Non-constant Coefficients Research Articles

Resolución de relaciones de recurrencia con apoyo de Wolfram Mathematica

This paper introduces some algorithms for solving linear relationships, homogeneous and non-homogeneous recurrence with constant and non-constant coefficients, using software as the main resource in solving processes. The Mathematica commercial application has provided the technical support necessary for the implementation of the methods used. It also presents other examples of recurrence relations, showing the effectiveness and limitations of the algorithms created by the author and programmed in Mathematica environment that provides.

Where Best to Place a Dirichlet Condition in an Anisotropic Membrane?

We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with nonconstant coefficients, over a fixed domain $\Omega$. Dirichlet conditions are imposed along $\partial \Omega$ and, additionally, along a set $\Sigma$ of prescribed length (one-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $\Sigma$ in order to maximize the first eigenvalue. The limit distribution of the optimal sets, as their prescribed length tends to infinity, is characterized via $\Gamma$-convergence of suitable functionals defined over varifolds; the use of varifolds, as opposed to probability measures, allows one to keep track of the local orientation of the optimal sets (which comply with the anisotropy of the problem), and not just of their limit distribution.

High-Order ADI Scheme for Option Pricing in Stochastic Volatility Models

We propose a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. Our approach combines different high-order spatial discretisations with Hundsdorfer and Verwer’s ADI time-stepping method, to obtain an efficient method which is fourth-order accurate in space and second-order accurate in time. Numerical experiments for the European put option pricing problem using Heston’s stochastic volatility model confirm the high-order convergence.

Study of fission dynamics of the excited nuclei produced in fusion reactions in the framework of the four-dimensional Langevin equations

The dynamics of fission of excited nuclei has been studied by solving four-dimensional Langevin equations with dissipation generated through the chaos-weighted wall and window friction formula. The projection of the total spin of the compound nucleus to the symmetry axis, K, was considered as the fourth dimension in Langevin dynamical calculations. The average pre-scission neutron multiplicities, mean kinetic energy of fission fragments and the variances of the mass and kinetic energy have been calculated in a wide range of fissile parameter for compound nuclei 162Yb, 172Yb, 215Fr, 224Th, 248Cf, 260Rf and results compared with the experimental data. Calculations were performed with a constant dissipation coefficient of K, γK (MeV zs)-1/2, and with a non-constant dissipation coefficient. Comparison of the theoretical results for the average pre-scission neutron multiplicities, mean kinetic energy of fission fragments and the variances of the mass and kinetic energy with the experimental data showed that the results of four-dimensional Langevin equations with a non-constant dissipation coefficient are in better agreement with the experimental data. Furthermore, the difference between the results of two models for compound nuclei with low fissile parameter is low whereas, for heavy compound nuclei, is high.

Study of fission time and prescission neutron multiplicity using four-dimensional Langevin equations

In this paper, four-dimensional stochastic approach is applied on dynamics of fission process to calculate mean fission time and prescission neutron multiplicity. Elongation, neck thickness, mass asymmetry and orientation degree of freedom (K-coordinate) are four dimensions in dynamical calculations. In order to investigate the influence of the dissipation coefficient of K-coordinate, the stochastic dynamics of the orientation degree of freedom using nonconstant dissipation coefficient was studied. Calculations were done for the 16 O +208 Pb and 16 O +232 Th reactions. Obtained results were compared with a constant dissipation coefficient (γK = 0.077( MeVzs )-1/2). The mean fission time and prescission neutron multiplicity for nonconstant dissipation case were lower than for the constant one.

A superconvergent fitted finite volume method for Black–Scholes equations governing European and American option valuation

We develop a superconvergent fitted finite volume method for a degenerate nonlinear penalized Black–Scholes equation arising in the valuation of European and American options, based on the fitting idea in Wang [IMA J Numer Anal 24 (2004), 699–720]. Unlike conventional finite volume methods in which the dual mesh points are naively chosen to be the midpoints of the subintervals of the primal mesh, we construct the dual mesh judiciously using an error representation for the flux interpolation so that both the approximate flux and solution have the second‐order accuracy at the mesh points without any increase in computational costs. As the equation is degenerate, we also show that it is essential to refine the meshes locally near the degenerate point in order to maintain the second‐order accuracy. Numerical results for both European and American options with constant and nonconstant coefficients will be presented to demonstrate the superconvergence of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1190–1208, 2015

Noetherian quotients of the algebra of partial difference polynomials and Gröbner bases of symmetric ideals

In this paper we develop a Gröbner bases theory for ideals of partial difference polynomials with constant or non-constant coefficients. In particular, we introduce a criterion providing the finiteness of such bases when a difference ideal contains elements with suitable linear leading monomials. This can be explained in terms of Noetherianity of the corresponding quotient algebra. Among these Noetherian quotients we find finitely generated polynomial algebras where the action of suitable finite dimensional commutative algebras and in particular finite abelian groups is defined. We obtain therefore a consistent Gröbner bases theory for ideals that possess such symmetries.

