Partial differential equations (PDEs) have been successful for solving many problems in computer vision. However, the existing PDEs are all crafted by people with skill, based on some limited and intuitive considerations. As a result, the designed PDEs may not be able to handle complex situations in real applications. Moreover, human intuition may not apply if the vision task is hard to describ...

Bivariate splines with various degrees are considered in this paper. A matrix form of the extended smoothness conditions for these splines is presented. Upon this form, the multivariate spline method for numerical solution of partial differential equations (PDEs) proposed by Awanou, Lai, and Wenston in [The multivariate spline method for scattered data fitting and numerical solutions of partial...

Standard books on the theory of differential equations deal with scalar equations and systems of equations in normal or Cauchy-Kovalevskaya form, i.e., systems that are solvable with respect to the highest-order derivative. Traditionally, one considers systems where a distinguished independent variable t exists such that the system can be written in evolution form, u t = φ(t, x, u, u x), where ...

Although most work to date on RBFs relates to scattered data approximation and in general to interpolation theory, there has recently been an increased interest in their use for solving partial differential equations (PDEs). This approach, which approximates the whole solution of the PDE directly using RBFs, is very attractive due to the fact that this is truly a mesh-free technique. Kansa [1] ...

Mathematical morphology (MM) operators can be defined in terms of algebraic sets or as partial differential equations. In this paper, we introduce a novel formulation of MM over weighted graphs of arbitrary topology. The proposed framework recovers local algebraic and PDEs-based formulations of MM and introduces nonlocal configurations. This enables to PDEs-based methods to process any discrete...

Based on an extended tanh-function method, a general method is suggested to obtain multiple travelling wave solutions for nonlinear partial differential equations (PDEs). The validity and reliability of the method is tested by its application to some nonlinear PDEs. The obtained results are compared with that of an extended tanh-function method and hyperbolic-function method. New exact solution...

This paper gives a preliminary description of DASOPT, a software system for the optimal control of processes described by time-dependent partial differential equations (PDEs). DASOPT combines the use of efficient numerical methods for solving differential-algebraic equations (DAEs) with a package for large-scale optimization based on sequential quadratic programming (SQP). DASOPT is intended fo...

Overview A new approach is presented for solving integral equations. It is a fundamental computational and theoretical advance that provides a unified, fully localized, and computationally efficient solution in closed-form that is useful in both symbolic and numeric computations. The approach is naturally suited for fine-grain parallel implementation. In practical problems such as shift-variant...

The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as expected value of a random time process. Using the latter, present interesting numerical based on Monte Carlo integration simulate solutions and partial equations. Thirdly, show that allows us find fundamental for (PDEs), which derivative Caputo sense space o...

We investigate conservative properties of Runge-Kutta methods for Hamiltonian PDEs. It is shown that multi-symplecitic Runge-Kutta methods preserve precisely norm square conservation law. Based on the study of accuracy of Runge-Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy of conservation laws of energy and momentum for H...