An iterative method based on Picard’s approach to ODEs’ initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examp...

We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs, as suggested in [4]. In particular, we provide a fully nonlinear extension of the Feynman-Kac formula. Unlike [4], the alternative formulation of this paper i...

The Multiquadric Radial Basis Function (MQ) Method is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. We descretize nonlinear elliptic PDEs by the MQ method. This results in modest size systems of nonlinear algebraic equations which can be eeciently continued by standard continuation software such as auto and c...

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the ZakharovKuznetsov and Kadomtsev-Petviashvili equations as examples. The method is algorithmic and has been implemented in Mathematica. The s...

Many large-scale scientific simulations require the solution of nonlinear partial differential equations (PDEs). The effective solution of such nonlinear PDEs depends to a large extent on efficient and robust sparse linear system solution. In this paper, we show how fast and reliable sparse linear solvers can be composed from several underlying linear solution methods. We present a combinatoria...

A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a ...

A method based on innnite parameter conservation laws is described to factor linear differential operators out of nonlinear partial diierential equations (PDEs) or out of diierential consequences of nonlinear PDEs. This includes a complete linearization to an equivalent linear PDE (system) if that is possible. Innnite parameter conservation laws can be computed, for example, with a computer alg...

General techniques are developed to obtain: (1) the completion of a systemof nonlinear first-order partid differential equations (PDES) which is an indepem dent set of further PDES derivable from the system by differentiation and elimination; and (2) simplifications of the system by choosing appropriate new independent and dependent variables using a result from Lie group theory The number of d...

We show that matrix Q × Q Self-dual type S-integrable Partial Differential Equations (PDEs) possess a family of lower-dimensional reductions represented by the matrix Q × n 0 Q quasilinear first order PDEs solved in [29] by the method of characteristics. In turn, these PDEs admit two types of available particular solutions: (a) explicit solutions and (b) solutions described implicitly by a syst...

Many physical applications require the solution of a system of coupled partial differential equations (PDEs). In most cases the analytic PDE solution does not exist for this system and we need to solve the problem numerically using the finite element method in COMSOL. This paper presents information on techniques needed in COMSOL 4 to enable computational studies of coupled systems of PDEs for ...