Cauchy Conditions Research Articles

On forward and backward SPDEs with non-local boundary conditions

We study linear stochastic partial differential equations of parabolic type with non-local in time or mixed in time boundary conditions. The standard Cauchy condition at the terminal time is replaced by a condition that mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. For the case of backward equations, this setting covers almost surely periodicity. Uniqueness, solvability and regularity results for the solutions are obtained. Some possible applications to portfolio selection are discussed.

Classical solution of a problem with an integral condition for the one-dimensional wave equation

We find a closed-form classical solution of the homogeneous wave equation with Cauchy conditions, a boundary condition on the lateral boundary, and a nonlocal integral condition involving the values of the solution at interior points of the domain. A classical solution is understood as a function that is defined everywhere in the closure of the domain and has all classical derivatives occurring in the equation and conditions of the problem. The derivatives are defined via the limit values of finite-difference ratios of the function and corresponding increments of the arguments.

Low-Resolution Modeling of Dense Drainage Networks in Confining Layers.

Groundwater-surface water (GW-SW) interaction in numerical groundwater flow models is generally simulated using a Cauchy boundary condition, which relates the flow between the surface water and the groundwater to the product of the head difference between the node and the surface water level, and a coefficient, often referred to as the "conductance." Previous studies have shown that in models with a low grid resolution, the resistance to GW-SW interaction below the surface water bed should often be accounted for in the parameterization of the conductance, in addition to the resistance across the surface water bed. Three conductance expressions that take this resistance into account were investigated: two that were presented by Mehl and Hill (2010) and the one that was presented by De Lange (1999). Their accuracy in low-resolution models regarding salt and water fluxes to a dense drainage network in a confined aquifer system was determined. For a wide range of hydrogeological conditions, the influence of (1) variable groundwater density; (2) vertical grid discretization; and (3) simulation of both ditches and tile drains in a single model cell was investigated. The results indicate that the conductance expression of De Lange (1999) should be used in similar hydrogeological conditions as considered in this paper, as it is better taking into account the resistance to flow below the surface water bed. For the cases that were considered, the influence of variable groundwater density and vertical grid discretization on the accuracy of the conductance expression of De Lange (1999) is small.

Shape Reconstruction of RF-Driven Divertor Plasma on QUEST

In the present RF-driven plasma with a lot of high-energy electrons, there may be anisotropic plasma pressure, which makes the usual equilibrium analysis difficult, but the Cauchy condition surface method can reconstruct the plasma shape precisely regardless of the anisotropy. In addition, the plasma current effect in the open magnetic surfaces outside of the closed magnetic surfaces is considered in the RF-driven divertor plasma. In the reconstruction process, singular value (SV) decomposition is used and optimal criterion function for generalized cross validation is estimated concerning truncation or reduction of the small-SV components.

Consistent boundary conditions at nonconducting surfaces of planetary bodies: Applications in a new Ganymede MHD model

AbstractThe interaction of planetary bodies with their surrounding magnetized plasma can often be described with the magnetohydrodynamic (MHD) equations, which are commonly solved by numerical models. For these models it is necessary to define physically correct boundary conditions for the plasma mass and energy density, the plasma velocity, and the magnetic field. Many planetary bodies have surfaces whose electrical conductivity is negligibly small and thus no electric current penetrates their surfaces. Magnetic boundary conditions, which consider that the associated radial electric current at the planetary surface is zero, are difficult to implement because they include the curl of the magnetic field. Here we derive new boundary conditions by a decomposition of the magnetic field in poloidal and toroidal parts. We find that the toroidal part of the magnetic field needs to vanish at the surface of the insulator. For the spherical harmonics coefficients of the poloidal part, we derive a Cauchy boundary condition, which also matches a possible intrinsic field by including its Gauss coefficients. Thus, we can additionally include planetary dynamo fields as well as time‐variable induction fields within electrically conductive subsurface layers. We implement the nonconducting boundary condition in the MHD simulation code ZEUS‐MP using spherical geometry and provide a numerical implementation in Fortran 90 as supporting information on the JGR website. We apply it to a model for Ganymede's plasma environment. Our model also includes a consistent set of boundary conditions for the other MHD variables density, velocity, and energy. With this model we can describe Galileo spacecraft observations in and around Ganymede's minimagnetosphere very well.

