Tiberiu Popoviciu Institute of Numerical Analysis

Background: There is an ongoing research for breast cancer diagnostic tools that are cheaper, more accurate and more convenient than mammography. Methods: In this study, we employed surface-enhanced Raman scattering (SERS) for analysing urine from n = 53 breast cancer patients and n = 22 controls, with the aim of discriminating between the two groups using multivariate data analysis techniques such as principal component analysis—linear discriminant analysis (PCA-LDA). The SERS spectra were acquired using silver nanoparticles synthesized by reduction with hydroxylamine hydrochloride, which were additionally activated with Ca2+ 10−4 M. Results: The addition of Ca(NO3)2 10−4 M promoted the specific adsorption to the metal surface of the anionic purine metabolites such as uric acid, xanthine and hypoxanthine. Moreover, the SERS spectra of urine were acquired without any filtering or processing step for removing protein traces and other contaminants. Using PCA-LDA, the SERS spectra of urine from breast cancer patients were classified with a sensitivity of 81%, a specificity of 95% and an overall accuracy of 88%. Conclusion: The results of this preliminary study contribute to the translation of SERS in the clinical setting and highlight the potential of SERS as a novel screening strategy for breast cancer.

Purpose – The purpose of this paper is to investigate the steady mixed convection boundary layer flow from a vertical frustum of a cone in water-based nanofluids. The problem is formulated to incorporate three kinds of nanoparticles: copper, alumina and titanium oxide. The working fluid is chosen as water with the Prandtl number of 6.2. The mathematical model used for the nanofluid incorporates the particle volume fraction parameter, the effective viscosity and the effective thermal diffusivity. The entire regime of the mixed convection includes the mixed convection parameter, which is positive for the assisting flow (heated surface of the frustum cone) and negative for the opposing flow (cooled surface of the frustum cone), respectively. Design/methodology/approach – The transformed non-linear partial differential equations are solved numerically for some values of the governing parameters. The derivatives with respect to? were discretized using the first order upwind finite differences and the resulting ordinary differential equations with respect to? were solved using bvp4c routine from Matlab. The absolute error tolerance in bvp4c was 1e-9. Findings – The features of the flow and heat transfer characteristics for different values of the governing parameters are analysed and discussed. The effects of the particle volume fraction parameter \phi, the mixed convection parameter \lambda and the dimensionless coordinate? on the flow and heat transfer characteristics are determined only for the Cu nanoparticles. It is found that dual solutions exist for the case of opposing flows. The range of the mixed convection parameter for which the solution exists increases in the presence of the nanofluids. Originality/value – The paper models the mixed convection from a vertical truncated cone using the boundary layer approximation. Multiple (dual) solutions for the flow reversals are obtained and the range of existence of the solutions was found. Particular cases for ?=0 (full cone), ? >>1 and (free convection limit) \lambda>>1were studied. To the authors best knowledge this problem has not been studied before and the results are new and original.

The present paper represents a continuation of Sofonea and Matei's paper (Sofonea, M. and Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math . 22 , 471–491). There a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact. In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the material's behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then we introduce the abstract variational inequalities together with their penalizations. We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.

The aim of this note is to prove an extension theorem for semi-Lipschitz real functions dened on quasi-metric spaces, similar to McShane extension theorem for real-valued Lipschitz functions dened on a metric space
 ([2], [4]).

This paper represents a continuation of our previous work, where a mathematical model which describes the equilibrium of an elastic body in frictional contact with a moving foundation was considered. An existence and uniqueness result was proved, together with a convergence result. The proofs were carried out by using arguments of elliptic variational inequalities. In this current paper, we complete our model by taking into account the wear of the foundation. This makes the problem evolutionary and leads to a new and nonstandard mathematical model, which couples a time-dependent variational inequality with an integral equation. We provide the unique weak solvability of the model by using a fixed point argument, among others. Then, we penalize the unilateral contact condition and prove that the penalized problem has a unique solution which converges to the solution of the original problem, as the penalization parameter converges to zero.

Whenever one uses translation invariant mean Green's functions to describe the behavior in the mean and to estimate dispersion coefficients for diffusion in random velocity fields, the spatial homogeneity of the transition probability of the transport process is implicitly assumed. This property can be proved for deterministic initial conditions if, in addition to the statistical homogeneity of the space-random velocity field, the existence of unique classical solutions of the transport equations is ensured. When uniqueness condition fails and translation invariance of the mean Green's function cannot be assumed, as in the case of nonsmooth samples of random velocity fields with exponential correlations, asymptotic dispersion coefficients can still be estimated within an alternative approach using the Itô equation. Numerical simulations confirm the predicted asymptotic behavior of the coefficients, but they also show their dependence on initial conditions at early times, a signature of inhomogeneous transition probabilities. Such memory effects are even more relevant for random initial conditions, which are a result of the past evolution of the process of diffusion in correlated velocity fields, and they persist indefinitely in case of power law correlations. It was found that the transition probabilities for successive times can be spatially homogeneous only if a long-time normal diffusion limit exits. Moreover, when transition probabilities, for either deterministic or random initial states, are spatially homogeneous, they can be explicitly written as Gaussian distributions.

