Class Of Solutions Research Articles

Singular solutions for space-time fractional equations in a bounded domain

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann–Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.

Faster classical boson sampling

Abstract Since its introduction boson has been the subject of intense study in the world of quantum computing. In the Fock protocol, the task is to sample independently from the set of all n by n submatrices built from possibly repeated rows of a larger m by n complex matrix according to a probability distribution related to the permanents of the submatrices. Experimental systems exploiting quantum photonic effects can in principle perform the task at high speed. In the framework of classical computing, Aaronson and Arkhipov (2011) showed that exact boson sampling problem cannot be solved in polynomial time unless the polynomial hierarchy collapses to the third level. Indeed for a number of years the fastest known exact classical algorithm ran in O(binomial(m+n-1, n) n 2^n ) time per sample, emphasising the potential speed advantage of quantum computation. The advantage was reduced by Clifford and Clifford (2018) who gave a significantly faster classical solution taking O(n 2^n + poly(m,n)) time and linear space, matching the complexity of computing the permanent of a single matrix when m is polynomial in n.
 
 We continue by presenting an algorithm for Fock boson sampling whose average-case time complexity is much faster when m is proportional to n. In particular, when m = n our algorithm runs in approximately O(n 1.69^n) time on average. This result further increases the problem size required to establish quantum computational advantage via the Fock scheme of boson sampling.

Global Well-Posedness of Classical Solutions to the Compressible Navier–Stokes–Poisson Equations with Slip Boundary Conditions in 3D Bounded Domains

Global Well-Posedness of Classical Solutions to the Compressible Navier–Stokes–Poisson Equations with Slip Boundary Conditions in 3D Bounded Domains

Convolutional double copy in (anti) de Sitter space

The double copy is a remarkable relationship between gauge theory and gravity that has been explored in a number of contexts, most notably scattering amplitudes and classical solutions. The convolutional double copy provides a straightforward method to bridge the two theories via a precise map for the fields and symmetries at the linearised level. This method has been thoroughly investigated in flat space, offering a comprehensive dictionary both with and without fixing the gauge degrees of freedom. In this paper, we extend this to curved space with an (anti) de Sitter background metric. We work in the temporal gauge, and employ a modified convolution that involves the Mellin transformation in the time direction. As an example, we show that the point-like charge in gauge theory double copies to the (dS-) Schwarzschild black hole solution.

The non-homogeneous Cauchy problem of semigroup of linear operators

This paper presents the results of ω-order reversing partial contraction mapping generated by the non-homogeneous Cauchy problem. We investigated several concepts of solution for non-homogeneous Cauchy problem where an analysis was arrived out on the relationship between C' strong and respectively C0-solution. We showed that Z is the infinitesimal generator of a C0-semigroup of contractions. Finally, we obtained that the problem has a unique classical solution and f, a member of L1(a,b; X) is continuous on (a,b).

Contribution of PRIDE VLBI products to the joint JUICE-Europa Clipper moons’ ephemerides solution

In the coming decade, JUICE and Europa Clipper radio-science will yield the most accurate estimation to date of the Galilean moons’ physical parameters and ephemerides. JUICE’s PRIDE (Planetary Radio Interferometry and Doppler Experiment) will help achieve such a solution by providing VLBI (Very Long Baseline Interferometry) observations of the spacecraft’s lateral position, complementing nominal radio-science measurements. In this paper, we quantify how PRIDE VLBI can contribute to the moons’ ephemerides determination, in terms of attainable solution improvement and validation opportunities. To this end, we simulated VLBI data for JUICE, but also investigated the possibility to perform simultaneous tracking of JUICE and Europa Clipper, thus ultimately generating both single- and dual-spacecraft VLBI. We considered various tracking and data quality scenarios for both VLBI types, and compared the formal uncertainties provided by covariance analyses with and without VLBI. These analyses were performed for both global and local (i.e. per-flyby) estimations of the moons’ states, as eventually achieving a global solution first requires proceeding arc-per-arc. We showed that both single- and multi-spacecraft VLBI measurements only bring limited improvement to the global state estimation, but significantly contribute to the moons’ normal points (i.e. local states at flyby times), most notably in the out-of-plane direction. Additionally, we designed a validation plan exploiting PRIDE VLBI to progressively validate the classical radio-science solution, whose robustness and statistical realism is sensitive to modelling inconsistencies. By improving the local state estimations and offering various validation opportunities, PRIDE will be invaluable in overcoming possible dynamical challenges. It can therefore play a key role in reconstructing a global solution for the Galilean moons’ dynamics with the uncertainty levels promised by JUICE-Europa Clipper analyses. This, in turn, is critical to the accurate characterisation of tidal dissipation in the Jovian system, holding the key to the long-term evolution of the Galilean moons.

