We present a new method based on functional tensor decomposition and dynamic approximation to compute the solution of high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea is project time derivative PDE onto tangent space low-rank manifold at each time. Such projection can be computed by minimizing convex energy over space. This minimization problem yields uniq...

The aim of this paper is to present a simple extension of the theory of linear, distributed, port-Hamiltonian systems to the nonlinear scenario. More precisely, an algebraic nonlinear skew-symmetric term has now been included in the PDE. It is then shown that the system can be equivalently written in terms of the scattering variables, and that these variables are strictly related with the Riema...

Nonlinear evolution wave equations (NEEs) are partial differential equations (PDEs) involving first or second order derivatives with respect to time. Such equations have been intensively studied for the past few decades [1-3] and several new methods to solve nonlinear PDEs either numerically or analytically are now available. Hirota's bilinear method is a powerful tool for obtaining a wide clas...

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily com...

The objective of this paper is to use the Pfaffian technique to construct different classes of exact Pfaffian solutions and N -soliton solutions to some of the generalized integrable nonlinear partial differential equations in (3+1) dimensions. In this paper, I will show that the Pfaffian solutions to the nonlinear PDEs are nothing but Pfaffian identities. Solitons are among the most beneficial...

A genetic algorithm procedure is demonstrated that refines the selection of interpolation points of the discrete empirical interpolation method (DEIM) when used for constructing reduced order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs) with proper orthogonal decomposition. The method achieves nearly optimal interpolation points with only a few g...

• Symmetry analysis of differential equations: classical and nonclassical symmetries, potential symmetries, group–invariant solutions, variational symmetries and conservation laws • Applications of symmetry analysis theory to mathematical models arising in mathematical physics, mathematical biology, image processing, engineering, financial mathematics and other research fields • Geometric appro...

In this paper, we prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. Our convergence analysis also provides a proof for the existence of the global weak...