Solution Of Equation Research Articles

Generalizations of Riemann–Liouville fractional integrals and applications

Generalizations of Riemann–Liouville fractional integrals and applications

Modeling of nanochannels in synthetic membranes

The behavior of a diluted electrolyte in a system of joint microchannel and nanochannel with charged dielectric walls under the action of external potential difference and external pressure is investigated numerically. The surface charge on the nanochannel walls prevents the ions of corresponding polarity from passing through it. Consequently, the system in question acquires ion-selective properties and can, under certain assumptions, be viewed as a fragment of an ion-selective membrane, including one synthesized by creating nanopores in a dielectric material. Such systems are used in experiments to control the movement of charged particles through concentration polarization. The objective of the work is to investigate the influence of a single pore on electrolyte flow and the possibilities to control that flow by changing the geometric and physical properties of the pore. The investigation relies on the specially developed simplified models based on cross-section-averaged Nernst-Planck, Poisson and Stokes equations that are subsequently reduced to a single nonlinear differential equation. The simplified models allow identifying the impact of different physical mechanisms of electrolyte movement: pressure-based (generated by the external mechanical action) and electroosmotic (generated by the electric field). A finite-difference method with semi-implicit time integration is used for the numerical solution of equations. It has been found that the behavior of the system qualitatively matches the behavior of a cell based on a non-ideally-selective ion-exchange membrane. In particular, the model correctly predicts the underlimiting and limiting electric current regimes, as well as vortex formation near the nanochannel inlet due to concurrency between electrolyte movement mechanisms. The proposed models can be extended to describe a channel with arbitrary geometry and an electrolyte with arbitrary number of charged species.

Study of Self-Excited Thermomechanical Oscillator with Shape Memory Alloys

In this paper, a new type of self-excited thermomechanical oscillator containing an oscillating shape memory alloy (SMA) filament with two symmetrically arranged spheres is investigated. The self-excitation of the oscillations is due to a heater of constant temperature, which causes periodic contractions of the filament when it approaches it. The contracted filament moves away from the heater a distance sufficient to cool it. Under the action of the weight of the spheres, the cooled filament re-approaches the heater, causing the above processes to repeat periodically. On the basis of experimental studies, approximating functions of the heater’s heat field distribution are derived. A dynamic model of the oscillator has been created, in which the minor and major hysteresis in the SMA alloy and the distribution of the heat field around the heater have been taken into account. Through numerical solutions of the differential equations, the laws of motion of the spheres are obtained. The displacements of the spheres in two perpendicular directions were measured using an experimental system. The obtained experimental results validate the proposed dynamic model and its assumptions with a high degree of confidence. Conclusions are drawn about the stochastic nature of the oscillations due to the hysteresis properties of the SMA and the temperature variation of the natural frequency of the oscillating system.

Multivariate Mittag-Leffler Solution for a Forced Fractional-Order Harmonic Oscillator

The harmonic oscillator is a fundamental physical–mathematical system that allows for the description of a variety of models in many fields of physics. Utilizing fractional derivatives instead of traditional derivatives enables the modeling of a more diverse array of behaviors. Furthermore, if the effect of the fractional derivative is applied to each of the terms of the differential equation, this will involve greater complexity in the description of the analytical solutions of the fractional differential equation. In this work, by using the Laplace method, the solutions to the multiple-term forced fractional harmonic oscillator are presented, described through multivariate Mittag-Leffler functions. Additionally, the cases of damped and undamped free fractional harmonic oscillators are addressed. Finally, through simulations, the effect of the fractional non-integer derivative is demonstrated, and the consistency of the result is verified when recovering the integer case.

Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces

AbstractWe provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.

