Piecewise Linear Valued CSPs Solvable by Linear Programming Relaxation

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of the complexity of infinite-domain VCSPs with piecewise linear homogeneous cost functions. Such VCSPs can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by reducing the problem to a finite-domain VCSP which can be solved using the basic linear program relaxation. It follows that VCSPs for submodular PLH cost functions can be solved in polynomial time; in fact, we show that submodular PLH functions form a maximally tractable class of PLH cost functions.

Algebraic stability modes in rotational shear flow

We investigate the two-dimensional (2D) stability of rotational shear flows in an unbounded domain. The eigenvalue problem is formulated by using a novel algebraic mode decomposition distinct from the normal modes with temporal evolution $\exp(\omega t)$. Based on the work of \citeasnoun{NoldOberlack2013}, we show how these new modes can be constructed from the symmetries of the linearized stability equation. For the azimuthal base flow velocity $V(r)=r^{-1}$ an additional symmetry exists, such that a mode with algebraic temporal evolution $t^s$ is found. $s$ refers to an eigenvalue for the algebraic growth or decay of the kinetic energy of the perturbations. An eigenvalue problem for the viscous and inviscid stability using algebraic modes is formulated on an infinite domain with $r \to \infty$. An asymptotic analysis of the eigenfunctions shows that the flow is linearly stable under 2D perturbations. We find stable modes with the algebraic mode ansatz, which can not be obtained by a normal mode analysis. The stability results are in line with Rayleigh's inflection point theorem.

A Comparison Study of Numerical Techniques for Solving Ordinary Differential Equations Defined on a Semi-Infinite Domain Using Rational Chebyshev Functions

A rational Chebyshev (RC) spectral collocation technique is considered in this paper to solve high-order linear ordinary differential equations (ODEs) defined on a semi-infinite domain. Two definitions of the derivative of the RC functions are introduced as operational matrices. Also, a theoretical study carried on the RC functions shows that the RC approximation has an exponential convergence. Due to the two definitions, two schemes are presented for solving the proposed linear ODEs on the semi-infinite interval with the collocation approach. According to the convergence of the RC functions at the infinity, the proposed technique deals with the boundary value problem which is defined on semi-infinite domains easily. The main goal of this paper is to present a comparison study for differential equations defined on semi-infinite intervals using the proposed two schemes. To demonstrate the validity of the comparisons, three numerical examples are provided. The obtained numerical results are compared with the exact solutions of the proposed problems.

Analysis of Free Vibration Characteristics of Cylindrical Shells with Finite Submerged Depth Based on Energy Variational Principle

Based on the principle of energy variation, a calculation model for the free vibration characteristics of a cylindrical shell with a finite submerged depth considering the influence of the free liquid surface is established in this paper. First, the Euler beam function is used instead of the shell axial displacement function to obtain the shell kinetic energy and potential energy. Then, by using the mirror image method, the analytical expression of the fluid velocity potential considering the free surface is obtained, and the flow field is added to the system energy functional in the form of fluid work. Then the energy functional is changed to obtain the shell–liquid coupled vibration equation. Solving the equation can obtain the natural frequencies and modes of the structure. The comparison with the finite element calculation results verifies the accuracy of the calculation model in this paper. The research on the influence of the free liquid surface shows that compared to the infinite domain, the free liquid surface destroys the symmetry of the entire system, resulting in a difference in the natural frequency of the positive and negative modes of the shell, and the circumferential mode shapes are no longer mutually uncoupled trigonometric functions. The existence of free liquid surface will also increase the natural frequency of the same order mode, and the closer to the free surface, the natural frequency is greater. As the immersion depth increases, the free vibration characteristics will quickly tend to the result of infinite domain. Additionally, when the immersion depth is equal to or greater than four times the radius of the shell structure, it can be considered that the free liquid surface has no effect. These law and phenomena have also been explained from the mechanism. The method in this paper provides a new analytical solution pattern for solving this type of problem.

Sinc-Self-consistent method to solve a class of nonlinear eigenvalue differential equation

In this paper, we combine the sinc and self-consistent methods to solve a class of non-linear eigenvalue differential equations. Some properties of the self-consistent and sinc methods required for our subsequent development are given and employed. Numerical examples are included to demonstrate the validity and applicability of the introduced technique and a comparison is made with the existing results. The method is easy to implement and yields accurate results. We show that the sinc-self-consistent method can solve the equations on an infinite domain and produces the smallest eigenvalue with the most accuracy

Reconstructing a point source from diffusion fluxes to narrow windows in three dimensions

We develop a computational approach to locate the source of a steady-state gradient of diffusing particles from the fluxes through narrow windows distributed either on the boundary of a three-dimensional half-space or on a sphere. This approach is based on solving the mixed boundary stationary diffusion equation with Neumann–Green’s function. The method of matched asymptotic expansions enables the computation of the probability fluxes. To explore the range of validity of this expansion, we develop a fast analytical-Brownian numerical scheme. This scheme accelerates the simulation time by avoiding the explicit computation of Brownian trajectories in the infinite domain. The results obtained from our derived analytical formulae and the fast numerical simulation scheme agree on a large range of parameters. Using the analytical representation of the particle fluxes, we show how to reconstruct the location of the point source. Furthermore, we investigate the uncertainty in the source reconstruction due to additive fluctuations present in the fluxes. We also study the influence of various window configurations: clustered versus uniform distributions on recovering the source position. Finally, we discuss possible applications for cell navigation in biology.

Approximate Analytical Solution of a Power-Law Pseudoplastic Fluid in a Boundary Layer over a Porous Plate: A New Application of Multi-Stage Parker-Sochacki Method

In this paper, based on Parker-Sochacki method for solving a system of differential equations,a multistage technique is developed for solving the nonlinear boundary layer equations of powerlawfluid on infinite domain. The problem domain is split into subintervals over which the boundaryvalue problem is replaced with a sequence of subproblems. In a shooting-like approach, the boundarycondition at infinity is converted to an equivalent initial condition. By recasting the problem as apolynomial system of first-order autonomous equations, the sub-problems are solved with Parker-Sochacki method with very high accuracy. The interval of convergence of the solution is deriveda-priorly in terms of the parameters of the polynomial system, which guides optimal choice of thediscretization parameter. The technique yielded a convergent piecewise continuous solution over theproblem domain. The results obtained, demonstrated graphically and in tables, compared well withexisting ones in the literature.

The vibrational response of a fluid-loaded baffled plate near a free surface

An analytical model to predict the vibrational response of a simply supported rectangular plate embedded in an infinite baffle with an upper free surface under heavy fluid loading and excited by a point force is presented. The equations of motion of a thin plate are solved using­­­­­­ modal decomposition technique by employing admissible functions for an in-vacuo plate and by directly solving the Helmholtz equation for acoustic waves in a fluid. The vibrational response for a flat plate in an infinite baffle and unbounded domain (semi-infinite domain) using analytical formulation available in literature is initially computed. These results are then compared against present results to observe the effect of a free surface. Predictions from analytical models are validated by comparison with results obtained by numerical models. The proposed analytical approach presents a novel formulation to describe a fluid-loaded flat plate in a waveguide and an efficient method for predicting its vibrational response.

Novel ANN Method for Solving Ordinary and Time-Fractional Black–Scholes Equation

The main aim of this study is to introduce a 2-layered artificial neural network (ANN) for solving the Black–Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine-tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black–Scholes model are reported.