General Cauchy Research Articles

Global conservative solutions of the nonlocal NLS equation beyond blow-up

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \mathrm{i}\partial_t q(x, t)+\partial_{x}^2q(x, t)+2\sigma q^{2}(x, t)\overline{q(-x, t)} = 0 $ with initial data $ q(x, 0)\in H^{1, 1}(\mathbb{R}) $. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $ H^{1, 1} $ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

On the Generalized Quadratic-Quartic Cauchy Functional Equation and its Stability over Non-Archimedean Normed Space

Functional equation plays a very important and interesting role in the area of mathematics, which involves simple algebraic manipulations and through which one can arrive an interesting solution. The theory of functional equations is also used in the development of other areas such as analysis, algebra, Geometry etc., the new methods and techniques are applied in solving problem in Information theory, Finance, Geometry, wireless sensor networks etc., In recent decades, the study of various types of stability of a functional equation such as HUS (Hyers-Ulam stability), HURS (Hyers-Ulam-Rassias stability) and generalized HUS of different types of functional equation and also for mixed type were discussed by many authors in various space. The problem of the stability of different functional equations has been widely studied by many authors, and more interesting results have been proved in the classical case (Archimedean). In recent years, the analogues results of the stability problem of these functional equations were investigated in non-Archimedean space. The aim of this study is to investigate the HUS of a mixed type of general Quadratic-Quartic Cauchy functional equation in non-Archimedean normed space. In this current article, we prove the generalized HUS for the following Quadratic-Quartic Cauchy functional equation over non-Archimedean Normed space.<img src=image/13428783_01.gif>

A Weierstrass type representation for translating solitons and singular minimal surfaces

In this paper we provide a Weierstrass representation formula for translating solitons and singular minimal surfaces in R3. As application we study when the euclidean Gauss map has a harmonic argument and solve a general Cauchy problem in this class of surfaces.

Graded damage in quasi‐brittle solids

AbstractA novel approach to damage modeling for quasi‐brittle solids is presented relying upon a differential inclusion that is closely related to the one of implicit gradient models. The proposed formulation naturally fits in the so‐called nonlocal standard approach, whereby the framework of generalized standard materials is extended to include gradients of internal variables to account for the physics of the fracture phenomenon in a regularized sense, that is, via extended constitutive equations in which a length scale parameter brings to the macro level information about material microstructure. This concept is fully embodied into the present approach to quasi‐brittle fracture, whereby progressive damage occurs in layers of finite thickness where the gradient of damage is bounded and a fully damaged region is understood as a fracture with no ambiguity. Key to the effective implementation of the model are the choice of two constitutive functions and the implicit tracking of regions in a state of progressive damage via Lagrange multipliers acting on internal constraints. The ideas are developed for a general Cauchy continuum and representative numerical simulations are included that demonstrate the model capabilities.

Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion

The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions {BtH,t≥0} and sub-fractional Brownian motions {ξtH,t≥0} with Hurst parameter H∈(12,1). We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.

Analytical and numerical investigation of the Hindmarsh-Rose model neuronal activity.

In this work, a set of nonlinear equations capable of describing the transit of the membrane potential's spiking-bursting process which is shown in experiments with a single neuron was taken into consideration. It is well known that this system, which is built on dynamical dimensionless variables, can reproduce chaos. We arrived at the chaotic number after first deriving the equilibrium point. We added different nonlocal operators to the classical model's foundation. We gave some helpful existence and uniqueness requirements for each scenario using well-known theorems like Lipchitz and linear growth. Before using the numerical solution on the model, we analyzed a general Cauchy issue for several situations, solved it numerically and then demonstrated the numerical solution's convergence. The results of numerical simulations are given.