RHDLPP: A multigroup radiation hydrodynamics code for laser-produced plasmas

In this paper, we introduce the RHDLPP, a flux-limited multigroup radiation hydrodynamics numerical code designed for simulating laser-produced plasmas in diverse environments. The code bifurcates into two packages: RHDLPP-LTP for low-temperature plasmas generated by moderate-intensity nanosecond lasers, and RHDLPP-HTP for high-temperature, high-density plasmas formed by high-intensity laser pulses. The core radiation hydrodynamic equations are resolved in the Eulerian frame, employing an operator-split method. This method decomposes the solution into two substeps: first, the explicit resolution of the hyperbolic subsystems integrating radiation and fluid dynamics; second, the implicit treatment of the parabolic part comprising stiff radiation diffusion, heat conduction, and energy exchange. Laser propagation and energy deposition are modeled through a hybrid approach, combining geometrical-optics ray-tracing in sub-critical plasma regions with a one-dimensional solution of the Helmholtz wave equation in super-critical areas. The thermodynamic states are ascertained using an equation of state, based on either the real gas approximation or the quotidian equation of state (QEOS). For ionization calculations, the code employs a steady-state collisional-radiation (CR) model using the screened-hydrogenic approximation. Additionally, RHDLPP includes RHDLPP-SpeIma3D, a three-dimensional spectral simulation post-processing module, for generating both temporally-spatially resolved and time-integrated spectra and imaging, facilitating direct comparisons with experimental data. The paper showcases a series of verification tests to establish the code's accuracy and efficiency, followed by application cases, including simulations of laser-produced aluminium (Al) plasmas, pre-pulse-induced target deformation of tin (Sn) microdroplets relevant to extreme ultraviolet lithography light sources, and varied imaging and spectroscopic simulations. These simulations highlight RHDLPP's effectiveness and applicability in fields such as laser-induced breakdown spectroscopy, extreme ultraviolet lithography sources, and high-energy-density physics.

Application of Model-Free and Model-Based Kinetic Methods in Evaluation of Reactions Complexity during Thermo-Oxidative Degradation Process: Case Study of [4-(Hydroxymethyl)phenoxymethyl] Polystyrene Resin

This work examined the possibilities and limitations of model-free and model-based methods related to decrypting the kinetic complexity of multi-step thermo-oxidative degradation processes (as a testing system, a [4-(hydroxymethyl)phenoxymethyl] polystyrene resin was used), monitored by thermal analysis (TGA-DTG-DTA) techniques. It was found that isoconversional methods could successfully determine the correct number of process stages and presence of multiple reactions based on derived Ea(α) profiles and identify the negative dependence of the rate constant on the temperature. These methods could not overcome the problem that arose due to mass transfer limitations. The model-based method overcame more successfully the problem associated with mass transfer because its calculation machinery had capabilities for the correct solution of the total mass balance equation. However, a perfect fit with the experimental data was not achieved due to the dependence on the thermal history of the contribution (ctb.) of a given reaction step inside a fitting procedure cycle. On the other hand, through this approach, it was possible to estimate the rate-controlling steps of the process regarding the influence of the heating rate. It was found that for consecutive reaction mechanisms, the production of benzaldehyde and gases in high yields was controlled by the heating rate, where low heating rates were strongly recommended (≤10 K/min). Also, it was shown that the transport phenomenon may be also the rate-determining step (within the set of “intrinsic” kinetic parameters). It was also established that external heat transfer controls the overall rate, where the “pure” kinetic control regime had not been reached but was approached when lowering the temperature and size of the resin particles.

Analysis of nonlinear fractional-order Fisher equation using two reliable techniques

Abstract In this article, the solution to the time-fractional Fisher equation is determined using two well-known analytical techniques. The suggested approaches are the new iterative method and the optimal auxiliary function method, with the fractional derivative handled in the Caputo sense. The obtained results demonstrate that the suggested approaches are efficient and simple to use for solving fractional-order differential equations. The approximate and exact solutions of the partial fractional differential equations for integer order were compared. Additionally, the fractional-order and integer-order results are contrasted using simple tables. It has been confirmed that the solution produced using the provided methods converges to the exact solution at the appropriate rate. The primary advantage of the suggested method is the small number of computations needed. Moreover, it may be used to address fractional-order physical problems in a number of fields.