For transport in statistically homogeneous random velocity fields with properties that are routinely assumed in stochastic groundwater models, the one‐particle dispersion (i.e., second central moment of the ensemble average concentration for a point source) is a “memory‐free” quantity independent of initial conditions. Nonergodic behavior of large initial plumes, as manifest in deviations of actual solute dispersion from one‐particle dispersion, is associated with a “memory term” consisting of correlations between initial positions and displacements of solute molecules. Reliable numerical experiments show that increasing the source dimensions has two opposite effects: it reduces the uncertainty related to the randomness of center of mass, but, at the same time, it yields large memory terms. The memory effects increase with the source dimension and depend on its shape and orientation. Large narrow sources oriented transverse to the mean flow direction yield ergodic behavior with respect to the one‐particle dispersion of the longitudinal dispersion and nonergodic behavior of the transverse dispersion, whereas for large longitudinal sources, the longitudinal dispersion behaves nonergodically, and the transverse dispersion behaves ergodically. Such memory effects are significantly large over hundreds of heterogeneity scales and should therefore be considered in practical applications, for instance, calibration of model parameters, forecasting, and identification of the contaminant source.

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and an obstacle, the so-called foundation. The contact is frictionless and is modeled with a new and nonstandard condition which involves both normal compliance, unilateral constraint and memory effects. We derive a variational formulation of the problem which is the form of a history-dependent variational inequality for the displacement field. Then, using a recent result obtained by Sofonea and Matei, we prove the unique weak solvability of the problem. Next, we study the continuous dependence of the weak solution with respect the data and prove a first convergence result. Finally, we prove that the weak solution of the problem represents the limit of the weak solution of a contact problem with normal compliance and memory term, as the stiffness coefficient of the foundation converges to infinity.

In this paper we introduce a class of positive linear operators by using the
 "umbral calculus", and we study some approximation properties of it. Let \(
 Q \) be a delta operator, and \(S\) an invertible shift invariant operator. For
 \(f\in C[0,1]\) we define
 \begin{equation*}
 (L_{n}^{Q,S}f)(x)=
 {\textstyle\frac{1}{s_{n}(1)}}
 \sum\limits_{k=0}^{n}{\textstyle\binom{n}{k}}p_{k}(x)s_{n-k}(1-x)f\left( \tfrac{k}{n}
 \right),
 \end{equation*}
 where \((p_{n})_{n\geq0}\) is a binomial sequence which is the basic
 sequence for \(Q,\) and \((s_{n})_{n\geq0}\) is a Sheffer set, \( s_{n}=S^{-1}p_{n} \). These operators generalize the binomial operators of T.
 Popoviciu.

The variance of the advection-diffusion processes with variable coefficients is exactly decomposed as a sum of dispersion terms and memory terms consisting of correlations between velocity and initial positions. For random initial conditions, the memory terms quantify the departure of the preasymptotic variance from the time-linear diffusive behavior. For deterministic initial conditions, the memory terms account for the memory of the initial positions of the diffusing particles. Numerical simulations based on a global random walk algorithm show that the influence of the initial distribution of the cloud of particles is felt over hundreds of dimensionless times. In case of diffusion in random velocity fields with finite correlation range the particles forget the initial positions in the long-time limit and the variance is self-averaging, with clear tendency toward normal diffusion.

Abstract This paper studies the steady, free convection boundary layer flow about a vertical, isothermal plate embedded in a non-Darcy bidisperse porous medium (BDPM). An appropriate mathematical model is proposed. The boundary layer analysis leads to a system of partial differential equations containing inertial, interphase momentum, thermal diffusivity ratio, thermal conductivity ratio, permeability ratio, modified thermal capacity, and convection parameters. These equations that govern the flow and heat transfer in the f-phase and the p-phase are solved numerically using an algorithm based on the bvp4c routine from matlab. The dependences of the dimensionless velocities and temperatures profiles, as well as of the Nusselt numbers on the governing parameters are investigated. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed in details.

This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers. Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on two-dimensional benchmark flow problems. Realizations of log-normal hydraulic conductivity fields are generated by Kraichnan algorithms in closed form as finite sums of random periodic modes, which allow direct code verification by comparisons with manufactured reference solutions. The quality of the methods is assessed for increasing number of random modes and for increasing variance of the log-hydraulic conductivity fields with Gaussian and exponential correlation. Experimental orders of convergence are calculated from successive refinements of the grid. The numerical methods are further validated by comparisons between statistical inferences obtained from Monte Carlo ensembles of numerical solutions and theoretical first-order perturbation results. It is found that while for Gaussian correlation of the log-conductivity field all the methods perform well, in the exponential case their accuracy deteriorates and, for large variance and number of modes, the benchmark problems are practically not solvable with reasonably large computing resources, for all the methods considered in this study.