On the Monge and Kantorovich problems for distributions of diffusion processes

We prove that, for the distributions of one-dimensional diffusions with nonconstant diffusion coefficients, the Monge and Kantorovich problems associated with the cost function generated by the Cameron-Martin norm have no nontrivial solutions, i.e., are solvable only when the considered measures coincide. In particular, this is true if the diffusion coefficient is real-analytic and nonconstant.

Ground states of a nonlinear Schrödinger system with nonconstant potentials

In the case where either the potentials Vj, µj and β are periodic or Vj are well-shaped and µj and β are anti-well-shaped, existence of a positive ground state of the Schrodinger system $$\left\{ \begin{gathered} - \Delta u_1 + V_1 (x)u_1 = \mu _1 (x)u_1^3 + \beta (x)u_1 u_2^2 in\mathbb{R}^N , \hfill \\ - \Delta u_2 + V_2 (x)u_2 = \beta (x)u_1^2 u_2 + u_2 (x)u_2^3 in\mathbb{R}^N , \hfill \\ u_j \in H^1 (\mathbb{R}^N ),j = 1,2, \hfill \\ \end{gathered} \right.$$ where N = 1, 2, 3, is proved provided that β is either small or large in terms of Vj and µj. The system with constant coefficients has been studied extensively in the last ten years, and the nonconstant coefficients case has seldom been studied. It turns out that new technical machineries in the setting of variational methods are needed in dealing with the nonconstant coefficients case.

Direct Measurements of Drag Forces in C. elegans Crawling Locomotion

With a simple and versatile microcantilever-based force measurement technique, we have probed the drag forces involved in Caenorhabditis elegans locomotion. As a worm crawls on an agar surface, we found that substrate viscoelasticity introduces nonlinearities in the force-velocity relationships, yielding nonconstant drag coefficients that are not captured by original resistive force theory. A major contributing factor to these nonlinearities is the formation of a shallow groove on the agar surface. We measured both the adhesion forces that cause the worm’s body to settle into the agar and the resulting dynamics of groove formation. Furthermore, we quantified the locomotive forces produced by C. elegans undulatory motions on a wet viscoelastic agar surface. We show that an extension of resistive force theory is able to use the dynamics of a nematode’s body shape along with the measured drag coefficients to predict the forces generated by a crawling nematode.

Integration by interpolation and look-up for Galerkin-based isogeometric analysis

Even though isogeometric analysis has a clear advantage regarding the number of degrees of freedom needed to achieve a certain level of accuracy, the time needed for matrix assembly (by means of numerical integration of stiffness or mass matrix entries) constitutes a severe bottleneck in the process. In order to overcome this difficulty, we propose the new approach of Integration by Interpolation and Lookup (IIL). Firstly, applying spline interpolation to the common factors in the occurring integrals approximately transforms them into integrals of piecewise polynomial functions, whose integrands are expressed in tensor-product B-spline form. The common factors represent the influence of the geometry mapping (i.e., the NURBS domain parameterization) and the contributions of possibly non-constant material coefficients. Secondly, these integrals are evaluated exactly using pre-computed look-up tables for integrals of tri-products of univariate B-splines and their derivatives. For the model case of elliptic problems, we perform a theoretical analysis to demonstrate that the IIL method maintains the overall approximation order of the Galerkin discretization, provided that the spline interpolation is sufficiently accurate. Moreover, we provide a comparison of the computational complexity of our method with that of a standard Gauss quadrature method. Finally, we present experimental results to illustrate the performance of the IIL method and to support our theoretical results.

Hodge-type Decomposition for Time-dependent First-order Parabolic Operators with Non-constant Coefficients: The Variable Exponent Case

In this paper we present a Hodge-type decomposition for variable exponent spaces. More concretely, we address some time-dependent parabolic firstorder partial differential operators with non-constant coefficients, where one of the components is the kernel of the parabolic-type Dirac operator. This decomposition is presented over different types of domains in the n-dimensional Euclidean space n-dimensional Euclidean space \({\mathbb{R}^{n}}\). The case of the time-dependent Schrodinger operator is included as a special case within this context.

A Short Note on the Local Solvability of the Quaternionic Beltrami Equation

In this paper we discuss the local solvability of the inhomogeneous Beltrami equations in Quaternionic Analysis. We give an example of a Beltrami equation with no distributional solution and deduce the compatibility condition. This study is closely linked to the study of Dirac operators with non-constant coefficients.

Robust error bounds for finite element approximation of reaction–diffusion problems with non-constant reaction coefficient in arbitrary space dimension

We present a fully computable a posteriori error estimator for piecewise linear finite element approximations of reaction–diffusion problems with mixed boundary conditions and piecewise constant reaction coefficient formulated in arbitrary dimension. The estimator provides a guaranteed upper bound on the energy norm of the error and it is robust for all values of the reaction coefficient, including the singularly perturbed case. The approach is based on robustly equilibrated boundary flux functions of Ainsworth and Oden (2000) and on subsequent robust and explicit flux reconstruction. This paper simplifies and extends the applicability of the previous result of Ainsworth and Vejchodský (2011) in three aspects: (i) arbitrary dimension, (ii) mixed boundary conditions, and (iii) non-constant reaction coefficient. It is the first robust upper bound on the error with these properties. An auxiliary result that is of independent interest is the derivation of new explicit constants for two types of trace inequalities on simplices.

