Solution Of Equation Research Articles

Blow-Up Analysis of L2-Norm Solutions for an Elliptic Equation with a Varying Nonlocal Term

This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of L2-norm solutions for the related equation when the potential function V(x) fulfills an appropriate choice.

The Parallel Compact Object CALculator: An Efficient General Relativistic Initial Data Solver for Compact Objects

Every numerical general relativistic investigation starts from the solution of the initial value equations at a given time. Astrophysically relevant initial values for different systems lead to distinct sets of equations that obey specific assumptions tied to the particular problem. Therefore, a robust and efficient solver for a variety of strongly gravitating sources is needed. In this work, we present the OpenMP version of the Compact Object CALculator (COCAL) on shared memory processors. We performed extensive profiling of the core COCAL modules in order to identify bottlenecks in efficiency, which we addressed. Using modest resources, the new parallel code achieves speedups of approximately one order of magnitude relative to the original serial COCAL code, which is crucial for parameter studies of computationally expensive systems such as magnetized neutron stars, as well as its further development towards more realistic scenarios. As a novel example of our new code, we compute a binary quark system where each companion has a dimensionless spin of 0.43 aligned with the orbital angular momentum.

A reference stator flux linkage calculation method for realizing maximum efficiency operation in direct‐torque‐controlled IPMSM drives

AbstractIn this paper, a reference stator flux linkage calculation method is proposed for realizing maximum efficiency operation in direct‐torque‐controlled interior permanent magnet synchronous motor (IPMSM) drives. The proposed method is based on the solutions of a quartic equation, a motor model in the d‐q rotating frame widely used for synchronous motor controllers, and a well‐known equation for maximum efficiency operation. Therefore, the proposed method uses neither any approximation function nor any look‐up table that depends on the parameters of the motor, which has to be created in advance. Simulation results validate that the proposed methods can achieve maximum efficiency operation in IPMSM drives.

Tensor Product Technique and Atomic Solution of Fractional Partial Differential Equations

In this paper, we investigate the atomic solution of a special type of fractional partial differential equations. Tensor product in Banach spaces, some properties of atom operators, and some properties of conformable fractional derivatives are utilized in such process.JEL Classification: 34G10, 34A55

Polytropic Dynamical Systems with Time Singularity

In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method that combines two simple methods; Euler's method and shooting method, is proposed to approximate the solution of this type of dynamic equations. We adopt the shooting method to reduce the boundary value problem, then we apply Euler's algorithm to the resulted initial value problem to get approximations for the solution of the Lane-Emden equation. Finally, numerical examples and simulation are provided to show the validity and efficiency of the proposed technique, as well as the convergence and error estimation are analyzed.

Dynamics of switching optical soliton in fiber with sixth order dispersion and inter modal dispersion

Abstract This work pertaining to the study of switching soliton in fiber for nonlinear Schr¨odinger (NLS) equation with the presence of higher order dispersion and inter modal dispersion (IMD). The nonlinear wave in the optical fiber are described by the NLS equation which having the second order, fourth order, fifth order, sixth order dispersion, higher order nonlinearity and IMD. The main goal of the paper is to examine the sixth order dispersion on nonlinear wave in the fiber with the conditions of IMD. Hence, we employ the B¨acklund transformation of the Riccati equation (BTRE) approach to NLS equation and obtain the soliton solution. By the use of soliton solution along with graphical snapshots, we provide the conditions for forming switching solition in optical fiber and also analysis the effect of sixth order dispersion in fiber. The stability of the solution of NLS equation is also addressed.

Deformations of solutions of algebraic differential equations

Deformations of solutions of algebraic differential equations

Topological defects on solutions of non-relativistic equation for extended double ring-shaped potential

Abstract In this study, we focus into the non-relativistic wave equation described by the Schrödinger equation, specifically considering angular-dependent potentials within the context of a topological defect background generated by a cosmic string. Our primary goal is to explore
quasi-exactly solvable problems by introducing an extended ring-shaped potential. We utilize the Bethe ansatz method to determine the angular solutions, while the radial solutions are obtained using special functions. Our findings demonstrate that the eigenvalue solutions
of quantum particles are intricately influenced by the presence of the topological defect of the cosmic string, resulting in significant modifications compared to those in a flat space background. The existence of the topological defect induces alterations in the energy spectra, disrupting degeneracy. Afterwards, we extend our analysis to study the same problem in the presence of a ring-shaped potential against the background of another topological defect geometry known as a point-like global monopole. Following a similar procedure, we obtain the eigenvalue solutions and analyze the results. Remarkably, we observe that the presence of a global monopole leads to a decrease in the energy levels compared to the flat space results. In both cases, we conduct a thorough numerical analysis to validate our findings.

