This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under idea data-driven methods, algorithms based on Gaussian processes are constructed to solve inverse problem, where we encode distribution information data into kernels ...

In this paper, the modified Korteweg-de Vries (mKdV) equation with variable coefficients (vc-mKdV equation) is investigated via two kinds of approaches and symbolic computation. On the one hand, we firstly reduce the vc-mKdV equation to a second-order nonlinear nonhomogeneous ODE using travelling wave-like similarity transformation. And then we obtain its many types of exact fractional solution...

In this paper, we study a new class of equations called mean-field backward stochastic differential (BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H > 1/2. First, the existence and uniqueness BSDEs are obtained. Second, comparison theorem solutions is established. Third, as an application, connect nonlocal partial equation (PDE, short), derive relationship between ...

Abstract We present a novel numerical scheme to approximate the solution map s ? u ( ) := ???? – f fractional PDEs involving elliptic operators. Reinterpreting as an interpolation operator allows us write integral including solutions parametrized family of local PDEs. propose reduced basis strategy on top finite element method its integrand. Unlike prior works, we deduce choice snapshots for pr...

Data-driven discovery of partial differential equations (PDEs) from observed data in machine learning has been developed by embedding the problem. Recently, traditional ODEs dynamics using linear multistep methods deep have discussed [Racheal and Du, SIAM J. Numer. Anal. 59 (2021) 429-455; Du et al. arXiv:2103.1148 ]. We extend this framework to data-driven time-fractional PDEs, which can effec...

Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It one most general methods when compared to other techniques. PDEs posed in a variational form over given space, say Hilbert are better numerically handled with FEM. The FEM algorithm used various applications which includes fluid flow, heat transfer, acoustics, structural mech...

We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension $ d\geq 1 $. Our approach relies a tree-based probabilistic representation based (2s) $-stable branching processes all s\in (0,1) $, and our results hold sufficiently small conditions nonlinearity coefficients. In comparison existing approaches, we con...

In this paper, we introduce two families of nontensorial generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient accurate spectral methods for solving PDEs with integral fractional Laplacian (IFL) and/or Schrödinger operators ? d . As a generalisation the G. Szegö’s family 1D (1939), first multivariate GHPs (resp. GHFs) are orthogonal respect to weig...

Most physical phenomena are formulated in the form of non-linear fractional partial differential equations to better understand complexity these phenomena. This article introduces a recent attractive analytic-numeric approach investigate approximate solutions for nonlinear time by means coupling Laplace transform operator and Taylor’s formula. The validity applicability used method illustrated ...