Dynamics in a Chemotaxis Model with Periodic Source

We consider a system of two differential equations modeling chemotaxis. The system consists of a parabolic equation describing the behavior of a biological species “u” coupled to an ODE patterning the concentration of a chemical substance “v”. The growth of the biological species is limited by a logistic-like term where the carrying capacity presents a time-periodic asymptotic behavior. The production of the chemical species is described in terms of a regular function h, which increases as “u” increases. Under suitable assumptions we prove that the solution is globally bounded in time by using an Alikakos-Moser iteration, and it fulfills a certain periodic asymptotic behavior. Besides, numerical simulations are performed to illustrate the behavior of the solutions of the system showing that the model considered here can provide very interesting and complex dynamics

A meshless numerical method for a system with intraspecific and interspecific competition

In this paper we study a novel mathematical model with intraspecific and interspecific competition between two species consisting of a non-linear parabolic–ODE–parabolic system. It describes the evolution of two populations in competition for a resource, one of which is subject to chemotaxis. We analyze the local stability of the constant equilibrium solutions and we obtain the periodic behavior of the solution for certain data of the problem. For this purpose, we apply the meshless numerical method of Generalized Finite Differences (GFDM) and we prove the conditional convergence of the discrete solution to the analytical one. The conditional convergence of the numerical method is demonstrated and, thought its implementation, we obtain numerical solutions whose asymptotic behavior agrees with the analytically one expected. We give several numerical examples on the applications of this meshless method over regularly and irregularly distributed nodes to illustrate its potential

Correction to the Moliere's formula for multiple scattering

The quasiclassical correction to the Moliere's formula for multiple scattering is derived. The consideration is based on the scattering amplitude, obtained with the first quasiclassical correction taken into account for arbitrary localized but not spherically symmetric potential. Unlike the leading term, the correction to the Moliere's formula contains the target density $n$ and thickness $L$ not only in the combination $nL$ (areal density). Therefore, this correction can be reffered to as the bulk density correction. It turns out that the bulk density correction is small even for high density. This result explains the wide region of applicability of the Moliere's formula.Comment: 6 pages, RevTe

A Novel Spatio-Temporal Fully Meshless Method for Parabolic PDEs

We introduce a meshless method derived by considering the time variable as a spatial variable without the need to extend further conditions to the solution of linear and non-linear parabolic PDEs. The method is based on a moving least squares method, more precisely, the generalized finite difference method (GFDM), which allows us to select well-conditioned stars. Several 2D and 3D examples, including the time variable, are shown for both regular and irregular node distributions. The results are compared with explicit GFDM both in terms of errors and execution time

On the numerical solution to a parabolic-elliptic system with chemotactic and periodic terms using Generalized Finite Differences

In the present paper we propose the Generalized Finite Difference Method (GFDM) for numerical solution of a cross-diffusion system with chemotactic terms. We derive the discretization of the system using a GFD scheme in order to prove and illustrate that the uniform stability behavior/ convergence of the continuous model is also preserved for the discrete model. We prove the convergence of the explicit method and give the conditions of convergence. Extensive numerical experiments are presented to illustrate the accuracy, efficiency and robustness of the GFDM

Effect of omeprazole on patient-reported outcome measures in uninvestigated heartburn: a multi-country, multi-center observational study

Background: Heartburn occurs predominantly in the upper gastrointestinal tract and is associated with gastroesophageal reflux disease (GERD) and gastritis. Omeprazole is the most prescribed proton pump inhibitor class of medication to treat heartburn related clinical conditions. To compare the efficacy of omeprazole 40 mg (as a total daily dose) and 20 mg using patient-reported outcome measures (PROMs) in patients with heartburn due to various aetiologies like non-erosive reflux disease, GERD, gastritis, dyspepsia, functional heartburn, gastro-duodenal ulcer.Methods: Naïve patients presenting heartburn symptoms were treated with omeprazole. PROMs were assessed based on short-form-leeds dyspepsia questionnaires (SF-LDQ), work productivity activity impairment (WPAI), relief obtained using medication and, treatment satisfactory questionnaires (TSQ).Results: A total of 18,724 patients with heartburn (GERD and gastritis; n=10,509) were treated with omeprazole (Dr. Reddy’s omeprazole [DO]/generic omeprazole [GO]/branded omeprazole [BO]) 40 mg (as a total daily dose) and 20 mg. Statistical comparative analysis showed significant improvement with omeprazole 40 mg (as a total daily dose) compared to omeprazole 20 mg in SF-LDQ, relief obtained using medication among patients with heartburn. DO 20 mg showed a greater improvement under the ‘a lot’ and ‘complete’ relief category.Conclusions: Omeprazole 40 mg (as a total daily dose) presented better efficacy as compared to omeprazole 20 mg in patient reported outcomes. This study highlights omeprazole 40 mg as the preferred intervention for improving PROMs and quality of life in the treatment of heartburn related clinical conditions

Solving a fully parabolic chemotaxis system with periodic asymptotic behavior using Generalized Finite Difference Method

This work studies a parabolic-parabolic chemotactic PDE's system which describes the evolution of a biological population “U” and a chemical substance “V”, using a Generalized Finite Difference Method, in a two dimensional bounded domain with regular boundary. In a previous paper [12], the authors asserted global classical solvability and periodic asymptotic behavior for the continuous system in 2D. In this continuous work, a rigorous proof of the global classical solvability to the discretization of the model proposed in [12] is presented in two dimensional space. Numerical experiments concerning the convergence in space and in time, and long-time simulations are presented in order to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms

Complex Ginzburg–Landau equation with generalized finite differences

In this paper we obtain a novel implementation for irregular clouds of nodes of the meshless method called Generalized Finite Difference Method for solving the complex Ginzburg–Landau equation. We derive the explicit formulae for the spatial derivative and an explicit scheme by splitting the equation into a system of two parabolic PDEs. We prove the conditional convergence of the numerical scheme towards the continuous solution under certain assumptions. We obtain a second order approximation as it is clear from the numerical results. Finally, we provide several examples of its application over irregular domains in order to test the accuracy of the explicit scheme, as well as comparison with other numerical methods