Generalized semi-analytical solution of advection–diffusion–reaction in finite and semi-infinite cylindrical ducts

Abstract A semi-analytical solution for the transient advection–diffusion–reaction problem within finite and semi-infinite ducts is derived. The solution allows for general radial- and time-dependent inlet/outlet conditions, complex boundary conditions on the duct wall including adsorption and decay, and arbitrary velocity profiles of the transporting fluid. The only numerical step of the solution is the inverse Laplace transform in the time variable. Therefore, the approach also produces fully analytical steady-state solutions. The solution is verified against computational fluid dynamics (CFD) simulations under various boundary conditions and velocity profiles (Newtonian and power-law), and in all cases good agreement is obtained. Although theoretically applicable to all regimes, the solution is computationally difficult at very high Peclet numbers and very early times due to numerical instabilities as a result of finite precision arithmetic of computers. A convergence analysis is conducted to delineate the boundaries of this limit for two important cases. The solution was derived using a new approach for solving two-dimensional partial differential equations (PDEs) with non-constant coefficients which parallels the Frobenius and power series methods for solving ordinary differential equations (ODEs). The approach reduces the original PDE to a single infinite-order ODE with constant coefficients. The approach is suspected to provide solutions to a large class of PDEs of this type. The solution may find applications in a number of engineering and/or biomedical fields, it can be used to verify numerical simulators, and serve as a simple and easy-to-implement alternative where access to numerical simulators is not available.

Stochastic Leontieff type equations with non-constant coefficients

Leontieff type equation is a differential-algebraic equation arising, in particular, in description of dynamically distorted signal in a radio device. In investigation of the signals, it is necessary to take into account noise that requires consideration of stochastic version of Leontieff type equations. For those equations, we apply the machinery of the so-called Nelson’s mean derivatives (more exactly, symmetric mean derivatives that is a natural analogue of physical velocity of deterministic processes) and obtain formulae of solutions in the case of non-constant coefficients of equations.

Stability of an implicit method to evaluate option prices under local volatility with jumps

In this paper, we consider a local volatility model with jumps under which the price of a European option can be derived by a partial integro-differential equation (PIDE) with nonconstant coefficients. In order to solve numerically the PIDE, we generalize the implicit method with three time levels which is constructed to avoid iteration at each time step. We show that the implicit method has the stability with respect to the discrete ℓ2-norm by using an energy method. We combine the implicit method with an operator splitting method to solve a linear complementarity problem (LCP) with nonconstant coefficients that describes the price of an American option. Finally we conduct some numerical simulations to verify the analysis of the method. The proposed method leads to a tridiagonal linear system at each time step and thus the option prices can be computed in a few seconds on a computer.

Evolution of superoscillatory data

Weak measurements and the theory of weak values have a very deep meaning in quantum mechanics, and new phenomena associated with them has recently been observed experimentally. This theory has also directly led to the notion of superoscillating sequences of functions. In this paper we consider Cauchy problems with superoscillatory initial conditions (in particular, the Cauchy problem for the Schrödinger equation and some of its variations), and we give conditions under which the superoscillations persist in time. Our work is based on results from the theory of formal solutions to Cauchy problems and on the study of the specific growth of superoscillatory sequences, when regarded as functions of a complex variable. There are two main aims of this paper: one is to explain the mathematical tools that are necessary to study superoscillations, also repeating a few results that we have already proved in other papers in order to clarify the strategy. The second aim is to show that our technique applies to a large class of problems, showing under which conditions the superoscillatory phenomenon persists. Finally, we point out that our strategy can be applied also to non-constant coefficients differential equations as the quantum harmonic oscillator.

Graphs whose Spectrum Determined by Non-constant Coefficients

Let G be a graph and M be a matrix associated with G whose characteristic polynomial is M(G,x)=∑i=0nαi(G)xn−i. We say that the spectrum of G is determined by non-constant coefficients (simply M-SDNC), if for any graph H with ai(H)=ai(G),0⩽i⩽n−1, then Spec(G)=Spec(H) (if M is the adjacency matrix or the Laplacian matrix of G, then G is called an A-SDNC graph or L-SDNC graph). In this paper, we study some properties of graphs which are A-SDNC or L-SDNC. Among other results, we prove that the path of order at least five is L-SDNC and moreover stars of order at least five are both A-SDNC and L-SDNC. Furthermore, we construct infinitely many trees which are not A-SDNC graphs. More precisely, we show that there are infinitely many pairs (T,T′) of trees such that A(T,x)−A(T′,x)=−1.

Stabilized Finite Elements for a Reaction-Dispersion Saddle-Point Problem with NonConstant Coefficients

We consider a system of two reaction-dispersion equations with nonconstant parameters. Both equations are coupled through the boundary conditions. We propose a mixed variational formulation that leads to a nonsymmetric saddle-point problem. We prove its well-posedness. Then, we develop a stabilized mixed finite element discretization of this problem and establish optimal a priori error estimates.