Some further results on ideal summability of nets in ( $$\ell $$ ℓ ) groups

In this paper we continue in the line of recent investigation of order summability of nets using ideals by Boccuto et al. (Czechoslovak Math. J. 62(137):1073–1083 2012; J. Appl. Anal. 20(1), 2014) where they had introduced the notions of \(\mathcal {I}\) and \(\mathcal {I}^*\) order convergence, \(\mathcal {I}\) and \(\mathcal {I}^*\) divergence of nets and its further extensions, namely the notions of \(\mathcal {I}^{\mathcal {K}}\)-order convergence and \(\mathcal {I}^{\mathcal {K}}\)-divergence of nets in a \((\ell )\)-group and investigate the relation between \(\mathcal {I}\) and \(\mathcal {I}^{\mathcal {K}}\)-concepts where a special class of ideals called \(\Lambda P(\mathcal {I},\mathcal {K})-\)ideals plays very important role. We also introduce, for the first time, the notion of \(\mathcal {I}^{\mathcal {K}}\)-order Cauchy condition and \(\mathcal {I}\)-order cluster points of nets in (\(\ell \)) groups and examine some of its characterizations and its consequences. In particular the role of \(\mathcal {I}\)-order cluster points in making the above mentioned Cauchy nets convergent is studied.

Validation of plasma shape reconstruction by Cauchy condition surface method in KSTAR

Cauchy Condition Surface (CCS) method is a numerical approach to reconstruct the plasma boundary and calculate the quantities related to plasma shape using the magnetic diagnostics in real time. It has been applied to the KSTAR plasma in order to establish the plasma shape reconstruction with the high elongation of plasma shape and the large effect of eddy currents flowing in the tokamak structures for the first time. For applying the CCS calculation to the KSTAR plasma, the effects by the eddy currents and the ferromagnetic materials on the plasma shape reconstruction are studied. The CCS calculation includes the effect of eddy currents and excludes the magnetic diagnostics, which is expected to be influenced largely by ferromagnetic materials. Calculations have been performed to validate the plasma shape reconstruction in 2012 KSTAR experimental campaign. Comparison between the CCS calculation and non-magnetic measurements revealed that the CCS calculation can reconstruct the accurate plasma shape even with a small IP.

КЛАССИЧЕСКОЕ РЕШЕНИЕ ЗАДАЧИ С ИНТЕГРАЛЬНЫМ УСЛОВИЕМ ДЛЯ ОДНОМЕРНОГО ВОЛНОВОГО УРАВНЕНИЯ

We find a closed-form classical solution of the homogeneous biwave equation with Cauchy conditions, a boundary condition on the lateral boundary, and a nonlocal integral condition involving the values of the solution at interior points of the domain. A classical solution is understood as a function that is defined everywhere in the closure of the domain and has all classical derivatives occurring in the equation and conditions of the problem.

Methods for limiting the calculation area during problem solving by the finite difference method

By using the integro-differential approach and classical boundary conditions (such as Dirichlet's, Neumann's or the very rarely used Cauchy boundary condition) for solving the two-dimensional problems in open space by the finite difference method, it is possible to - in the numerically exact way - close the calculation area to finite distance. Thus, one of great limitations of the finite difference method is overcome.