Lax pairs informed neural networks solving integrable systems

Lax pairs are one of the most important features of integrable system. In this work, we propose the Lax pairs informed neural networks (LPNNs) tailored for the integrable systems with Lax pairs by designing novel network architectures and loss functions, comprising LPNN-v1 and LPNN-v2. The most noteworthy advantage of LPNN-v1 is that it can transform the solving of complex integrable systems into the solving of relatively simple Lax pairs, and it not only efficiently solves data-driven localized wave solutions, but also obtains spectral parameter and corresponding spectral function in spectral problems of the integrable systems. On the basis of LPNN-v1, we additionally incorporate the compatibility condition/zero curvature equation of Lax pairs in LPNN-v2, its major advantage is the ability to solve and explore high-accuracy data-driven localized wave solutions and associated spectral problems for integrable systems with Lax pairs. The numerical experiments focus on studying abundant localized wave solutions for very important and representative integrable systems with Lax pairs, including the soliton solution of the Korteweg-de Vries (KdV) equation and modified KdV equation, rogue wave solution of the nonlinear Schrödinger equation, kink solution of the sine-Gordon equation, non-smooth peakon solution of the Camassa-Holm equation and pulse solution of the short pulse equation, as well as the line-soliton solution of Kadomtsev-Petviashvili equation and lump solution of high-dimensional KdV equation. The innovation of this work lies in the pioneering integration of Lax pairs informed of integrable systems into deep neural networks, thereby presenting a fresh methodology and pathway for investigating data-driven localized wave solutions and spectral problems.

Bounded in the mean and stationary solutions of second-order difference equations with operator coefficients

We study the question of the existence of a unique bounded in the mean solution for the second-order difference equation with piecewise constant operator coefficients and of the stationary solution of the corresponding difference equation with constant operator coefficients. The case is considered when the corresponding “algebraic” operator equations have separated roots.

The data-driven rogue waves of the Hirota equation by using Mix-training PINNs approach

Recently, solving differential equations by deep neural networks has attracted lots of attention, however, when the solution of differential equations has localized areas such as the rogue waves, the classical physics-informed neural networks (PINNs) cannot guarantee its prediction accuracy. In this paper, we propose two neural networks methods: mix-training physics-informed neural networks (MTPINNs) and prior information mix-training physics-informed neural networks (PMTPINNs), two deep learning models with more approximation ability based on PINNs. A series of numerical experiments show that the learning efficiency and absolute error accuracy of our proposed model can be improved significantly. The PMTPINNs model can so as to make up for the shortcomings of PINNs in this aspect. Furthermore, by testing the robustness of these two methods, for two kinds of rogue waves of the Hirota equation, it is surprising that these models also have good performance and thereby may provide more possibilities for solving mathematical physics problems better.

Global existence and stability of the wave equation with boundary variable damping

AbstractIn this paper, we present the result of global existence of solution for the wave equation with boundary variable damping term. Then, we prove that this global solution is stable. Our study is based on the semi‐groups theory and some integral inequalities.

Global higher regularity and decay estimates for positive solutions of fractional equations in RN *

In the paper, we study the global higher regularity and decay estimates of the positive solutions for the following fractional equations {(−Δ)su+u=|u|p−2uin RN,lim|x|→∞u(x)=0,u∈Hs(RN),(0.1) where s∈(0,1) , N>2s , 2<p<2s∗:=2NN−2s and (−Δ)s is the fractional Laplacian. Let Q be a positive solution of (). We prove that Q∈Ck,γ(RN)∩Hk(RN) and obtain the decay estimates of D k Q as |x|→∞ for all k∈N+ and γ∈(0,1) . The argument relies on the Bessel kernel, comparison principle, Fourier analysis and iteration methods.

Application of Perturbation Method to Approximate the Solutions of Differential Equation

We investigate the core features of the perturbation method with the help of some simple but sophisticated problems and demonstrate how much accurately it predicts the solutions of the problems. To fulfill the target, we use the method for getting the solution of differential equations with initial and boundary conditions. Then the results obtained are compared with the series solution and the exact/numerical solution by using Mathematica and Fortran Programming. The comparisons are shown graphically. Also, the perturbation series approximation and the exact or numerical solution are in good agreement. Our investigation shows that a certain number of terms of the perturbation series gives an excellent approximation than the same number of terms of the numerical solution.

The classical double copy in curved spacetimes: perturbative Yang-Mills from the bi-adjoint scalar

We formulate a version of the double copy for classical fields in curved spacetimes. We provide a correspondence between perturbative solutions to the bi-adjoint scalar equations and those of the Yang-Mills equations in position space. At the linear level, we show that there exists a map between these solutions for maximally symmetric spacetime backgrounds, that provides every Yang-Mills solution by the action of an appropriate differential operator on a bi-adjoint scalar solution. Given the existence of a linearized map, we show that it is possible to cast the solutions of the Yang-Mills equations at arbitrary perturbation order in terms of the corresponding bi-adjoint scalar solutions. This all-order map is reminiscent of the flat space BCJ double copy, and works for any curved spacetime where the perturbative expansion holds. We show that these results have the right flat space limit, and that the correspondence is agnostic to the choice of gauge.

