We developed parallel time domain decomposition methods to solve systems of linear ordinary differential equations (ODEs) based on the Aitken-Schwarz [5] or primal Schur complement domain decomposition methods [4]. The methods require the transformation of the initial value problem in time defined on ]0,T ] into a time boundary values problem. Let f (t,y(t)) be a function belonging to C 1(R+,Rd...

We develop a theoretical foundation for the application of Nesterov’s accelerated gradient descent method (AGD) to approximation solutions wide class partial differential equations (PDEs). This is achieved by proving existence an invariant set and exponential convergence rates when its preconditioned version (PAGD) applied minimize locally Lipschitz smooth, strongly convex objective functionals...

In the article we discuss notion of generalized invariant manifold introduced in our previous study. literature method differential constraints is well known as a tool for constructing particular solutions nonlinear partial equations. Its essence adding to PDE, much simpler, rule ordinary, equation, compatible with given one. Then any solution ODE PDE well. However main problem find this ODE. O...

We consider a limit case of a system of two equations arising in magnetic recording for a one-dimensional domain. The system models the tape deflection when it is driven over a magnetic head profile, which is a known function. The system is reduced to a second order nonlinear equation, where the unknown u appears evaluated in a finite set of distinguished points {xi}i=1 of the domain.

In this paper we improve the quasilinearization method by barycentric Lagrange interpolation because of its numerical stability and computation speed to achieve a stable semi analytical solution. Then we applied the improved method for solving the Fin problem which is a nonlinear equation that occurs in the heat transferring. In the quasilinearization approach the nonlinear differential equatio...

In this article, a reduced vector potential formulation for transient magneto-quasistatic field problems is presented. The finite element method spatial discretization yields system of differential algebraic equations (DAE). Applying generalized Schur complement the DAE decomposes into ordinary (ODE) and an equation. order to avoid computationally expensive linearization schemes solve nonlinear...

In this paper we consider a nonlinear strongly damped wave equation as a model for a controlled spring–mass–damper system and give some results concerning its large time behaviour. It can be seen that the infinite dimensional system admits a twodimensional attracting manifold where the equation is well represented by a classical nonlinear oscillations ODE, which can be exhibited explicitly. In ...

Finite Element Analysis (FEA) has been widely used to generate simulations of complex nonlinear systems. Despite its strength and accuracy, FEA usually two limitations: (i) running high-fidelity often requires high computational cost consumes a large amount time; (ii) is deterministic method that insufficient for uncertainty quantification when modeling systems with various types uncertainties....