# Applied Mathematics and Numerical Analysis Seminar

## Past sessions

### Oscillatory behavior of a mixed type difference equation with variable coefficients

MathJax TeX Test PageIn this talk, we present a study on the oscillatory behaviour of the mixed type difference equation with variable coefficients$$\Delta x(n) = \sum_{i=1}^l p_i(n)x(\tau_i(n)) + \sum_{j=1}^m q_j(n)x(\sigma_j(n)),\, n \geq n_0.$$

### An alternative stabilization in numerical simulations of Oldrod-B type fluids

The numerical simulation of non-Newtonian viscoelastic fluids flow is a challenging problem. One of the approaches being often adopted to stablize the numerical simulations is based on addition of stress diffusion term into the transport equations for viscoelastic stress tensor. The additional term affect the solution of the problem and special care should be taken to keep the modified model consistent with the original problem.

In this work it was analyzed in detail the influence of numerical stabilization using artificial stress diffusion and it was presented a new arternative. Instead of the classical addition of artificial stress diffusion term it was used the modified additional term which is only present during the transient phase and should vanish in when approaching the stationary case. The steady solution is not affected by such vanishing artificial term, however the stability of the numerical method is improved.

This is joint work with Tomás Bodnár (Institute of Mathematics, Czech Academy of Sciences and Faculty of Mechanical Engineering, Czech Technical University in Prague, Czech Republic).

### Exact solution for a Benney-Lin equation type (Gain in Regularity)

In this seminar we will show a exact solution for a Benney-Lin equation type using mainly the Ince Transformation. We establish exact traveling waves solution to the nonlinear evolution equation Benney-Lin type.

### Very high-order finite volume schemes with curved domains

Finite volume methods of third or higher order require a specific treatment of the boundary conditions when dealing with a non-polygonal domain that does not exactly fit with the mesh. We also face a similar situation with internal smooth interfaces sharing two subdomains. To address this issue, several technologies have been developed since the 90's such as the isoparamatric elements, the (ghost cells) immersed boundary and the inverse Lax-Wendroff boundary treatment among others. We propose a quick overview of the traditional methods and introduce the new Reconstruction of Off-site Data (ROD) method. Basically, the idea consists, first in definitively distinguishing the computational domain (cells or nodes where the solution is computed) to the physical one and, secondly, in "transporting" the boundary conditions prescribed on the real boundary to the computing domain. To this end, specific local polynomial reconstructions that contains a fingerprint of the boundary conditions are proposed and used to schemes that achieve up to sixth-order of accuracy. Several applications will be proposed in the context of the finite volume (flux reconstruction) and finite difference (ghost cells) for the convection diffusion equation and the Euler system.

### Renormalized transport of inertial particles

We study how an imposed fluid flow — laminar or turbulent — modifies the transport properties of inertial particles (e.g. aerosols, droplets or bubbles), namely their terminal velocity, effective diffusivity, and concentration following a point-source emission.

Such quantities are investigated by means of analytical and numerical computations, as functions of the control parameters of both flow and particle; i.e., density ratio, inertia, Brownian diffusivity, gravity (or other external forces), turbulence intensity, compressibility degree, space dimension, and geometric/temporal properties.

The complex interplay between these parameters leads to the following conclusion of interest in the realm of applications: any attempt to model dispersion and sedimentation processes — or equivalently the wind-driven surface transport of floaters — cannot avoid taking into account the full details of the flow field and of the inertial particle.

### On the existence of a solution of a class of non-stationary free boundary problems

We consider a class of parabolic free boundary problems with heterogeneous coefficients. We establish existence of a solution for this problem. We use a regularized problem for which we prove existence of a solution by applying the Tychonoff fixed point theorem. Then we pass to the limit to get a solution of our problem.

### On the time fractional differential equation with integral conditions

We study the existence and uniqueness of a solution for time order partial fractional differential equations with integral conditions. By using the method of energy inequalities, we find a priori estimates and the density of the range of the operator generated by given the problem.