A fractional Darcian model‐based analytical solution for non‐Darcian flow toward a fully penetrating well in a confined aquifer

AbstractThe Forchheimer and Izbash equations have been long employed to investigate the behavior of non‐Darcian flow toward a well in various aquifer systems, but both two equations inevitably introduce problems such as more or less empirical nature, and dimensional unbalance. Therefore, this work makes the attempt to introduce the fractional Darcian model for characterizing the non‐Darcian behavior flow toward a fully penetrating well in a confined aquifer instead of the Forchheimer and Izbash equations. In this study, a fractional Darcian model‐based analytical solution in the time domain is obtained by means of Laplace transform and linearization approximation. The proposed analytical solution of this study can be readily reduced to the classical Theis solution for Darcian flow. Meanwhile, the late‐times and steady‐state analytical solutions for non‐Darcian flow described using the fractional Darcian model are also developed. Moreover, a comparison with previous analytical solutions shows that the newly derived analytical solution in this study is sufficiently accurate at later times. The influences of different parameters on transient drawdown are investigated. The results indicate that the fractional derivative order and hydraulic conductivity have a large influence on drawdown compared with other parameters. The introduction of fractional Darcian model in this study could also provide potential application for further investigations of non‐Darian flow behavior toward fully or partially penetrating wells in different aquifer systems, which can be very beneficial for hydrology and other related fields.

Conditions for the existence of power solutions to a linear difference equation with constant coefficients

With the help of the formula for the general solution of a difference equation with constant coefficients, it is shown that the set of solutions to this equation contains classical solutions of the type km\λk. We present necessary and sufficient conditions on the coefficients of the equation and the initial parameters under which such solutions are obtained.

Three solutions for a discrete fourth-order boundary value problem with four parameters

This paper presents several sufficient conditions for the existence of at least three classical solutions of a boundary value problem for a fourth-order difference equation. Fourth-order boundary value problems act as models for the bending or deforming of elastic beams. In different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, economics and many others,the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. Our technical approach is based on variational methods. An example is included in the paper.Numerical computations of the example confirm our theoretical results.

More holographic M5 branes in AdS7 × S4

We study classical M5 brane solutions in the probe limit in the AdS7×S4 spacetime geometry with worldvolume 3-form flux. These solutions describe the holography of codimension-4 defects in the 6d boundary dual N=(0,2) supersymmetric gauge theories. Starting from some half-BPS solutions which follow stricter BPS conditions, we find a general 116-BPS solution. We show how some of the giant-like M5 brane solutions here may be related to the codimension-2 surface operators in the 4d gauge theory.

Competition between self- and other-regarding preferences in resolving social dilemmas

Abstract Evolutionary game theory assumes that individuals maximize their benefits when choosing strategies. However, an alternative perspective proposes that individuals seek to maximize the benefits of others. To explore the relationship between these perspectives, we develop a model where self- and other-regarding preferences compete in public goods games. We find that other-regarding preferences are more effective in promoting cooperation, even when self-regarding preferences are more productive. Cooperators with different preferences can coexist in a new phase where two classic solutions invade each other, resulting in a dynamical equilibrium. As a consequence, a lower productivity of self-regarding cooperation can provide a higher cooperation level. Our results, which are also valid in a well-mixed population, may explain why other-regarding preferences could be a viable and frequently observed attitude in human society.

General decay of solutions for a viscoelastic porous system with Kelvin-Voigt damping

This study investigates the asymptotic stability of a viscoelastic porous system with Kelvin-Voigt damping. The system under consideration involves coupled equations describing the displacement of the solid elastic material, the volume fraction, and a viscoelastic or memory term. Previous studies have shown that introducing control to one equation of these systems can lead to uniform stability for weak solutions when the wave speeds are equal. However, when the wave propagation speeds differ, the decay rate of classical solutions becomes slower. In this work, we improve previous findings by considering the viscoelastic porous elasticity system with Kelvin-Voigt damping applied to the volume fraction equation. By employing the multiplier method in conjunction with the characteristics of convex functions, we establish a general stability estimate independent of the wave speeds. Our results provide valuable insights into the stability properties of viscoelastic porous systems. This study contributes to the existing literature on the subject and has implications for the analysis and control of such systems in various engineering applications.

On an inverse boundary-value problem for the pseudohyperbolic equation with nonclassical boundary conditions

In this paper, we consider an inverse boundary-value problem for a fourth-order pseudohyperbolic equation with nonclassical boundary conditions. The primary purpose of the work is to study the existence and uniqueness of the classical solution of the considered inverse boundary-value problem. To investigate the solvability of the considered problem, we carried out a transformation from the original problem to some auxiliary equivalent problem with trivial boundary conditions. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness of the classical solution of the original problem are shown.

