# Results in Mathematical and Numerical Modeling, obtained at the Institute

## Time series theory and quantitative finance

An automatic algorithm for monotone trend estimation from a noisy time series was designed and a Monte Carlo method to estimate the errors associated with detrending has been obtained. A new numerical method to estimate the financial volatility from the returns time series was obtained.

- C. Vamos, M. Craciun,
*Numerical demodulation of a Gaussian white noise modulated in amplitude by a deterministic volatility*, The European Physical Journal B 86: 166, pp. 8, 2013. (pdf) - C. Vamos, M. Craciun,
*Automatic trend estimation*, Springer, Dordrecht, 2012, pp. 131, ISBN: 978-94-007-4824-8 (Print), 978-94-007-4825-5 (Online), DOI 10.1007/978-94-007-4825-5 - C. Vamos, M. Craciun,
*Separation of components from a scale mixture of Gaussian white noises*, Physical Review E 81 (2010) no. 5, article no. 051125. (pdf) - C. Vamos, M. Craciun,
*Serial correlation of detrended time series*, Physical Review E 78 (2008) no. 3, article no. 036707. (pdf) - C. Vamos,
*Automatic algorithm for monotone trend removal*, Physical Review E 75 (2007) no. 3, article no. 036705. (pdf)

## Mathematical modeling in hydrology

Transport processes in groundwater were modeled as diffusion in random velocity fields in terms of trajectories (Ito equations) and probability distributions (Fokker-Planck equations). The focus was mainly on ergodicity issue, memory effects, and modeling probability density functions of solute concentrations. The theoretical results were supported by accurate numerical simulations.

- N. Suciu,
*Diffusion in random velocity fields with applications to contaminant transport in groundwater*, Adv. Water Resour., 69, 114-133, doi:10.1016/j.advwatres.2014.04.002, (2014). - N. Suciu, C. Vamos, J. Vanderborght, H. Hardelauf, H. Vereecken,
*Numerical investigations on ergodicity of solute transport in heterogeneous aquifers*, Water Resources Research 42 (2006) W04409. (pdf) - N. Suciu, C. Vamos, H. Vereecken, K. Sabelfeld, P. Knabner,
*Memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media*, Water Resources Research 44 (2008) W08501. (pdf) - N. Suciu, C. Vamos, F. A. Radu, H. Vereecken, P. Knabner,
*Persistent memory of diffusing particles*, Physical Review E 80 (2009) 061134. (pdf) - N. Suciu,
*Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields*, Physical Review E 81 (2010) 056301.

## Mathematical modeling in contact mechanics

New models of quassistatic contact between deformable bodies were introduced. A nonstandard contact condition was provided, modelled by normal compliance, unilateral constraint and memory term.

- M. Sofonea, F. Pătrulescu, A. Farcaş,
*A Viscoplastic Contact Problem with Normal Compliance, Unilateral Constraint and Memory Term*, Appl. Math. and Opt., vol. 62 (2014), pp. 175-198. - M. Barboteu, F. Pătrulescu, A. Ramadan, M. Sofonea,
*History-dependent contact models for viscoplastic materials*, IMA Journal of Applied Mathematics, DOI:10.1093/imamat/hxt024. - M. Sofonea, F. Pătrulescu,
*Analysis of a history-dependent frictionless contact problem*, Mathematics and Mechanics of Solids, vol. 18, no.4 (2013), pp. 409-430. - A. Farcaş, F. Pătrulescu, M. Sofonea,
*A history-dependent contact problem with unilateral constraint*, Ann. Acad. Rom. Sci. Ser. Math. Appl. 4 (2012) no. 1, 90-96.

## Continuous models of corpuscular systems

The balance equations of continuum fields expressed as space-time coarse-grained averages was derived using the condition that the microscopic particles preserve their identity over a finite time interval within which their motion can be described by piecewise analytical time functions. Applications of this approach were obtained for financial markets, granular flows, natural porous media, lipid bilayers, etc.

- C. Vamos, A. Georgescu, N. Suciu, I. Turcu,
*Balance equations for physical systems with corpuscular structure*, Physica A 227 (1996) 81-92. - C. Vamos, N. Suciu, A. Georgescu,
*Hydrodynamic equations for one-dimensional systems of inelastic particles*, Physical Review E 55 (1997) 6277-6280. (pdf) - C. Vamos, N. Suciu, W. Blaj,
*Derivation of one-dimensional hydrodynamic model for stock price evolution*, Physica A 287 (2000) 461-467. (pdf)

## Numerical modeling of the diffusion with application in technological processes and transport in porous materials

A new algorithm for complex diffusion processes was designed and successfully applied to large scale simulations of transport in groundwater as well as to diffusion in an epidermic structure.

- Vamos, C., N. Suciu, H. Vereecken,
*Generalized random walk algorithm for the numerical modeling of complex diffusion processes*, Journal of Computational Physics 186(2) (2003) 527-544. - N. Suciu, C. Vamos,
*Evaluation of overshooting errors in particle methods for diffusion by biased global random walk*, Rev. Anal. Numer. Theor. Approx., 35 (2006), 119-126. - N. Suciu, C. Vamos, I. Turcu, C.V.L. Pop, L.I. Ciortea,
*Global random walk modeling of transport in complex systems*, Computing and Visualization in Science 12 (2009) 77-85.

## Mathematical modeling of some physico-chemical hydrodynamical phenomena

The viscous flow due to gravitation and gradients of the superficial tension (Marangoni effect) was mathematically modeled. We also introduced a pseudospectral method (collocation type) to solve high order eigenvalue problems in electrohydrodynamics.

- C.I. Gheorghiu, I.F. Dragomirescu,
*Spectral methods in linear stability. Applications to thermal convection with variable gravity field*, Applied Numerical Mathematics 59 (2009) 1290-1302. - C.I. Gheorghiu, I.F. Dragomirescu,
*Analytical and numerical solutions to an electrohydrodynamic stability problem*, Applied Mathematics and Computation 216 (2010) 3718-3727.

## Numerical modeling of metals and alloys casting

A numerical model for the flow and solidification of the metals and alloys during casting was designed and it was applied to real industrial casting processes.

- V. Soporan, C. Vamos, C. Pavai,
*Numerical modeling of the cavities during the alloys solidification*, Metalurgia, Vol. III, No. 4, 53-55 (1998). - V. Soporan, C. Vamos, C. Pavai,
*Numerical simulation for technological interpretation of solidification process castings*, Metalurgia, Vol. III, No. 4, 56-60 (1998). - V. Soporan, C. Vamos, C. Pavai,
*Numerical Modeling of Solidification*, Ed. Dacia, Cluj-Napoca, 2003 (in Romanian).

## Mathematical modeling in speleology

A mathematical model for the propagation of temperature waves, equivalent with an inductive-capacitive electrical circuit, was used to estimate bulk thermal properties of a natural limestone formation from temperature measurements. The results were found to be consistent with cave-microclimate measurements.

- N. Suciu,
*La determination “in situ” de quelques grandeurs thermiques des calcaires*, Trav. Inst Speol. “Emile Racovitza”, vol. XX, 179-185, 1981. - V. Decu, W. Herdlicka, N. Suciu,
*Mensuration indirecte de l’eau de condensation de deux grottes des monts de Mehedinti (Carpates Meridionales). Implications ecologiques de la condensation endokarstique*, Trav. Inst Speol. “Emile Racovitza”, vol. XXI, 43-52, 1982.