Reconstruction of dynamically changing boundary of multilayer heat conduction composite walls

Abstract This paper presents an inverse reconstruction procedure to determine the inner boundary location of heat conduction composite walls from the measurement data of temperature and heat flux on the exterior boundary. Our procedure uses a meshless forward solver that was developed for solving inhomogeneous heat transfer problems across a multilayer composite wall with Cauchy conditions. The forward solver uses the radial basis functions (RBFs) approximation in both time and space in a unified fashion, and hence is well suited for inverse problems. In this work, we consider that the length of the inner layer of the composite wall may vary caused by the material erosion at very high temperature such as in iron-making blast furnaces. In order to mitigate the ill-posed inverse problem, we use the Tikhonov regularization technique to obtain a stable and accurate numerical approximation of the moving boundary. Numerical experiments for a number of examples are presented to demonstrate the effectiveness of our inverse procedure. It can be observed that the error of the inverse solution is smaller or at the same level of noises in the simulated measurement data, demonstrating that our inverse procedure is effective and stable with respect to noisy data for moving boundary problems.

Extending asymmetric convergence and Cauchy condition using ideals

Abstract In this paper we use the notion of ideals to extend the convergence and Cauchy conditions in asymmetric metric spaces. The asymmetry (or rather, absence of symmetry) of these spaces makes the whole treatment different from the metric case and we use a genuinely asymmetric condition called (AMA) to prove many results and show that certain classic results fail in the asymmetric context if the assumption is dropped.

Use of a twisted 3D Cauchy condition surface to reconstruct the last closed magnetic surface in a non-axisymmetric fusion plasma

The three-dimensional (3D) Cauchy condition surface (CCS) method code, ‘CCS3D’, is now under development to reconstruct the 3D magnetic field profile outside a non-axisymmetric fusion plasma using only magnetic sensor signals. A new ‘twisted CCS’ is introduced, whose elliptic cross-section rotates with the variation in plasma geometry in the toroidal direction of a helical-type device. Independent of the toroidal angle, this CCS can be placed at a certain distance from the last closed magnetic surface (LCMS). With this new CCS, it is found through test calculations for the Large Helical Device that the numerical accuracy in the reconstructed field is improved. Furthermore, the magnetic field line tracing indicates the LCMS more precisely than with the use of the axisymmetric CCS. A new idea to determine the LCMS numerically is also proposed.

Solution of the cauchy problem for a hyperbolic equation with constant coefficients in the case of two independent variables

On the plane, we consider a linear partial differential equation of arbitrary order of hyperbolic type. The operator in the equation is a composition of first-order differential operators. The equation is accompanied with Cauchy conditions. For the equation, we obtain an analytic form of the general solution, from which we single out the unique classical solution of the Cauchy problem.

Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters

We are concerned with the inverse problem of detecting sources in a coupled diffusion-reaction system. This problem arises from the Biochemical Oxygen Demand-Dissolved Oxygen model($^1$) governing the interaction between organic pollutants and the oxygen available in stream waters. The sources we consider are point-wise and simulate stationary or moving pollution sources. The ultimate objective is to obtain their discharge location and recover their output rate from accessible measurements of DO when BOD measurements are difficult and time consuming to obtain. It is, as a matter of fact, the most realistic configuration. The subject to address here is the identifiability of these sources, in other words to determine if the observations uniquely determine the sources. The key tool is the study of coupled parabolic systems derived after restricting the global model to regions at the exterior of the observations. The absence of any prescribed condition on the BOD density is compensated by data recorded on the DO which provide over-determined Cauchy boundary conditions. Now, the first step toward the identifiability of the sources is precisely to recover the BOD at the observation points (of DO). This may be achieved by handling and solving the coupled systems. Unsurprisingly, they turn out to be ill-posed. That issue is investigated first. Then, we state a uniqueness result owing to a suitable saddle-point variational framework and to Pazy's uniqueness Theorem. This uniqueness complemented by former identifiability results proved in [2011, Inverse problems] for scalar reaction-diffusion equations yields the desired identifiability for the global model.

