相关论文: Effective Particle Methods for Fisher-Kolmogorov E…
We study a time-fractional Fisher-KPP equation involving a Riemann-Liouville fractional derivative acting on the diffusion term, as derived by Angstmann and Henry (Entropy, 22:1035, 2020). The model captures memory effects in diffusive…
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…
In this paper is described the general aspect of a numerical method for piecewise determin-istic Markov processes with boundary. Under very natural hypotheses, a crucial result about uniqueness of solution of a generalized Kolmogorov…
We construct double layered solutions to the extended Fisher-Kolmogorov P.D.E., under the assumption that the set of minimal heteroclinics of the corresponding O.D.E. satisfies a separation condition. The aim of our work is to provide for…
Recent advances in deep learning have significantly improved brain tumour segmentation techniques; however, the results still lack confidence and robustness as they solely consider image data without biophysical priors or pathological…
We give a very short introduction to discrete and continuum models for the evolutionary and spatial dynamics of cancer through two case studies: a model for the evolutionary dynamics of cancer cells under cytotoxic therapy and a model for…
Physical laws governing population dynamics are generally expressed as differential equations. Research in recent decades has incorporated fractional-order (non-integer) derivatives into differential models of natural phenomena, such as…
New kind of differential equations, called local fractional differential equations, has been proposed for the first time. They involve local fractional derivatives introduced recently. Such equations appear to be suitable to deal with…
Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering, and mathematical problems involving functions of several variables, such as the propagation of heat…
I analyse a generalised Burger's equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show…
The Chapman-Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Fractional integrals can be used to describe the fractal media. Using…
A large class of physically important nonlinear and nonhomogeneous evolution problems, characterized by advection-like and diffusion-like processes, can be usefully studied by a time-differential form of Kolmogorov's solution of the…
Motivated by applications in systems biology, we seek a probabilistic framework based on Markov processes to represent intracellular processes. We review the formal relationships between different stochastic models referred to in the…
We investigate the evolution of tumor growth relying on a nonlinear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application.…
The solution of option-pricing problems may turn out to be computationally demanding due to non-linear and path-dependent payoffs, the high dimensionality arising from multiple underlying assets, and sophisticated models of price dynamics.…
This work (Part (I)) together with its companion (Part (II) [45]) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past…
We develop a unified theory for continuous in time finite element discretisations of partial differential equations posed in evolving domains including the consideration of equations posed on evolving surfaces and bulk domains as well…
Mechanical models for tumor growth have been used extensively in recent years for the analysis of medical observations and for the prediction of cancer evolution based on imaging analysis. This work deals with the numerical approximation of…