Anisotropy Induced by Electric Charge: A Computational Analytical Approach

This paper presents a novel class of interior solutions for anisotropic stars under the imposition of a self-similar symmetry. This means proposing exact solutions to the Einstein field equations to describe charged matter distribution with radiation flow. The Einstein–Maxwell system by employing specific choices of mass function is formulated to describe the gravitational collapse of charged, anisotropic, spherically symmetric distributions using the Schwarzschild metric. Two ordinary differential equations governing the dynamics are derived by matching a straightforward solution of the symmetry equations to the charged exterior (Reissner–Nordström–Vaidya). Models with satisfactory physical behavior are constructed by extensively exploring self-similar solutions for a set of parameters and initial conditions. Finally, the paper presents the evolution of physical variables and the collapsing radius, demonstrating the inevitable collapse of the matter distribution.

Riemann-Hilbert approach and double-pole solutions for the third-order flow equation of DNLS-type equation with nonzero boundary conditions

In this paper, the Riemann-Hilbert approach is applied to study a third-order flow equation of derivative nonlinear Schrödinger-type equation with nonzero boundary conditions. By utilizing the analytical, symmetric, and asymptotic properties of eigenfunctions, a generalized Riemann-Hilbert problem is formulated for the third-order flow equation of derivative nonlinear Schrödinger-type equation with nonzero boundary conditions. The formulas of N-soliton solutions for cases of single pole and double poles are given. We present some kinds of soliton solutions of these two cases according to different distributions of spectral parameters to study the dynamical behavior of them.

SOME COUNTING FORMULAE FOR -QUIDDITIES OVER THE RINGS

Abstract The $\lambda $ -quiddities of size n are n-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter’s friezes. Their number and properties are closely linked to the structure and the cardinality of the chosen set. Our main objective is an explicit formula giving the number of $\lambda $ -quiddities of odd size, and a lower and upper bound for the number of $\lambda $ -quiddities of even size, over the rings ${\mathbb {Z}}/2^{m}{\mathbb {Z}}$ ( $m \geq 2$ ). We also give explicit formulae for the number of $\lambda $ -quiddities of size n over ${\mathbb {Z}}/8{\mathbb {Z}}$ .

Deep bed filtration and formation damage by particles with distributed properties

Current models for deep bed filtration describe particles with uniform properties. Yet, the sizes, densities, and mineral composition of particles vary significantly in the same injection well. The aim of this work is to provide an effective mathematical model for water injection of particles with distributed properties and formation damage prediction. We average the set of traditional population balance equations for single-property particles and obtain one upscaled equation. The upscaled equation for particle retention rate contains a non-linear function of suspended concentration, which we call the 'suspension function'. We derive analytical solutions for the upscaled equation for linear (coreflood) and radial (well injectivity) flows. Then we treat lab coreflood data to determine the model suspension function and provide a model for well injectivity prediction. The retention profile for the flow of uniform particles has an exponential form. Frequently reported in the literature, hyper-exponential forms have been hypothetically explained by multiple particle properties. The inverse solution allows revealing the individual filtration coefficients for binary mixtures from total breakthrough concentrations during coreflood. Treatment of the data from lab experiments reveals individual filtration coefficients that belong to common intervals. For the first time, deep bed filtration of particles with distributed properties is upscaled and presented using a single equation that reflects the particle property distribution. This equation provides an effective mathematical model for tuning lab coreflood data, determines the model function, and uses it for injectivity decline prediction.

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

Abstract In this paper, we consider the multi-dimensional distributed-order time-fractional diffusion equation with the unit density function. We introduce the new Volterra-Bessel function and give the integral representations of fundamental solutions of equations in terms of this function in the whole- and half-space. The fractional moments of fundamental solutions are also provided in the higher dimensions using the Mellin transforms. We further apply steepest descent method to find the asymptotic behaviors of solutions using the Schl"{a}fli integral of the Volterra-Bessel function. In this respect, we study the asymptotic analysis of the Volterra-Bessel function with the large parameters, and subsequently obtain the asymptotic behaviors of fundamental solutions with a discussion on the large space variable, large time variable, higher dimensions and small diffusivity constant.