Following the idea of T. Wongyat and W. Sintunavarat, we obtain some existence and uniqueness results for the solution of an integral equation with supremum. The paper ends with the study of Gronwall-type theorems, comparison theorems and a result regarding a Ulam–Hyers stability result for the corresponding fixed point problem.

In this article, we present new random walk methods to solve flow and transport problems in saturated/unsaturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the L-scheme developed in finite element/volume approaches. The resulting GRW L-schemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solvers are validated by comparisons with mixed finite element and finite volume solvers in one- and two-dimensional benchmark problems. They include Richards’ equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes.

In this paper, two control problems for a symmetric model of cell dynamics related to leukemia are considered. The first one, in connection with classical chemotherapy, is that the evolution of the disease under treatment should follow a prescribed trajectory assuming that the drug works by increasing the cell death rates of both malignant and normal cells. In the case of the second control problem, as for targeted therapies, the drug is assumed to work by decreasing the multiplication rate of leukemic cells only, and the control objective is that the disease state reaches a desired endpoint. The solvability of the two problems as well as their stability are proved by using a general method of analysis. Some numerical simulations are included to illustrate the theoretical results and prove their applicability. The results can possibly be used to design therapeutic scenarios such that an expected clinical evolution can be achieved.

We consider a mathematical model which describes the quasistatic contact between a viscoelastic body and an obstacle, the so-called foundation. The material's behavior is modelled with a constitutive law with long memory. The contact is with normal compliance, unilateral constraint, memory effects and adhesion. We present the classical formulation of the problem, then we derive its variational formulation and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities and fixed point.

The aim of this paper is to draw attention to a general principle for solving control problems for operator equations with the help of fixed point techniques. Three distinct applications are presented: a control problem related to the Lotka–Volterra system, the nonlinear Stokes system, and radial solutions of the Neumann problem for ϕ-Laplace equations. Only for the first application, the control is an explicit external one, while for the next two applications, the control is dependent on the models and arises from the necessity to conform to the actual modeled process or to a certain boundary condition. From the perspective of those interested in applications, the three examples of problems of such a different nature, we believe have been able to suggest the wide applicability of our method thus paving the way for new applications. From a theoretical perspective, the method leading to fixed point equations with composed operators is suitable to be related to advanced research in fixed point theory for single-valued and multi-valued operators.

Abstract Transport processes in groundwater systems with spatially heterogeneous properties often exhibit anomalous behavior. Using first-order approximations in velocity fluctuations we show that anomalous superdiffusive behavior may result if velocity fields are modeled as superpositions of random space functions with correlation structures consisting of linear combinations of short-range correlations. In particular, this corresponds to the superposition of independent random velocity fields with increasing integral scales proposed as model for evolving scale heterogeneity of natural porous media [Gelhar, L. W. Water Resour. Res. 22 (1986), 135S-145S]. Monte Carlo simulations of transport in such multi-scale fields support the theoretical results and demonstrate the approach to superdiffusive behavior as the number of superposed scales increases.

SUMMARY The aim of this paper is to show that the Jacobi–Davidson (JD) method is an accurate and robust method for solving large generalized algebraic eigenvalue problems with a singular second matrix. Such problems are routinely encountered in linear hydrodynamic stability analysis of flows that arise in various areas of continuum mechanics. As we use the Chebyshev collocation as a discretization method, the first matrix in the pencil is nonsymmetric, full rank, and ill‐conditioned. Because of the singularity of the second matrix, QZ and Arnoldi‐type algorithms may produce spurious eigenvalues. As a systematic remedy of this situation, we use two JD methods, corresponding to real and complex situations, to compute specific parts of the spectrum of the eigenvalue problems. Both methods overcome potentially severe problems associated with spurious unstable eigenvalues and are fairly stable with respect to the order of discretization. The real JD outperforms the shift‐and‐invert Arnoldi method with respect to the CPU time for large discretizations. Three specific flows are analyzed to advocate our statements, namely a multicomponent convection–diffusion in a porous medium, a thermal convection in a variable gravity field, and the so‐called Hadley flow. Copyright © 2012 John Wiley & Sons, Ltd.

We analyze a general class of coupled systems of stationary Navier–Stokes type equations with variable coefficients and non-homogeneous terms of reaction type in the incompressible case. Existence of solutions satisfying the homogeneous Dirichlet condition in a bounded domain in [Formula: see text], [Formula: see text], and localization results for the corresponding kinetic energy and enstrophy are obtained by using a variational approach and the fixed point index theory.