Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning associated Green's function $G$. By exploiting hierarchical low-rank structure $G$, show that one can construct approximant to $G$ converges almost surely and achieves a relative error $\mathcal{O}(\Gamma_\epsilon^{-1/2}\log^3(1/\epsilon...

abstract. the main contribution of the current paper is to propose a new effective numerical method for solving the first-order linear matrix differential equations. properties of the legendre basis operational matrix of integration together with a collocation method are applied to reduce the problem to a coupled linear matrix equations. afterwards, an iterative algorithm is examined for solvin...

In this article, a numerical investigation of magnetohydrodynamic non-Newtonian nanofluid flow on a stretching sheet through an isotropic porous medium. The effects of both non-linear thermal radiation and heat generation/absorption were studied on distributions of velocity, temperature and concentration. On the other side, the governing partial differential equations have been transformed by u...

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiencyand convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partia...

A historical development of the theory of linear partial differential equation is reviewed with comments. A recent development in the theory of linear partial differential equations is discussed.

The linearization of complex ordinary differential equations is studied by extending Lie’s criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations implies the linearizability of systems of partial differential equations corresponding to those complex ordinary differential equations. The invertible comp...

This paper presents a simple and computationally-efficient algorithm for solving steady two-dimensional subsonic and transonic compressible flow over an airfoil. This work uses an interactive viscous-inviscid solution by incorporating the viscous effects in a thin shear-layer. Boundary-layer approximation reduces the Navier-Stokes equations to a parabolic set of coupled, non-linear partial diff...

The paper is devoted to an application of Lie group theory to differential equations. The basic infinitesimal method for calculating symmetry group is presented, and used to determine general symmetry group of some differential equations. We include a number of important applications including integration of ordinary differential equations and finding some solutions of partial differential equa...

In this research, Fischer Tropsch synthesis (FTS) has been modeled in the fixed bed chromatographic reactor for the first time by applying a rather complex dispersed plug flow model for fluid phase and linear driving force (LDF) model for adsorbent. Model equations are dynamic, multi-component, non-linear and heterogeneous including reaction and adsorption simultaneously Complex kinetics fo...