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A new iterative technique is presented for solving of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations. Proposed iterative scheme does not require any discretization,…

In this contribution we present recent developments in the formulation and solution of Initial Boundary Value Problems (IBVPs). Building upon a modern variational action formulation of classical dynamics, we treat Initial Boundary Value…

The PAMPA (Point-Average-Moment PolynomiAl-interpreted) method, proposed in [R. Abgrall, Commun. Appl. Math. Comput., 5: 370-402, 2023], combines conservative and non-conservative formulations of hyperbolic conservation laws to evolve cell…

Taking insight from the theory of general relativity, where space and time are treated on the same footing, we develop a novel geometric variational discretization for second order initial value problems (IVPs). By discretizing the dynamics…

In this study, an implicit-explicit local differential transform method (IELDTM) based on Taylor series representations is produced for solving 2D and 3D advection-diffusion equations. The parabolic advection-diffusion equations are reduced…

The Alternating Direction Method of Multipliers (ADMM) and its distributed version have been widely used in machine learning. In the iterations of ADMM, model updates using local private data and model exchanges among agents impose critical…

A compact version of the variation evolving method (VEM) is developed in the primal variable space for optimal control computation. Following the idea that originates from the Lyapunov continuous-time dynamics stability theory in the…

A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…

A new implicit-explicit local differential transform method (IELDTM) is derived here for time integration of the nonlinear advection-diffusion processes represented by (2+1)-dimensional Burgers equation. The IELDTM is adaptively constructed…

In this paper, we present a novel pseudospectral (PS) method for solving a new class of initial-value problems (IVPs) of time-dependent one-dimensional fractional partial differential equations (FPDEs) with variable coefficients and…

The compact Variation Evolving Method (VEM) that originates from the continuous-time dynamics stability theory seeks the optimal solutions with variation evolution principle. It is further developed to be more flexible in solving the…

Here the Integral Value Transformations (IVTs) are considered to be Discrete Dynamical System map in the space\mathbb{N}_(0). In this paper, the dynamics of IVTs is deciphered through the light of Topological Dynamics.

A notion of dimension preservative map, \textit{Integral Value Transformations} (IVTs) is defined over $\mathbb{N}^k$ using the set of $p$-adic functions. Thereafter, two dimensional \textit{Integral Value Transformations} (IVTs) is…

On the basis of additive schemes (splitting schemes) we construct efficient numerical algorithms to solve approximately the initial-boundary value problems for systems of time-dependent partial differential equations (PDEs). In many applied…

In [13], an Inexact variant of Stochastic Dual Dynamic Programming (SDDP) called ISDDP was introduced which uses approximate (instead of exact with SDDP) primal dual solutions of the problems solved in the forward and backward passes of the…

In this paper we report a few numerical tests by using a slight extension of the Matlab code fhbvm in [8], implementing Fractional HBVMs, a recently introduced class of numerical methods for solving Initial Value Problems of Fractional…

In this paper we define a discrete dynamical system that governs the evolution of a population of agents. From the dynamical system, a variant of Differential Evolution is derived. It is then demonstrated that, under some assumptions on the…

In this work, we present a machine learning approach for reducing the error when numerically solving time-dependent partial differential equations (PDE). We use a fully convolutional LSTM network to exploit the spatiotemporal dynamics of…

Physics-informed neural networks (PINNs) have shown promising potential for solving partial differential equations (PDEs) using deep learning. However, PINNs face training difficulties for evolutionary PDEs, particularly for dynamical…

The Variation Evolving Method (VEM) that originates from the continuous-time dynamics stability theory seeks the optimal solutions with variation evolution principle. After establishing the first and the second evolution equations within…