Gülnihal Meral

This study focuses on the numerical approximation of singular initial value problems (SIVPs) defined by the generalized Emden–Fowler equation, which appears in many physical and engineering models such as astrophysics and heat conduction. The proposed method is based on the Differential Quadrature Method (DQM), a powerful domain discretization technique that approximates derivatives by weighted sums of function values at discrete points. By employing a relatively small number of collocation points, DQM offers highly accurate approximations even for problems with singularities or steep solution profiles. In this work, the resulting nonlinear system of algebraic equations derived from DQM is solved using Picard iteration, which provides a simple yet effective approach to handle the nonlinear terms. The method is applied to a range of test problems that involve different types of nonlinearities and source terms, including exponential and logarithmic forms. Numerical experiments are conducted to compare the approximate solutions with known exact solutions, and the maximum and minimum absolute errors are calculated to evaluate the performance of the method. The results confirm that the proposed DQM–Picard approach yields highly accurate solutions using significantly fewer discretization points than many traditional numerical techniques. This makes it a promising tool for efficiently solving a broad class of nonlinear singular problems governed by Emden–Fowler type equations. Overall, the study demonstrates the reliability and effectiveness of the method in handling nonlinear singular models with various complexities.

In this study, the finite propogation speed properties investigated for a two dimensional exterior problem defined by nonlinear Klein-Gordon equation. Under some assumptions on the initial data and the nonlinearity, the solution is shown to have a finite propogation speed. Furthermore, it is demonstrated that the problem has a unique solution, and accurate numerical solutions have been produced by the use of the dual reciprocity boundary element approach with linear radial basis functions.

Cancer cell invasion is one of the most important stages of metastasis. In this paper, the local existence and uniqueness of the cancer cell invasion model is proved using an iterative procedure. Moreover the numerical simulations are performed using a combination of a nonstandard finite difference scheme and backward and forward Euler methods. It is seen that the results agree well with the expected behaviour of the invasion.

It is known that the pH level of the extracellular tumour environment directly effects the progression of the tumour. In this study, the mathematical model for the acid-mediated tumour cell invasion consisting of a system of nonlinear reaction diffusion equations describing the interaction between the density of the tumour cells, normal cells and the concentration of protons produced by the tumour cells is solved numerically using the combined application of dual reciprocity boundary element method (DRBEM) and finite difference method. The space derivatives in the model are discretised by DRBEM using the fundamental solution of Laplace equation considering the time derivative and the nonlinearities as the nonhomogenity. The resulting systems of ordinary differential equations after the application of DRBEM are then discretised using forward difference. Because of the highly nonlinear character of the model, there arises difficulties in solving the model especially for two-dimensions and the boundary-only nature of DRBEM discretisation gives the advantage of having solutions with a lower computational cost. The proposed method is tested with different kinds of carrying capacities which also depend on time. The results of the numerical simulations are compared among each case and seen to confirm the expected behaviour of the model.

Cancer cell migration is an essential feature in the process of tumor spread and establishing of metastasis. It characterizes the invasion observed on the level of the cell population, but it is also tightly connected to the events taking place on the subcellular level. These are conditioning the motile and proliferative behavior of the cells, but are also influenced by it. In this work we propose a multiscale model linking these two levels and aiming to assess their interdependence. On the subcellular, microscopic scale it accounts for integrin binding to soluble and insoluble components present in the peritumoral environment, which is seen as the onset of biochemical events leading to changes in the cell's ability to contract and modify its shape. On the macroscale of the cell population this leads to modifications in the diffusion and haptotaxis performed by the tumor cells and implicitly to changes in the tumor environment. We prove the (local) well posedness of our model and perform numerical simulations in order to illustrate the model predictions.

We propose a model for acid-mediated tumour invasion involving two different scales: the microscopic one, for the dynamics of intracellular protons and their exchange with their extracellular counterparts, and the macroscopic scale of interactions between tumour cell and normal cell populations, along with the evolution of extracellular protons. We also account for the tactic behaviour of cancer cells, the latter being assumed to biase their motion according to a gradient of extracellular protons (following Bartel et al. (2012) pH-taxis: directional tumor cell migration along pH-gradients. Acta Physiol., 204, 113; Paradise et al. (2013) Directional cell migration in an extracellular pH gradient: a model study with an engineered cell line and primary microvascular endothelial cells. Exp. Cell Res., 319, 487–497; we call this pH taxis). A time dependent (and also time delayed) carrying capacity for the tumour cells in response to the effects of acidity is considered as well. The global well posedness of the resulting multiscale model is proved with a regularization and fixed point argument. Numerical simulations are performed to illustrate the behaviour of the model.