Hopf-type theorem for self-shrinkers

Abstract In this paper, we prove that a two-dimensional self-shrinker, homeomorphic to the sphere, immersed in the three-dimensional Euclidean space ℝ 3 {{\mathbb{R}}^{3}} is a round sphere, provided its mean curvature and the norm of the its position vector have an upper bound in terms of the norm of its traceless second fundamental form. The example constructed by Drugan justifies that the hypothesis on the second fundamental form is necessary. We can also prove the same kind of rigidity results for surfaces with parallel weighted mean curvature vector in ℝ n {{\mathbb{R}}^{n}} with radial weight. These results are applications of a new generalization of Cauchy’s Theorem in complex analysis which concludes that a complex function is identically zero or its zeroes are isolated if it satisfies some weak holomorphy.

Semi-analytic treatment of mixed hyperbolic–elliptic Cauchy problem modeling three-phase flow in porous media

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

General Cauchy Conjugate Gradient Algorithms Based on Multiple Random Fourier Features

The general Cauchy loss (GCL) criterion has been successfully proposed to improve the performance of the Cauchy loss (CL) criterion for linear adaptive filtering in the presence of complex non-Gaussian noise. However, the commonly used adaptive filtering algorithms based on the GCL criterion utilize the stochastic gradient descent (SGD) method to update their weights with slow convergence rate and poor steady-state performance. To overcome these issues, a general Cauchy-loss conjugate gradient (GCCG) method is first developed by solving the proposed half-quadratic general Cauchy loss (HQGCL) with the conjugate gradient method. To further tackle complex nonlinear issues, novel multiple random Fourier features (MRFF) spaces are then constructed in finite-dimensional features spaces, which is proven effective for approximation of multi-kernel adaptive filter (MKAF), theoretically. The GCCG method is thus applied into the constructed MRFF spaces to generate novel multiple random Fourier features GCCG (MRFGCG) algorithms, curbing the linear growth structures of kernel adaptive filters (KAFs) and MKAFs. The proposed MRFGCG algorithms in fixed networks have lower computational complexity and higher filtering accuracy than sparsification KAFs in non-Gaussian environments. Monte Carlo simulations on the prediction of synthetic and real-world time-series and the identification of nonlinear system confirm the superiorities of the proposed MRFGCG algorithms.

A class of time-fractional Dirac type operators

By using a Witt basis, a new class of time-fractional Dirac type operators with time-variable coefficients is introduced. These operators lead to considering a wide range of fractional Cauchy problems. Solutions of the considered general fractional Cauchy problems are given explicitly. The representations of the solutions can be used efficiently for analytic and computational purposes. We apply the obtained representation of a solution to recover a variable coefficient solution of an inverse fractional Cauchy problem. Some concrete examples are given to show the diversity of the obtained results.

Multidimensional inverse Cauchy problems for evolution equations

We discuss inverse problems to finding the time-dependent coefficient for the multidimensional Cauchy problems for both strictly hyperbolic equations and polyharmonic heat equations. We also extend our techniques to the general inverse Cauchy problems for evolution equations.

New numerical approximation for Chua attractor with fractional and fractal-fractional operators

In this study, we present new numerical scheme for modified Chua attractor model with fractional operators. However we give numerical solution of the considered model with fractal-fractional operators. Also, we offer error analysis for a general Cauchy problem with fractional and fractal-fractional operators. For numerical solution of the considered equation, we use new numerical scheme which is established with an efficient polynomial known as Newton interpolation polynomial. The results are discussed with some examples and simulations.

Modified numerical methods for fractional differential equations

A new modified numerical scheme is proposed. This idea is inspired through the general idea of the Lagrange and the Adams–Bashforth. We present the derivation of the general steps for a modified numerical scheme for an equation with two different fractional derivatives. We present an application of such method in a general nonlinear Cauchy problem. The application is presented to some fractional differential equations. Additionally we realized that when the non-integer order is one, the well-known multi-step-Adams–Bashforth numerical scheme for classical differential equations is recovered.

