Related papers: Explicit solutions of PDE via Vessiot theory and s…

We have recently solved the inverse spectral problem for integrable PDEs in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter $\lambda$. The associated inverse problem, in…

Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…

A {\it Lie system} is a nonautonomous system of first-order differential equations admitting a {\it superposition rule}, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants.…

Distributed algorithms for solving coupled semidefinite programs (SDPs) commonly require many iterations to converge. They also put high computational demand on the computational agents. In this paper we show that in case the coupled…

Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…

In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there…

This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into…

In this article, we propose a new numerical approach to high-dimensional partial differential equations (PDEs) arising in the valuation of exotic derivative securities. The proposed method is extended from Reisinger and Wittum (2007) and…

We study one-dimensional stochastic integral equations with non-smooth dispersion coefficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss…

We develop a unified PDE-probabilistic framework for pointwise gradient and Hessian estimates of Markov semigroups associated with stochastic differential equations with singular and unbounded coefficients. Under mild local structural…

We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when…

Partial differential equations (PDEs) are often computationally challenging to solve, and in many settings many related PDEs must be be solved either at every timestep or for a variety of candidate boundary conditions, parameters, or…

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs.…

We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo's time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can…

Modeling physical phenomena like heat transport and diffusion is crucially dependent on the numerical solution of partial differential equations (PDEs). A PDE solver finds the solution given coefficients and a boundary condition, whereas an…

Second order integrals of motion for 3d quantum mechanical systems with position dependent masses (PDM) are classified. Namely, all PDM systems are specified which, in addition to their rotation invariance, admit at least one second order…

We extend the notion of viscosity solutions for path-dependent PDEs introduced by Ekren et al. [Ann. Probab. 42 (2014), no. 1, 204-236] to path-dependent integro-differential equations and establish well-posedness, i.e., existence,…

We consider divergence form uniformly parabolic SPDEs with bounded and measurable leading coefficients and possibly growing lower-order coefficients in the deterministic part of the equations. We look for solutions which are summable to the…

A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…