### Mathematical modeling, analysis and simulation of biological, bio-inspired and engineering systems

Over the last decade, there have been dramatic advances in mathematical modeling, analysis and simulation techniques to understand fundamental mechanisms underlying multidisciplinary applications that involve multi-physics interactions. This work will present the results from projects that evolved from multidisciplinary applications of differential equations for multi-physics problems in biological, bio-inspired and engineering systems. Specifically, mathematical modeling and numerical methods for efficient computation of nonlinear interaction for coupled differential equation models that arise from applications such as flow-structure interactions to understand rupture of aneurysms to dynamics of micro-air vehicles as well as modeling dynamics of infectious disease to modeling social dynamics will be presented. Some theoretical results that validate the reliability and robustness of the computational methodology employed will also be presented. We will also discuss how such projects can provide opportunities for students and faculty at all levels to employ transformative research in multidisciplinary areas. Upcoming opportunities for undergraduate and graduate fellowships as well as research opportunities for faculty for collaborative proposals will also be discussed.

### A new numerical method based on the modified hat functions for solving fractional optimal control problems

In the present work, a numerical method based on the modified hat functions is introduced for solving a class of fractional optimal control problems. In this scheme, the control and the fractional derivative of state function are considered as linear combinations of the modified hat functions. The Riemann–Liouville integral operator is used to give approximations for the state function and some of its derivatives. Using the properties of the considered basis functions, the fractional optimal control problem is easily reduced to solving a system of nonlinear algebraic equations. An error bound is presented for the approximate optimal value of the performance index obtained by the proposed method. Then the method is developed for solving a class of optimal control problems with inequality constraints. Finally, some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.

### Solving integral and integro-diﬀerential equations using Collocation and Wavelets methods

The main objective of this work is to study some classes of integral and integro-diﬀerential equations with regular and singular kernels. We introduce a wavelets method to solve a new class of Fredholm integral equations of the second kind with non symmetric kernel; we also apply a collocation method based on the airfoil polynomial to numerically solve an integro-diﬀerential equation of second order with Cauchy kernel.

### Numerical simulations of Stokes flow in 3D using the MFS

In this work we present the method of fundamental solutions (MFS) in several contexts. We start with a more intuitive example and then we extend it to vectorial PDE's. We present the density results that justify our approach and then we use the MFS to solve the Stokes system in 2D and 3D. We deal with non-trivial domains and with different types of boundary conditions, namely the mixed Dirichlet-Neumann conditions. We will present several simulations that show the strengths of the method and some of the numerical difficulties found.

(Joint work with C. J. S. Alves and A. L. Silvestre)

### Modeling microbial interactions, dynamics and interventions: from data to processes

Controlling the evolution and spread of antimicrobial resistance is a major global health priority. While the discovery of new antibiotics does not follow the rate at which new resistances develop, a more rational use of available drugs remains critical. In my work, I explore the role of host immunity in infection dynamics and control, hence as an important piece in the puzzle of antibiotic resistance management.

I will present two studies from my research on microbial dynamics, addressing processes that occur at the within- and between-host level.

The first study examines antibiotic resistance and treatment optimization for bacterial infections, quantifying the crucial role of host immune defenses at the single host level.

The second study presents a multi-strain epidemiological model applied to pneumococcus data before and after vaccination.

This framework allows for retrospective inference of strain interactions in co-colonization and vaccine efficacy parameters, and can be useful for comparative analyses across different immunized populations.

### Suitable Far-field Boundary Conditions for Wall-bounded Stratified Flows

This talk presents an alternative boundary conditions setup for the numerical simulations of stable stratified flow. The focus of the tested computational setup is on the pressure boundary conditions on the artificial boundaries of the computational domain. The simple three dimensional test case deals with the steady flow of an incompressible, variable density fluid over a low smooth model hill. The Boussinesq approximation model is solved by an in-house developed high-resolution numerical code, based on compact finite-difference discretization in space and Strong Stability Preserving Runge-Kutta method for (pseudo-) time stepping.

This is a joint work with Philippe Fraunie, University of Toulon, France.

### Necessary and sufficient conditions for existence of minimizers for vector problems in the Calculus of Variations

We report our work about the main necessary and sufficient conditions for weak lower semi-continuity of integral functionals in vector Calculus of Variations.