On an attraction-repulsion chemotaxis model involving logistic source

This paper is concerned with the attraction-repulsion chemotaxis system involving logistic source: u_{t}=Δu-χ∇⋅(u∇υ)+ξ∇⋅(u∇ω)+f(u), ρυ_{t}=Δυ-α₁υ+β₁u, ρω_{t}=Δω-α₂ω+β₂u under homogeneous Neumann boundary conditions with nonnegative initial data (u₀,υ₀,ω₀)∈ (W^{1,∞}(Ω))³, the parameters χ, ξ, α₁, α₂, β₁, β₂>0, ρ≥0 subject to the non-flux boundary conditions in a bounded domain Ω⊂ℝ^{N}(N≥3) with smooth boundary and f(u)≤au-μu² with f(0)≥0 and a≥0, μ>0 for all u>0. Based on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a globally bounded classical solution provided that χ+ξ0 is sufficiently small for all β₁, β₂

Optimization for stochastic model arisen from investment problem associated with default risk

In this paper, we study the optimal strategy arisen from the excess-of-loss reinsurance and asset allocation in defaultable markets under a general stochastic model. By developing the corresponding dynamic programming approach, we establish the optimal investment approach through two sub-problems: a pre-default case and a post-default case, respectively, characterized by the obtained Hamilton–Jacobi–Bellman (HJB) equations. We show the existence of a classical solution to the pre-default case via super-sub solution techniques and give an explicit characterization of the optimal reinsurance and investment policy that maximizes the common used utility associated with the terminal wealth. Verification theorem is further established to show the uniqueness of the corresponding solution of HJB equation that is critical to the desired optimal solution.

Surface Green’s functions of a horizontally graded elastic half-plane

High-throughput mechanical testing based on functionally graded specimens is very promising for accelerating the development of new materials. However, due to the inhomogeneity-induced complexity, most existing analyses on functionally graded materials have recourse to numerical methods to predict their mechanical responses in reaction to external stimuli. This work investigates the surface Green’s functions for an inhomogeneous half-plane with horizontal exponential material gradient subject to both normal and tangential concentrated forces acting on the surface. The governing equations are first simplified by introducing appropriate potential functions, which facilitates the mathematical derivation of displacements via the Fourier transform technique. In the case of normal force, the vertical surface displacement is derived explicitly under the weak gradient assumption while the horizontal surface displacement is derived directly without the same assumption. In particular, the Meijer G-function and Fox H-function are introduced to express and simplify the vertical displacement. In the case of tangential force, the analytical expressions of surface displacements are also derived similarly. It is noted that the surface Green’s functions not only exhibit singularity and asymmetry properties as expected, but also can be reduced to the classical Boussinesq-Flamant solutions for a homogeneous half-plane. In addition, the analytical results are verified through comparison with the finite element analyses. The surface Green’s functions derived here could be a theoretical basis for developing high-throughput mechanical testing methods which use specimens made of functionally graded materials.

Global Solution and Stability of a Haptotaxis Mathematical Model for Complex MAP

A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing “third-party” molecular anchors to crosslink nanoparticulates to the matrix is an effective strategy, and a trapped nanoparticulate formed a desired complex MAP that is necessary to keep the nanoparticulate immobilized at any given time. In this paper, the global solution and stability of a parabolic–ordinary-parabolic haptotaxis system to complex MAP are studied. First, the existence of a local classical solution to system () has been observed using fixed point argument and parabolic Schauder estimates. Furthermore, some a priori estimates that can raise the regularity estimate of the solution for the relatively complicated first equation of system () from Lρ to L2ρ (ρ≥1) are given; then, the local classic solution can thus extend to the global classic solution when the space dimension N≤3. Lastly, by using various analytical methods, a threshold value ξ00(ξ00<0) is found, such that positive constant steady state (u*,v*,w*) becomes unstable when ξ<ξ00. Our results show that the haptotaxis plays a crucial role in determining the stability to the model (), that is, it can have a destabilizing effect.

Classical solution of the Cauchy problem for a semilinear hyperbolic equation in the case of two independent variables

In the upper half-plane, we consider a semilinear hyperbolic partial differential equation of order higher than two. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. The solution is constructed in an implicit analytical form as a solution of some integral equation. The local solvability of this equation is proved by the Banach fixed point theorem and/or the Schauder fixed point theorem. The global solvability of this equation is proved by the Leray-Schauder fixed point theorem. For the problem in question, the uniqueness of the solution is proved and the conditions under which its classical solution exists are established.

Coefficient inverse problem for an equation of mixed parabolic-hyperbolic type with a non-characteristic line of type change

In this paper, we study the direct and two inverse problems for a model equation of mixed parabolic-hyperbolic type. In the direct problem, the Tricomi problem for this equation with a non-characteristic line of type change is considered. The unknown of the inverse problem is the variable coefficient at the lowest derivative in the parabolic equation. To determine it, two inverse problems are studied: with respect to the solution defined in the parabolic part of the domain, the integral overdetermination condition (inverse problem 1) and one simple observation at a fixed point (inverse problem 2) are given. Theorems on the unique solvability of the formulated problems in the sense of classical solution are proved.

Classical Solutions of Rayleigh–Taylor instability for inhomogeneous incompressible viscous fluids in bounded domains

Classical Solutions of Rayleigh–Taylor instability for inhomogeneous incompressible viscous fluids in bounded domains