Related papers: Partial and complete linearization of PDEs based o…
In this paper, we present a new method for estimating the $L_2$-gain of systems governed by 2nd order linear Partial Differential Equations (PDEs) in two spatial variables, using semidefinite programming. It has previously been shown that,…
We represent an algorithm reducing a big class of systems of ($M+1$)-dimensional nonlinear partial differential equations (PDEs) to the systems of $M$-dimensional first order PDEs. Thus, we integrate the original system with respect to only…
Using the symmetry approach, we find a class of integrable nonlinear PDEs with dispersion law $\omega(k)=k^{\frac32}$. All these equations turn out to be linearizable by means of a differential parametrization.
Conservation laws are formulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not…
It is well-known that the controllability of finite-dimensional nonlinear systems can be established by showing the controllability of the linearized system. However, this classical result does not generalize to infinite-dimensional…
We prove that under certain assumptions a partial differential equation can be derived from a variational principle. It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the…
The combination of machine learning and physical laws has shown immense potential for solving scientific problems driven by partial differential equations (PDEs) with the promise of fast inference, zero-shot generalisation, and the ability…
For nonlinear Schroedinger equations with a power nonlinearity, a new approach to derive the conservation law of the momentum and the pseudo conformal conservation law is obtained. Since this approach does not contain approximating…
Conservation laws are an inherent feature in many systems modeling real world phenomena, in particular, those modeling biological and chemical systems. If the form of the underlying dynamical system is known, linear algebra and algebraic…
We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including…
Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law…
A new integration technique is presented for systems of linear partial differential equations (PDEs) for which syzygies can be formulated that obey conservation laws. These syzygies come for free as a by-product of the differential Groebner…
We introduce a robust first order accurate meshfree method to numerically solve time-dependent nonlinear conservation laws. The main contribution of this work is the meshfree construction of first order consistent summation by parts…
Using standard calculus, explicit formulas for one-, two- and three-dimensional homotopy operators are presented. A derivation of the one-dimensional homotopy operator is given. A similar methodology can be used to derive the…
State-dependent parameter identification, where unknown model parameters depend on one or more state variables in partial differential equations (PDEs) or coupled PDE systems, is fundamental to a wide range of problems in physics,…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
An alternative method of constructing the formal diagonalization for the discrete Lax operators is proposed which can be used to calculate conservation laws and in some cases generalized symmetries for discrete dynamical systems. Discrete…
For the hyperbolic system of quasilinear first-order partial differential equations, linearizable by hodograph transformation, the conservation laws are used to solve the Cauchy problem. The equivalence of the initial problem for…
The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite…
Nonlinear dynamics is a pervasive phenomenon observed in scientific and engineering disciplines. However, the task of deriving analytical expressions to describe nonlinear dynamics from limited data remains challenging. In this paper, we…