Related papers: Explicit solutions of PDE via Vessiot theory and s…
Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of…
The paper deals with second order abstract linear partial differential equations (LPDE) over a partial differential field with two commuting differential operators. In terms of usual differential equations the main content can be presented…
We generalize our earlier results concerning meshfree collocation methods for semilinear elliptic second order problems to the quasilinear case. The stability question, however, is treated differently, namely by extending a paper on…
Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions with…
We explore variational Poisson-Nijenhuis structures on nonlinear PDEs and establish relations between Schouten and Nijenhuis brackets on the initial equation with the Lie bracket of symmetries on its natural extensions (coverings). This…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…
In this present paper, we study geometric structures of rank two prolongations of implicit second-order partial differential equations (PDEs) for two independent and one dependent variables and characterize the type of these PDEs by the…
We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…
We represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it…
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous…
We investigate the relation between pluri-Lagrangian hierarchies of $2$-dimensional partial differential equations and their variational symmetries. The aim is to generalize to the case of partial differential equations the recent findings…
A family of solutions of the Jacobi PDEs is investigated. This family is $n$-dimensional, of arbitrary nonlinearity and can be globally analyzed (thus improving the usual local scope of Darboux theorem). As an outcome of this analysis it is…
In this article, we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation (2BDSDE). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator,…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
For the class of systems of PDEs, for which infinitesimal translations (with respect to some (in)dependent variables) possess specific finite-dimensional invariant subspaces of the space of generalized symmetries of the system considered.…
For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the…
The first part of the book is devoted to the symmetry approach to classification of scalar integrable evolution PDEs with two independent variables. In the second part systems of evolution equations with polynomial homogeneous right-hand…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
A template-based generic programming approach was presented in a previous paper that separates the development effort of programming a physical model from that of computing additional quantities, such as derivatives, needed for embedded…