In particular we provide tools to investigate rank-one convexity of functions defined on $2\times 2$-matrices. Furthermore, we explore some consequences and examples.

We also explore the quasiconvexity condition in the case where the integrand of an integral functional is a fourth-degree homogeneous polynomial.

### Reconstruction of PDE coefficients with overprescripton of Cauchy data at the boundary

Frequently incomplete information about coefficients in partial differential equations is compensated by overprescribed Cauchy data on the boundary.

We analyse this kind of boundary value problems in an elliptic system in Lipschitz domains. Main techniques are variational formulation, boundary integral equations and the Calderon projector. To estimate those coefficients we propose a variational formulation based on internal discrepancy observed in the mixed boundary value problem, obtained by splitting the overprescribed Cauchy data. Some numerical experiments are presented.

### Models for sustainable biodiesel production

Several countries have already begun to invest in alternative energies due to smaller and smaller fossil fuel resources. In particular, for biodiesel production the Jatropha curcas appears to be a possible resource, in that it thrives even in harsh and very dry conditions. From its seeds a relevant quantity of oil can be extracted, for production of high grade biodiesel fuel.

But this plant is subject to parasitism from a mosaic virus, the Begomovirus, that is carried by white-flies Bemisia tabaci. The talk is centered on the investigation of two models for the fight of this plant infection.

In the case of large plantations we investigate the optimal insecticide spraying policy. Here the most relevant parameters of the ecosystem appear to be the infection transmission rate from vectors to plants and the vector mortality. The results indicate that spraying should be administered only after 10 days of the epidemics insurgence, relentlessly continued for about three months, after which disease eradication is obtained, [2].

At the small scale instead, we consider possible production by individuals, that cultivate this plant in small plots that would be otherwise be left wild and unproductive, [1]. We consider the effects of media campaigns that keep people aware of this plant disease, and indicate means for fighting it. The model shows that awareness campaigns should be implemented rather intensively, in order to effectively reduce or completely eradicate the infection.

References

[1] Priti Kumar Roy, Fahad Al Basir, Ezio Venturino (2017) Effects of Awareness Program for Controlling Mosaic Disease in Jatropha curcas Plantations, to appear in MMAS.

[2] Ezio Venturino, Priti Kumar Roy, Fahad Al Basir, Abhirup Datta (2016) A model for the control of the Mosaic Virus disease in Jatropha Curcas plantations, to appear in Energy, Ecology and Environment, doi:10.1007/s40974-016-0033-8.

(work in collaboration with Fahad Al Basir and Priti K. Roy, Javadpur University, India)

### A hierarchy of models for the flow of fluids through porous solids

The celebrated equations due to Fick and Darcy are approximations that can be obtained systematically on the basis of numerous assumptions within the context of Mixture Theory; these equations however not having been developed in such a manner by Fick or Darcy. Relaxing the assumptions made in deriving these equations via mixture theory, selectively, leads to a hierarchy of mathematical models and it can be shown that popular models due to Forchheimer, Brinkman, Biot and many others can be obtained via appropriate approximations to the equations governing the flow of interacting continua. It is shown that a variety of other generalizations are possible in addition to those that are currently in favor, and these might be appropriate for describing numerous interesting technological applications.

### Perturbative methods in thermal imaging

I will discuss shortly three different inverse problems in the field of thermal nondestructive testing. In all cases we have a thin metallic plate $\Omega_0$ whose top boundary $S_{top}$ is not accessible while we are able to operate on the opposite surface $S_{bot}$.

We heat the specimen by applying a heat flux of density $\phi$ on $S_{bot}$ and measure a sequence of temperature maps.

Problem 1. Recover a perturbation of the heat transfer coefficient on $S_{top}$. [Inglese, Olmi - 2017]

Problem 2. Recover a surface damage on the inaccessible side $S_{top}$. [Inglese, Olmi - in progress]

Problem 3. Recover a nonlinear heat transfer coefficient on $S_{top}$. [Clarelli, Inglese - 2016]

Older session pages: Previous 2 3 4 5 6 7 8 9 10 11 12 13 14 Oldest