Bargmann–Radon transform for axially monogenic functions

ABSTRACT In this paper, we study the Bargmann–Radon transform and a class of monogenic functions called axially monogenic functions. First, we compute the explicit formula of the Bargmann–Radon transform for axially monogenic functions, by making use of the Funk–Hecke theorem. Then we present the explicit form of the general Cauchy–Kowalewski extension for radial function. Finally, by making use of the results we obtained, we give an application of the Bargmann–Radon transform for Cauchy–Kowalewski extension.

Validity of fractal derivative to capturing chaotic attractors

Abstract Suggested independently with different definitions, fractal derivative and conformable derivative are α proportional. They have been applied in quit a few problems in many field of sciences in the last few years with great success. However, some researchers have pointed out some criticisms and even concluded that they were flawed. In this paper, while confirm the validity of the conformable and fractal derivatives and we present their applications to chaotic attractors. We considered a general non-linear Cauchy problem where the differential operator is that of fractal and conformable and present the derivation of conditions for which the existence and the uniqueness of the exact solution are reached. Several examples are considered, solved and numerical simulations depicting real world observations.

Existence results for Isaacs equations with local conditions and related semilinear Cauchy problems

Our goal is to prove existence results for classical solutions to some general nondegenerate Cauchy problems which are natural generalizations of Isaacs equations. For the latter we are able to extend our results by admitting local conditions for coeffici

Hypergeometric Cauchy numbers and polynomials

For positive integers N and M, the general hypergeometric Cauchy polynomials c M,N,n (z) (M, N ≥ 1; n ≥ 0) are defined by $$\frac{1}{(1+t)^z} \frac{1}{{}_2F_1(M,N;N+1;-t)}=\sum_{n=0}^\infty c_{M,N,n}(z)\, \frac{t^n}{n!}\,, $$ where \({{}_2 F_1(a,b;c;z)}\) is the Gauss hypergeometric function. When M = N = 1, c n = c 1,1,n are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In the aspect of determinant expressions, hypergeometric Cauchy numbers are the natural extension of the classical Cauchy numbers, though many kinds of generalizations of the Cauchy numbers have been considered by many authors. In this paper, we show some interesting expressions of generalized hypergeometric Cauchy numbers. We also give a convolution identity for generalized hypergeometric Cauchy polynomials.

Refractive index retrievals for polystyrene latex spheres in the spectral range 220–420 nm

ABSTRACTThe recent development of an Aerosol Extinction Differential Optical Absorption Spectrometer (AE-DOAS) has allowed for the retrieval of wavelength dependent complex refractive indices for polystyrene latex spheres (PSL). The AE-DOAS is a white-type multi-pass gas cell coupled to a UV-Vis spectrometer. Refractive index values are retrieved for wavelengths between 220 and 420 nm by minimizing the χ2 goodness-of-fit between measured extinction for five diameters of PSL and model Mie Theory predictions. Comparison to literature shows agreement at wavelengths >360 nm demonstrating the validity of this new instrumental approach while expanding the known refractive index range for PSLs further into the UV where it is previously unreported. In the studied wavelength range, coefficients for the general Cauchy dispersion relationship (A = 1.538(11); B = 0.0043(16) μm2; C = 0.00094(5) μm4) for PSLs were determined using the retrieved real portion of the refractive index and the wavelength in microns. In addi...

Isentropic fluid dynamics in a curved pipe

In this paper we study isentropic flow in a curved pipe. We focus on the consequences of the geometry of the pipe on the dynamics of the flow. More precisely, we present the solution of the general Cauchy problem for isentropic fluid flow in an arbitrarily curved, piecewise smooth pipe. We consider initial data in the subsonic regime, with small total variation about a stationary solution. The proof relies on the front-tracking method and is based on [1].

Noncommutative Cauchy Integral Formula

The aim of this paper is to provide and prove the most general Cauchy integral formula for slice regular functions and for $$C^1$$ functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions.