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Cell migration is fundamental in a wide variety of physiological and pathological phenomena, being exploited in biomedical engineering as well. In this respect, we here present a hybrid non-local integro-differential model where a representative cell, reproduced by a point particle with an orientation, moves on a planar domain upon signals coming from environmental variables. From a numerical point of view, non-locality implies the need to evaluate integral terms which may present non-regular integrand functions because of heterogeneities in the environmental conditions and/or in cell sensing region. Having in mind multicellular applications, we here propose a robust computational method able to handle such non-regularities. The procedure is based on low order Runge–Kutta methods and on an ad hoc application of the Gauss–Legendre quadrature rule. The accuracy and efficiency of the resulting computational method is then tested by selected benchmark settings. In this context, the ad hoc application of the quadrature rule reveals to be crucial to obtain a high accuracy with a remarkably low number of quadrature nodes with respect to the standard Gauss–Legendre quadrature formula, and which thus results in a reduced overall computational cost. Finally, the proposed method is further coupled with the cubic spline interpolation scheme which allows to deal also with possible poor (i.e., point-wise defined) molecular spatial information. The performed simulations (which accounts also for different scenarios) show how the interpolation of the molecular variables affects the efficiency of the overall method and further justify the proposed procedure.

In this paper we consider the 2D exterior problem for the non homogeneous wave equation, with mixed type boundary conditions and non homogeneous initial conditions, these latter having local supports. First we derive a space-time boundary integral equation formulation to solve the problem. Then we propose a numerical approach for its solution, which couples a Lubich discrete convolution quadrature with a classical collocation method. For the computation of the extra "volume" integrals generated by the initial data, having local supports, we propose a new numerical approach. Finally, we solve some test problems and present some of the numerical results we have obtained.

In this paper we examine the computation of the potential generated by space-time BIE representations associated with Dirichlet and Neumann problems for the 2D wave equation. In particular, we consider the efficient evaluation of the (convolution) time integral that appears in the potential representation. For this, we propose two simple quadrature rules which appear more efficient than the currently used ones. Both are of Gaussian type: the classical Gauss-Jacobi quadrature rule in the case of a Dirichlet problem, and the classical Gauss-Radau quadrature rule in the case of a Neumann problem. Both of them give very accurate results by using a few quadrature nodes, as long as the potential is evaluated at a point not very close to the boundary of the PDE problem domain. To deal with this latter case, we propose an alternative rule, which is defined by a proper combination of a Gauss-Legendre formula with one of product type, the latter having only 5 nodes. The proposed quadrature formulas are compared with the second order BDF Lubich convolution quadrature formula and with two higher order Lubich formulas of Runge-Kutta type. An extensive numerical testing is presented. This shows that the new proposed approach is very competitive, from both the accuracy and efficiency points of view.

In this paper we propose an efficient strategy to compute nearly singular integrals over planar triangles in R^3, arising in 3D Galerkin BEM. The strategy is based on a proper use of various nonlinear transformations, which smooth, or move away or quite eliminate all the singularities close to the domain of integration. We will deal with near singularities of the form 1/r and 1/r^3, [email protected][email protected]? being the distance between two generic points x and y lying on triangles which are disjoint, but very close to each other. The approach proposed here is demonstrated by numerical experiments.