An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed.

Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black–Scholes Equation

In this paper, we use a numerical method for solving the nonlinear Black–Scholes partial differential equation of the European option under transaction costs, which is based on the nonstandard discretization of the spatial derivatives. The proposed scheme, in addition to the unconditional positivity, is stable, consistent, and monotone. In order to illustrate the efficiency of the new method, numerical results have been performed by four models.

Different facets of attitudes towards having children: The Procreation Attitude Scale (PrAttS)

The attitudes towards children are more complex than simple positive/negative distinctions. In the present study, we sought in two studies (N = 445) to provide a tool to explore different facets of attitudes towards babies and procreation by developing and validating a questionnaire regarding attitudes towards procreation. Study 1 was conducted to develop an English language scale tapping into adults’ attitudes toward having offspring. A larger number of statements were formulated that expressed an emotional value and motivational attitudes towards having babies. The sample consisted of n = 157 participants. Participants’ responses were subjected to a preliminary principal components analysis with an oblique rotation. Study 1 obtained a relatively economical scale to tap into three facets of attitudes toward having children. Study 2 sought to validate a German language version of the same scale as in the previous study. A total of n = 288 participants were recruited. The 13 items of the scale were subjected to a confirmatory factor analysis. Study 2 confirmed the three-dimensional structure. The Procreation Attitude Scales (PrAttS) consists of 13 items representing three underlying dimensions: (1) unconditional positivity, (2) anticipated annoyance and (3) contingent willingness. The present investigations showed twice a gender difference in emotional attitudes, revealing that women have a stronger emotional interest in procreation and babies. The PrAttS provides an explicit method of interest for children, providing an alternative to more recently criticized implicit measures.

Internal disorder pattern in mental development disorder: features and dynamics in childhood and adolescence

The article presents the results of theoretical and empirical analysis of the ontogenetic dynamics of the internal disorder pattern in mental development disorder. It is revealed that the self-attitude and self-esteem of children with mental development disorder depends on the form of education, has a special pace, goes from dependence to independence, from unconditional positivity to criticality. The study on 60 children and adolescents revealed an increase in the level of awareness of qualities, the development of gender and family identity, maintaining a high and medium level of self-relationship. The age dynamics of the cognitive and physical component of the internal disorder pattern is determined. Recognition of the violation and the level of knowledge increases in primary school, in adolescence decreases. It is suggested that the denial of their own violations occur. In adolescents with mental retardation there is a decrease in sensitivity, suggestibility, fixation on their own bodily sensations.

A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model

In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr–Debye model. By using the approach first introduced by Zhang and Shu in [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004) 641–666.] with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit–explicit (IMEX) Runge–Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr–Debye model. The new IMEX Runge–Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge–Kutta method and have second-order accuracy. The numerical results validate our analysis.