Fokas transform method for a brain tumor invasion model with heterogeneous diffusion in 1 + 1 dimensions

Gliomas are among the most aggressive forms of brain tumors. Over the last years mathematical models have been well developed to study gliomas growth. We consider a simple and well established mathematical model focused on proliferation and diffusion. Due to the heterogeneity of the brain tissue (white and grey matter) the diffusion coefficient is considered to be discontinuous. Fokas transform approach for the solution of linear PDE problems, apart from the fact that it avoids solving intermediate ODE problems, yields novel integral representations of the solution in the complex plane that decay exponentially fast and converge uniformly at the boundaries. To take advantage of these properties for the solution of the model problem at hand, we have successfully implemented Fokas transform method in the multi-domain environment induced by the interface discontinuities of our problem's domain. The fact that the integral representation of the solution at any time–space point of our problem's domain is independent on any other points of the domain, except of course on initial data, coupled with a simple composite trapezoidal rule, implemented on appropriately chosen integration contours, yields a fast and efficient analytical–numerical technique capable of producing directly high-order approximations of the solution at any point of the domain requiring no prior knowledge of the solution at any other time instances or space information.