Monotone and nonmonotone trust-region-based algorithms for large scale unconstrained optimization problems

Two trust regions algorithms for unconstrained nonlinear optimization problems are presented: a monotone and a nonmonotone one. Both of them solve the trust region subproblem by the spectral projected gradient (SPG) method proposed by Birgin, Martinez and Raydan (in SIAM J. Optim. 10(4):1196---1211, 2000). SPG is a nonmonotone projected gradient algorithm for solving large-scale convex-constrained optimization problems. It combines the classical projected gradient method with the spectral gradient choice of steplength and a nonmonotone line search strategy. The simplicity (only requires matrix-vector products, and one projection per iteration) and rapid convergence of this scheme fits nicely with globalization techniques based on the trust region philosophy, for large-scale problems. In the nonmonotone algorithm the trial step is evaluated by acceptance via a rule which can be considered a generalization of the well known fraction of Cauchy decrease condition and a generalization of the nonmonotone line search proposed by Grippo, Lampariello and Lucidi (in SIAM J. Numer. Anal. 23:707---716, 1986). Convergence properties and extensive numerical results are presented. Our results establish the robustness and efficiency of the new algorithms.

I-uniform continuity and I-uniform boundedness of a function

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Matrix-based Chebyshev spectral approach to dynamic analysis of non-uniform Timoshenko beams

A Chebyshev spectral method (CSM) for the dynamic analysis of non-uniform Timoshenko beams under various boundary conditions and concentrated masses at their ends is proposed. The matrix-based Chebyshev spectral approach was used to construct the spectral differentiation matrix of the governing differential operator and its boundary conditions. A matrix condensation approach is crucially presented to impose boundary conditions involving the homogeneous Cauchy conditions and boundary conditions containing eigenvalues. By taking advantage of the standard powerful algorithms for solving matrix eigenvalue and generalized eigenvalue problems that are embodied in the MATLAB commands, chebfun and eigs, the modal parameters of non-uniform Timoshenko beams under various boundary conditions can be obtained from the eigensolutions of the corresponding linear differential operators. Some numerical examples are presented to compare the results herein with those obtained elsewhere, and to illustrate the accuracy and effectiveness of this method.

Controllability of fractional impulsive neutral integrodifferential systems with a nonlocal Cauchy condition in Banach spaces

In this work the controllability of fractional impulsive neutral functional integrodifferential systems with a nonlocal Cauchy condition in a Banach space has been addressed. Sufficient conditions for the controllability are established using fractional powers of operators and the Banach contraction mapping theorem.

When [formula omitted]-Cauchy nets in complete uniform spaces are [formula omitted]-convergent

In this paper we continue our investigation of nets using ideals in line of our earlier work where we had studied I -Cauchy nets and asked when I -Cauchy nets in complete uniform spaces are I -convergent in line of a problem by Di Maio and Kočinac who asked in 2008 when statistically Cauchy sequences are statistically convergent in uniform spaces. To answer this, here we consider another type of Cauchy condition of nets, namely I ⁎ -Cauchy condition and examine its basic properties and in particular its relation with the concept of I -Cauchy nets. This helps us to give an answer to the above mentioned open question.

Eddy current-adjusted plasma shape reconstruction by Cauchy condition surface method on QUEST

CCS (Cauchy Condition Surface) method is a numerical approach to reproduce plasma shape, which has good precision in conventional tokamaks. In order to apply it in plasma shape reproduction of ST (Spherical Tokamak), the calculation precision of the CCS method in a spherical tokamak CPD (Compact PWI experimental Device) ( B t = 0.25 T, R = 0.3 m, a = 0.2 m) has been analyzed. The precision was confirmed also in ST and decided to be applied to a spherical tokamak QUEST ( B t = 0.25 T, R = 0.68 m, a = 0.40 m).. In present stage from the magnetic measurement, it is known that the eddy current effect is large in QUEST experiment, and there are no special magnetic measurements for eddy current now, so some proper model should be selected to evaluate the eddy current effect. The eddy current density by not only CS (Center Solenoid) coil but also plasma current is calculated using EDDYCAL (JAEA). The eddy current magnitudes are taken as unknown variables and solved together with plasma shape reconstruction in ohmic discharge and ECCD (Electron Cyclotron Current Drive) discharge.