Related papers: Partial and complete linearization of PDEs based o…
The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of $n$ linear…
This article demonstrates how variation of parameters can be successfully implemented in combination with other classical techniques, such as the method of characteristics, to derive novel classes of solutions to nonlinear partial…
The hierarchy of the integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of the linearization of these equations and their conservation law in the terms of the solutions of the…
This work presents structure-preserving Lift & Learn, a scientific machine learning method that employs lifting variable transformations to learn structure-preserving reduced-order models for nonlinear partial differential equations (PDEs)…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is…
Partial-differential-equation (PDE)-constrained optimization is a well-worn technique for acquiring optimal parameters of systems governed by PDEs. However, this approach is limited to providing a single set of optimal parameters per…
We propose a conservation-dissipation formalism (CDF) for coarse-grained descriptions of irreversible processes. This formalism is based on a stability criterion for non-equilibrium thermodynamics. The criterion ensures that non-equilibrium…
In this paper we propose a method to couple two or more explicit numerical schemes approximating the same time-dependent PDE, aiming at creating new schemes which inherit advantages of the original ones. We consider both advection equations…
All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted…
We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without…
Conservation laws related to the gauge invariance of Lagrangians and Euler-Lagrange operators in finite and infinite order Lagrangian formalisms are analyzed.
A computational method for numeric resolution of a PDEs system, based on a Finite Differences schema integrated by interpolations of partial results, and an estimate of the error of its solution respect to the normal FD solution.
Symmetry- and conservation law-preserving finite difference discretizations are obtained for linear and nonlinear one-dimensional wave equations on five- and nine-point stencils, using the theory of Lie point symmetries of difference…
This paper develops one of the methods for study of nonlinear Partial Differential equations. We generalize Sato equation and represent the algorithm for construction of some classes of nonlinear Partial Differential Equations (PDE)…
A simple trick is illustrated, whereby nonlinear evolution equations can be modified so that they feature a lot - or, in some cases, only -- periodic solutions. Several examples (ODEs and PDEs) are exhibited.
A novel structure-preserving numerical method to solve random hyperbolic systems of conservation laws is presented. The method uses a concept of generalized, measure-valued solutions to random conservation laws. This yields a linear partial…
Using the principle of structural analogy of solutions, approaches have been developed for constructing exact solutions of complex nonlinear PDEs, including PDEs with delay, based on the use of special solutions to auxiliary simpler related…
Partial differential equations (PDEs) describing thermodynamically isolated systems typically possess conserved quantities (like mass, momentum, and energy) and dissipated quantities (like entropy). Preserving these conservation and…
Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large…