Continuous data assimilation for the three dimensional primitive equations with magnetic field

In this paper, the problem of continuous data assimilation of three dimensional primitive equations with magnetic field in thin domain is studied. We establish the well-posedness of the assimilation system and prove that the H 2 -strong solution of the assimilation system converges exponentially to the reference solution in the sense of L 2 as t → ∞. We also study the sensitivity analysis of the assimilation system and prove that a sequence of solutions of the difference quotient equation converge to the unique solution of the formal sensitivity equation.

Regularity for critical fractional Choquard equation with singular potential and its applications

Abstract We study the following fractional Choquard equation ( − Δ ) s u + u ∣ x ∣ θ = ( I α * F ( u ) ) f ( u ) , x ∈ R N , {\left(-\Delta )}^{s}u+\frac{u}{{| x| }^{\theta }}=({I}_{\alpha }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ⩾ 3 N\geqslant 3 , s ∈ 1 2 , 1 s\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) , α ∈ ( 0 , N ) \alpha \in \left(0,N) , θ ∈ ( 0 , 2 s ) \theta \in \left(0,2s) , and I α {I}_{\alpha } is the Riesz potential. The main purpose of this article is twofold. We first study the regularity of weak solutions for the aforementioned equation with critical nonlinearity, which extends the results of θ = 0 \theta =0 in Moroz-Van Schaftingen [Existence of groundstates for a class of nonlinear Choquardequations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579]. Then, as an application of the regularity results, we establish the existence of ground state solutions for above equation with the nonlinearity involving embedding top and bottom indices, which is related to the Hardy-Littlewood-Sobolev inequality and singular term 1 ∣ x ∣ θ \frac{1}{{| x| }^{\theta }} . It is worth noting that our approach is not involving the concentration-compactness principle.

Computationally Efficient and Interface Accurate Dual-Grid Phase-Field Simulation of Turbulent Drop-Laden Flows

Abstract In this work, we develop a dual-grid approach for the direct numerical simulations (DNS) of tur- bulent multiphase flows in the framework of the phase-field method (PFM). With the dual-grid approach, the solution of the Navier-Stokes equations (flow-field) and of the Cahn-Hilliard equa- tion (phase-field) are performed on two different computational grids. In particular, a base grid - fine enough to resolve the flow down to the Kolmogorov scale - is used for the solution of the Navier-Stokes equations, while a refined grid - required to improve the description of small interfacial structures - is used for the solution of the Cahn-Hilliard equation (phase-field method). The proposed approach is validated, and its computational efficiency is evaluated considering the deformation of a drop in a two-dimensional shear flow. Analyzing the computational time and memory usage, we observe a reduction between ≃30% and ≃40% (with respect to the single-grid approach), depending on the grid refinement factor employed for the phase-field variable. The applicability of the approach to a realistic three-dimensional case is also discussed, by focusing on the breakage of a thin liquid sheet inside a turbulent channel flow. Indications on the grid resolution representing a good compromise between accuracy and computational efficiency in drop-laden turbulence are also provided.

Transcendental meromorphic solutions of certain nonlinear difference equations

In this paper, we mainly explore the following finite order meromorphic solutions of the nonlinear difference equationfn(z)+Pd(z,f)=p1eα1zk+p2eα2zk, where n≥2, k≥1 are two integers, Pd(z,f) is a difference polynomial in f of degree d≤n−1 with polynomials as its coefficients, and pj, αj(j=1,2) are nonzero constants satisfying certain conditions. Furthermore, we obtain the exact forms of meromorphic solutions of the above difference equation. The result of this paper extends and improves previous results, and some examples are given to show the accuracy of the result.

Pogorelov estimates for semi-convex solutions of 𝑘-curvature equations

In this paper, we consider k k -curvature equations σ k ( κ [ M u ] ) = f ( x , u , ∇ u ) \sigma _k(\kappa [M_u])=f(x,u,\nabla u) subject to ( k + 1 ) (k+1) -convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J. 123 (2004), pp. 235–264]. By using the crucial concavity inequality for Hessian operator of Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23], we derive Pogorelov estimates of semi-convex admissible solutions for these k k -curvature equations.