in this paper, the ritz-galerkin method in bernstein polynomial basis is applied for solving the nonlinear problem of the magnetohydrodynamic (mhd) flow of third grade fluid between the two plates. the properties of the bernstein  polynomials together with the ritz-galerkin method are used to reduce the solution of the mhd couette flow of non-newtonian fluid in a porous medium to the solution o...

BACKGROUND Although the Netherlands currently has its own procedure for evaluating chemical compounds and setting occupational exposure limits (OELs), most of these limits were originally adopted in the 1970s from threshold limit values (TLVs) set by the American Conference of Governmental Industrial Hygienists (ACGIH). However, beginning in the late 1980s, criticism about non-scientific consid...

File List 97 Notation 98 References 100 Index 103 2 Preface This work is about the numerical evaluation of the expression f (A)b, where A ∈ C N ×N is an arbitrary square matrix, b ∈ C N is a vector and f is a suitable matrix function. This task is of very high importance in all applied sciences since it is a generalization of the following problems, to name just a few: • Solve the linear system...

On the convergence properties of the orthogonal similarity transformations to tridiagonal and semiseparable (plus diagonal) form. Abstract In this paper, we will compare the convergence properties of three basic reduction methods, by placing them in a general framework. It covers the reduction to tridiagonal, semiseparable and semiseparable plus diagonal form. These reductions are often used as...

We are concerned with eigenvalue problems for definite and indefinite symmetric matrix pencils. First, Rayleigh-Ritz methods are formulated and, using Krylov subspaces, a convergence analysis is presented for definite pencils. Second, generalized symmetric Lanczos algorithms are introduced as a special Rayleigh-Ritz method. In particular, an a posteriori convergence criterion is demonstrated by...

We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in d...