower solutions method. An interesting feature of this paper is that f may be singular in both the time and space variables, and the Riemann–Stieltjes integral may involve signed measures. Full article

In this research, we establish a precise correspondence between the theory of logarithmic connections on principal G-bundles over compact Riemann surfaces and the geometric formulation of control systems on curved manifolds, providing a novel differential–geometric framework for analyzing optimal control problems with non-holonomic constraints. By characterizing control systems through the geometric structure of flat connections with logarithmic singularities at marked points, we demonstrate that optimal trajectories correspond precisely to horizontal lifts with respect to the connection. These horizontal lifts project onto geodesics on the punctured surface, which is equipped with a Riemannian metric uniquely determined by the monodromy representation around the singularities. The main geometric result proves that the isomonodromic deformation condition translates into a compatibility condition for the control system. This condition preserves the conjugacy classes of monodromy transformations under variations of the marked points, and ensures the existence and uniqueness of optimal trajectories satisfying prescribed boundary conditions. Furthermore, we analyze systems with non-holonomic constraints by relating the constraint distribution to the kernel of the connection form, showing how the degree of non-holonomy can be measured through the failure of integrability of the associated horizontal distribution on the principal bundle. As an application, we provide computational implementations for $\mathrm{SL}(2,\mathbb{C})$ connections over hyperbolic Riemann surfaces with genus $g\ge 2$, explicitly constructing the monodromy-induced metric via the Poincaré uniformization theorem and deriving closed-form expressions for optimal control strategies that exhibit robust performance characteristics under perturbations of initial conditions and system parameters. Full article

In this research, we study the geometry of the moduli space of $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundles over a compact Riemann surface through the analysis of singular spectral curves and the triality automorphism of $\mathrm{Spin}(8,\mathbb{C})$. We establish a characterization of triality invariance, proving that a $\mathrm{Spin}(8,\mathbb{C})$-Higgs bundle admits a reduction to the exceptional group $G_2$ if and only if its spectral curve is invariant under the induced triality action. This transforms the problem of detecting $G_2$-structures into a question about spectral data. We decompose the discriminant locus of the Hitchin fibration into two disjoint strata: a fixed stratum arising from $G_2$-Higgs bundles with singular spectral curves and a free stratum consisting of orbits of size three under triality. We prove the existence of non-abelian spectral data compatible with triality symmetry, showing that non-abelian phenomena persist in free triality orbits. To quantify symmetry breaking, we introduce a triality defect invariant, which measures the dimension of the quotient of the Prym variety by its triality-invariant sublocus, and we prove that Higgs bundles with positive defects form a Zariski open dense subset. Full article

The nonlinear Schrödinger equation is a classical nonlinear evolution equation with wide applications. This paper explores the asymptotic behavior of solutions to the nonlinear Schrödinger equation with non-zero boundary conditions in the presence of a pair of second-order discrete spectra. We analyze the Riemann–Hilbert problem in the inverse scattering transform by the Deift–Zhou nonlinear steepest descent method. Then we propose a proper deformation to deal with the growing time term and give the conditions for the series in the process of deformation by the Laurent expansion. Finally, we provide the characterization of the interactions between the solitary waves corresponding to second-order discrete spectra and the coherent oscillations produced by the perturbation. Numerical verifications are also performed. Full article

The paper is devoted to the study of a class of singular high-order fractional integro-differential equations with p-Laplacian operator, involving both the Riemann–Liouville fractional derivative and the Caputo fractional derivative. First, we investigate the problem by the method of reducing the order of fractional derivative. Then, by using the Schauder fixed point theorem, the existence of solutions is proved. The upper and lower bounds for the unique solution of the problem are established under various conditions by employing the Banach contraction mapping principle. Furthermore, four numerical examples are presented to illustrate the applications of our main results. Full article

This paper develops a generalized Laplace transform theory within weighted function spaces tailored for the analysis of fractional differential equations involving the $\psi$-Hilfer derivative. We redefine the transform in a weighted setting, establish its fundamental properties—including linearity, convolution theorems, and action on ${\delta }_{\psi }$ derivatives—and derive explicit formulas for the transforms of $\psi$-Riemann–Liouville, $\psi$-Caputo, and $\psi$-Hilfer fractional operators. The results provide a rigorous analytical foundation for solving hybrid fractional Cauchy problems that combine classical and fractional derivatives. As an application, we solve a hybrid model incorporating both ${\delta }_{\psi }$ derivatives and $\psi$-Hilfer fractional derivatives, obtaining explicit solutions in terms of multivariate Mittag-Leffler functions. The effectiveness of the method is illustrated through a capacitor charging model and a hydraulic door closer system based on a mass-damper model, demonstrating how fractional-order terms capture memory effects and non-ideal behaviors not described by classical integer-order models. Full article