The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in...

An integration formula for calculating the initial transient response of stiff systems exhibiting highly oscillatory solutions, such as quartz oscillators, is derived. The method reformulates the original system of ordinary differential-algebraic equations (DAEs) as a system of partial differential-algebraic equations (PDEs). The PDEs solve the original DAEs for a family of initial conditions. ...

Partial Differential Equations (PDEs)-based morphology offers a wide range of continuous operators to address various image processing problems. Most of these operators are formulated as Hamilton–Jacobi equations or curve evolution level set and morphological flows. In our previous works, we have proposed a simple method to solve PDEs on point clouds using the framework of PdEs (Partial differe...

Transfer functions are a standard description of onedimensional linear and time-invariant systems. They provide an alternative to the conventional representation by ordinary differential equations and are suitable for computer implementation. This article extends that concept to multidimensional (MD) systems, normally described by partial differential equations (PDEs). Transfer function modelin...

There are many evolution partial differential equations which can be cast into Hamiltonian form. Conservation laws of these equations are related to one–parameter Hamiltonian symmetries admitted by the PDEs [1]. The same result holds for semidiscrete Hamiltonian equations [2]. In this paper we consider semidiscrete canonical Hamiltonian equations. Using symmetries, we find conservation laws for...

The Overture framework is an object-oriented environment for solving partial differential equations on overlapping grids. We describe some of the tools in Overture that can be used to generate grids and solve partial differential equations (PDEs). Overture contains a collection of C++ classes that can be used to write PDE solvers either at a high level or at a lower level for efficiency. There ...

We show that any classical matrix Q × Q S-integrable Partial Differential Equation (PDE) obtainable as commutativity condition of a certain type of vector fields possesses a family of lower-dimensional reductions represented by the matrix Q × n 0 Q quasilinear first order PDEs solved in [29] by the method of characteristics. In turn, these PDEs admit two types of available particular solutions:...

The standard numerical algorithms for solving time-dependent partial differential equations (PDEs) are inherently sequential in the time direction. This paper makes the observation that algorithms exist for the time-accurate solution of certain classes of linear hyperbolic and parabolic PDEs that can be parallelized in both time and space and have serial complexities that are proportional to th...

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under idea data-driven methods, algorithms based on Gaussian processes are constructed to solve inverse problem, where we encode distribution information data into kernels ...

In this study, the optimal control problem of nonlinear parabolic partial differential equations (PDEs) is investigated. Optimal control of nonlinear PDEs is an open problem with applications that include fluid, thermal, biological, and chemically-reacting systems. Model predictive control with a fast numerical solution method has been well established to solve the optimal control problem of no...