The maximum principle is an important property of solutions to PDE. Correspondingly, it’s of great interest for people to design a high order numerical scheme solving PDE with this property maintained. In this thesis, our particular interest is solving convection-dominated diffusion equation. We first review a nonconventional maximum principle preserving(MPP) high order finite volume(FV) WENO s...

This paper presents a method to extend PDE surfaces to high dimensional spaces. We review a common existing analytic solution, and show how it can be used straightforwardly to increase the dimension of the space the surface is embedded within. We then further develop a numerical scheme suitable for increasing the number of variables that parametrise the surface, and investigate some of the prop...

In this paper, we introduce a new method for image smoothing based on a fourth-order PDE model. The method is tested on a broad range of real medical magnetic resonance images, both in space and time, as well as on nonmedical synthesized test images. Our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail, even at poor signal-to-noise r...

Abstract. In this paper, we consider an equivalence problem of second order partially differential equations (PDE) and a duality of the flat differential equation. For the equivalence problem, explicit form of invariants (curvatures) are given. In particular, if all of the curvatures vanish, then PDE are equivalent to the flat equation. We also investigate a duality associated with the flat equ...

In the present paper a bicriterial optimal control problem governed by a parabolic partial differential equation (PDE) and bilateral control constraints is considered. For the numerical optimization the reference point method is utilized. The PDE is discretized by a Galerkin approximation utilizing the method of proper orthogonal decomposition (POD). POD is a powerful approach to derive reduced...

Through assuming that nonlinear superposition principles NLSPs are embedded in a Lie group, a class of 3rd-order PDEs is derived from a general determining equation that determine the invariant group. The corresponding NLSPs and transformation to linearize the nonlinear PDE are found, hence the governing PDE is proved C-integrable. In the end, some applications of the PDEs are explained, which ...

Shape optimization problems frequently take the form of finding parameters that describe the shape of an object or a region in order to minimize a given design objective (such as weight or drag). In many practical situations, the behavior of states in the system can be modeled as the solution to a partial differential equation (PDE). Calculating the dependence of the state solution on these sha...

This work focuses on the study of partial differential equation (PDE) based basis function for Discontinuous Galerkin methods to solve numerically wave-related boundary value problems with variable coefficients. To tackle constant coefficients, wave-based have been widely studied in literature: they rely concept Trefftz functions, i.e. local solutions governing PDE, using oscillating functions ...