Critic learning control via zero-sum differential game for affine formation maneuver of multi-agent systems with cyber-attacks

The affine formation maneuver (AFM) control problem using the zero-sum differential game is studied for a second-order multi-agent system (MAS) against cyber-injection attacks on the actuators. First and foremost, the distributed control signals and the cyber-injected attack signals are viewed as two competing teams. Then, the leader-follower tracking error system is established where the followers aim to track the AFMs of the leaders. The leader-follower distributed optimal control policies and the cyber-attack attenuation policies are derived from the Nash equilibrium solution of the Hamilton-Jacobi-Isaac (HJI) equation. The solution of the HJI equation is estimated with the aid of the critic neural network. The weights of the critic system are learned online based on the adaptive dynamic programming (ADP) scheme. The uniformly ultimately bounded stability of the closed-loop system under the formulated zero-sum differential game control strategy has been proven by the Lyapunov stability theorem. A simulation example illustrates that the presented control method can attenuate cyber-injection attacks on the actuators and maintain the AFMs of the agents.

One-dimensional, multi-fluid model of the plasma-wall transition. II. Negative ions

The plasma-wall transition is investigated by a one-dimensional steady-state multifluid model, which was presented in detail in Part I [T. Gyergyek et al., AIP Adv. 14, 045201 (2024)]. In this work, the plasma-wall transition is analyzed for the case where the plasma consists of singly charged positive ions, electrons, and singly charged negative ions. When the temperature and initial density of the negative ions are varied, a transition between two types of solutions of the model is observed. We call them the low and high solution, with respect to the absolute value of the potential drop. When the density and temperature of the negative ions are above a critical value, the low solution is observed. As the mass of the positive ions increases, these critical values also increase, but only until the ion mass is below about 1000 electron masses. With larger ion masses, the critical density of the negative ions and the temperature no longer change. In the low solution, the potential drop in front of the sheath is determined by the negative ions and is smaller in absolute terms than in the case of the high solution, where the potential drop in front of the sheath is determined by the electrons. If the problem is analyzed on the pre-sheath scale, the transition between the low and high solution is very sharp. However, when the neutrality condition is replaced by the Poisson equation, this transition becomes blurred and the solutions of the model equations exhibit oscillations. The role of the smallness parameter is highlighted. It is shown how the initial electric field is determined. Deviation of the negative ion density profile from the Boltzmann relation is discussed.

LOCAL FRACTIONAL SUMUDU DECOMPOSITION METHOD TO SOLVE FRACTAL PDEs ARISING IN MATHEMATICAL PHYSICS

In this paper, we investigate solutions of telegraph, Laplace and wave equations within the local fractional derivative operator by using local fractional Sumudu decomposition method. This method is coupled by the Sumudu transform and decomposition method. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the presented method.

SUMMABILITY AND ASYMPTOTICS OF POSITIVE SOLUTIONS OF AN EQUATION OF WOLFF TYPE

Abstract We use potential analysis to study the properties of positive solutions of a discrete Wolff-type equation $$ \begin{align*} w(i)=W_{\beta,\gamma}(w^q)(i), \quad i \in \mathbb{Z}^n. \end{align*} $$ Here, $n \geq 1$ , $\min \{q,\beta \}>0$ , $1<\gamma \leq 2$ and $\beta \gamma <n$ . Such an equation can be used to study nonlinear problems on graphs appearing in the study of crystal lattices, neural networks and other discrete models. We use the method of regularity lifting to obtain an optimal summability of positive solutions of the equation. From this result, we obtain the decay rate of $w(i)$ when $|i| \to \infty $ .

A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition

Abstract In this work, we present a sophisticated operating algorithm, the reproducing kernel Hilbert space method, to investigate the approximate numerical solutions for a specific class of fractional Begley-Torvik equations (FBTE) equipped with fractional integral boundary condition. Such fractional integral boundary condition allows us to understand the non-local behavior of FBTE along with the given domain. The algorithm methodology depends on creating an orthonormal basis based on reproducing kernel function that satisfies the constraint boundary conditions so that the solution is finally formulated in the form of a uniformly convergent series in ϖ 3 [ a , b ] {\varpi }_{3}\left[a,b] . From a numerical point of view, some illustrative examples are provided to determine the appropriateness of algorithm design and the effect of using non-classical boundary conditions on the behavior of solutions approach.

A new approach for the regularity of weak solutions of the 3D Boussinesq system

We address here the problem of regularity for weak solutions of the 3D Boussinesq equation. By introducing the new notion of partial suitable solutions, which imposes some conditions over the velocity field only, we show a local gain of regularity for the two variables u→ and θ.

Optimal decay for solutions of nonlocal semilinear equations with critical exponent in homogeneous groups

In this paper, we establish the sharp asymptotic decay of positive solutions of the Yamabe type equation $\mathcal {L}_s u=u^{\frac {Q+2s}{Q-2s}}$ in a homogeneous Lie group, where $\mathcal {L}_s$ represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.