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This paper studies a model of two-phase flow with an immersed material viscous interface and a finite element method for numerical solution of the resulting system of PDEs. The interaction between the bulk and surface media is characterized…
In this work, we will study a numerical method that allows finding an approximation of the exact solution for a in-situ combustion model using the nonlinear mixed complementary method, which is a variation of the Newtons method for solving…
Fractional Differential Equations (FDEs) are essential tools for modelling complex systems in science and engineering. They extend the traditional concepts of differentiation and integration to non-integer orders, enabling a more precise…
We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.
These notes are an introduction to fluctuating hydrodynamics (FHD) and the formulation of numerical schemes for the resulting stochastic partial differential equations (PDEs). Fluctuating hydrodynamics was originally introduced by Landau…
Numerical methods: mimetic finite differences and finite elements, are analyzed from a numerical point of view. It seeks to conclude on the efficiency, order of convergence and computational cost of these methods. The analysis is done in…
This article presents a new finite element method for convection-diffusion equations by enhancing the continuous finite element space with a flux space for flux approximations that preserve the important mass conservation locally on each…
In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded…
This paper presents a new numerical method which approximates Neumann type null controls for the heat equation and is based on the Fokas method. This is a direct method for solving problems originating from the control theory, which allows…
In this note we shall introduce a simple, effective numerical method for solving partial differential equations for scalar and vector-valued data defined on surfaces. Even though we shall follow the traditional way to approximate the…
In this paper, we establish several local and global gradient estimates for the positive solution of Porous Medium Equations (PMEs) and Fast Diffusion Equations (FDEs). Our proof is probabilistic and uses martingale techniques.
Fractional order models have proven to be a very useful tool for the modeling of the mechanical behaviour of viscoelastic materials. Traditional numerical solution methods exhibit various undesired properties due to the non-locality of the…
The article describes various aspects of mathematical modeling of fluid flows, both in general and with reference to hydraulic machinery. The article reviews historical development of corresponding methods of mathematical modeling.…
In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing…
Feedforward neural networks offer a promising approach for solving differential equations. However, the reliability and accuracy of the approximation still represent delicate issues that are not fully resolved in the current literature.…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
Numerical solutions for flows in partially saturated porous media pose challenges related to the non-linearity and elliptic-parabolic degeneracy of the governing Richards' equation. Iterative methods are therefore required to manage the…
We introduce the multivariate decomposition finite element method (MDFEM) for solving elliptic PDEs with uniform random diffusion coefficients. We show that the MDFEM can be used to reduce the computational complexity of estimating the…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
Despite their ubiquity throughout science and engineering, only a handful of partial differential equations (PDEs) have analytical, or closed-form solutions. This motivates a vast amount of classical work on numerical simulation of PDEs and…