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This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of…
We study a nonlinear, pseudomonotone, stochastic diffusion-convection evolution problem on a bounded spatial domain, in any space dimension, with homogeneous boundary conditions and reflection. The additive noise term is given by a…
The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time-space dependent variable coefficients is challenging, especially when the classical weak initial singularities are…
In this paper, we consider a time-fractional reaction-diffusion system with the same nonlinearities of the Newton-Leipnik chaotic system. Through analytical tools and numerical results, we derive sufficient conditions for the asymptotic…
This paper focuses on systems of nonlinear second-order stochastic differential equations with multi-scales. The motivation for our study stems from mathematical physics and statistical mechanics, for examples, Langevin dynamics and…
We investigate a system of nonlinear partial differential equations modeling the unsteady flow of a shear-thinning non-Newtonian fluid with a concentration-dependent power-law index. The system consists of the generalized Navier-Stokes…
In this paper, we develop an efficient numerical solver for unsteady diffusion-type partial differential equations with random coefficients. A major computational challenge in such problems lies in repeatedly handling large-scale linear…
A model has two main aims: predicting the behavior of a physical system and understanding its nature, that is how it works, at some desired level of abstraction. A promising recent approach to model building consists in deriving a…
We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion system. The purpose of this article is twofold: first, it aims at a…
In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that is, backward stochastic evolution equations, stochastic Volterra type evolution…
This paper is devoted to the study of a fractional version of non-linear $\mathpzc{M}^\nu(t)$, $t>0$, linear $M^\nu (t)$, $t>0$ and sublinear $\mathfrak{M}^\nu (t)$, $t>0$ death processes. Fractionality is introduced by replacing the usual…
In this paper we study pattern formation arising in a system of a single reaction-diffusion equation coupled with subsystem of ordinary differential equations, describing spatially-distributed growth of clonal populations of precancerous…
We consider an evolution equation with the regularized fractional derivative of an order $\alpha \in (0,1)$ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables.…
Cross-diffusion systems are systems of nonlinear parabolic partial differential equations that are used to describe dynamical processes in several application, including chemical concentrations and cell biology. We present a space-time…
We present and analyze a structure-preserving method for the approximation of solutions to nonlinear cross-diffusion systems, which combines a Local Discontinuous Galerkin spatial discretization with the backward Euler time-stepping scheme.…
We study a class of semilinear diffusion equations on infinite, connected, weighted graphs, focusing on two types of nonlinearities: monotone decreasing and Lipschitz continuous. Under minimal structural assumptions on the graph, we…
We introduce an $L_q(L_p)$-theory for the quasi-linear fractional equations of the type $$ \partial^{\alpha}_t u(t,x)=a^{ij}(t,x)u_{x^i x^j}(t,x)+f(t,x,u), \quad t>0, \,x\in \mathbf{R}^d. $$ Here, $\alpha\in (0,2)$, $p,q>1$, and…
A non-autonomous evolution semi-linear differential system under non-instantaneous impulses, delays, and perturbed by non-local conditions is studied. Its piece-wise continuous solutions belong to a finite-dimensional Banach space. The…
Many physical systems characterized by nonlinear multiscale interactions can be effectively modeled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative…
The method of generalized modeling has been applied successfully in many different contexts, particularly in ecology and systems biology. It can be used to analyze the stability and bifurcations of steady-state solutions. Although many…