Then we employ the direct technique based on the operational matrices for integration, differentiation, and the product of the orthonormal polynomials together with the Ritz–Galerkin technique to reduce the main problem to the solution of a system of nonlinear algebraic equations. Numerical simulations...
Differential problems Rational approximation Functions of matrices and operatorsUse our pre-submission checklist Avoid common mistakes on your manuscript. Associated Content Part of a collection: CAM23 Sections Figures References Abstract Introduction The abstract nonlocal differential problem The matrix case...
This section introduces methods based on multiplying high powers of the matrix times a vector, which usually will turn into an eigenvector as the power is raised. 12.1.1 Power Iteration(也称作Power Method) The motivation behind Power Iteration is that multiplication by a matrix tends to move ...
We will derive and discuss the conditions under which a strip\(C_1\)is a characteristic strip in this section. These conditions will be called the characteristic conditions. We will impose conditions on\(C_1\), based on our starting Eq. (3), such that (at least one of the) second an...
Computational fluid dynamics modeling is based on the principles of fluid mechanics, utilizing numerical methods and algorithms to solve problems that involve fluid flows. Models can integrate chemical reactions—combustion processes—with fluid flows to provide a three-dimensional understanding of boiler pe...
Some problems are inherently ill-conditioned, which means that a small change in the data results in a large change in the solution. A simple example is the system of twolinear equationsrepresenting the intersection of twostraight linesthat are nearly parallel, such as ...
©2002PublishedbyElsevierScienceB.V.onbehalfofIMACS. MSC:Primary73T05,73C50,92A06;Secondary73P99,73C60,90C20,65N30 Keywords:Non-linearelasticity;Contactproblems;Variationalinequality;Finiteelementmethod;Wrist;Spine;Fracture; Biomechanics 1.Introduction Inthispaper,theanalysesofaweight-bearingwristandaChance...
Gradient-Based Optimization Most deep learning algorithms involve optimization of some sort. Optimization refers to the task of either minimizing or maximizing some function f ( x ) f(\boldsymbol{x}) f(x) by altering x \boldsymbol{x} x. We usually phrase most optimization problems in terms ...
Variational inequalitiesandnonlinear complementarity problems, which benefit from availability of effective methods to deal with systems of nonsmooth equations. Bilevel problems, based on the existence, as in Stackelberg’s games, of a hierarchy of two autonomous decision makers. The related optimization ...
Based on matrices QL and QR in Eq. (3.38), we can examine the property of the upwind CESE scheme at the limiting case of ν = 0. Recall that the a–α scheme becomes very diffusive when ν→ 0 and cannot guarantee that qnj → qnj−1 as Δt → 0 (but Δx is a finite ...