The present work extends a previous paper where an agent-based and two-dimensional partial differential diffusion model was introduced for describing immune cell dynamics (leukocytes) in cancer-on-chip experiments. In the present work, new features are introduced for the dynamics of leukocytes and for their interactions with tumor cells, improving the adherence of the model to what is observed in laboratory experiments. Each system's solution realization is a family of biased random walk trajectories, affected by the chemotactic gradients and in turn affecting them.

During melting under gravity in the presence of a horizontal thermal gradient, buoyancy-driven convection in the liquid phase affects significantly the evolution of the liquid-solid interface. Due to the obvious engineering interest in understanding and controlling melting processes, fluid dynamicists and applied mathematicians have spent many efforts to model and simulate them numerically. Their endeavors concentrated in the twenty-five years period between the publication of the paper by Brent, Voller & Reid (1988) and that by Mansutti & Bucchignani (2011).

In this paper, an algorithm for the numerical evaluation of hypersingular finite-part integrals with rapidly oscillating kernels is proposed. The method is based on an interpolatory procedure at zeros of the orthogonal polynomials with respect to the first kind Chebyshev weight. Bounds of the error and of the amplification factor are also provided. Numerically stable procedure are obtained and the corresponding algorithms can be implemented in a fast way.

Functionally graded materials (FGMs), possessing properties that vary smoothly from one region to another, have been receiving increasing attention in recent years, particularly in the aerospace, automotive and biomedical sectors. However, they have yet to reach their full potential. In this paper, we explore the potential of FGMs in the context of drug delivery, where the unique material characteristics offer the potential of finetuning drug-release for the desired application.

We deal with integral equations with a singular kernel of Carlman type. A method to approach to the solution of these equations is given. Infinite matrix theory is used to determine the Fourier coefficients of the solution in the expansion in a series of orthogonal polynomials.

Recent literature confirms the crucial influence of non-viscous deformations together with temperature impact on glacier and rock glacier flow numerical simulation. Along this line, supported by the successful test on a one-dimensional set-up developed by two of the author, we propose the numerical solution of a two-dimensional rock-glacier flow model based on an ice constitutive law of second grade differential type .

We tackle the problem of coupling a spatiotemporal model for simulating the spread and control of an invasive alien species with data coming from image processing and expert knowledge. In this study, we implement a spatially explicit optimal control model based on a reaction-diffusion equation which includes an Holling II type functional response term for modeling the density control rate. The model takes into account the budget constraint related to the control program and searches for the optimal effort allocation for the minimization of the invasive alien species density.

A new fractional q-order variation of the RothC model for the dynamics of soil organic carbon is introduced. A computational method based on the discretization of the analytic solution along with the finite-difference technique are suggested and the stability results for the latter are given. The accuracy of the scheme, in terms of the temporal step size h, is confirmed through numerical testing of a constructed analytic solution. The effectiveness of the proposed discrete method is compared with that of the classical discrete RothC model.