Weak Formulation

To express this in more formal mathematical terms, we cast a partial differential equation in variational form (or integral form or weak formulation) by multiplying by a suitable function, integrating over the domain where the equation is posed and applying Greens theorem. Thus, if we consider the reaction-diffusion equation... 

Various formulations of hnite element methods have been proposed. For an exhaustive account on hnite element methods, the reader is referred to Chen (2005), Donea (2003), Reddy (2005), etc. We present here one of the popular formulations known as the weak formulation of the governing differential equation that, instead of requiring the solution to be twice continuously differentiable, requires that the derivative of the solution be square integrable. We illustrate the weak formulation of the boundary-value problem. Equation (2.157). 

To derive the weak formulation, multiplying Equation (2.157) by a test function, (()(x) (any member of a family of suitably smooth functions), and integrating by parts over an arbitrary element (Xi,X2) we get... 

Selecting ( ) = / and substituting the approximate solution into the weak formulation, one obtains the local Galerkin finite element equation,... 

The Galerkin method is applied with linear finite elements. The weak formulation of Eqs. (la) and (lb) is obtained by taking simultaneously the products of the equations with appropriate test functions and integration by parts of the spatial derivatives. We use a Lagrangian interpolation of the approximate solutions C for the aqueous solute concentration C, and S for the sorbed phase concentration Si for every species ... 

A weak formulation is an equation derived from the original (strong) formulation of a physical problem by alleviating the strict differentiability requirements for the unknown function. This is often achieved by lowering the order of partial derivatives appearing in the equation, for instance, using integration by parts. 

The solution to a weak formulation associated with an ordinary or partial differential equation is called the weak (or generalized) solution. There are many different definitions of a weak solution, appropriate for different classes of equations. 

Weak solutions are important in engineering because a large number of differential equations that are used to describe real-world phenomena do not admit smooth enough solutions. Weak formulations provide a way to solve such problems. 

Implementation of the FEM requires integration of the differential equations once (by parts), to produce what is often referred to as the weak formulation (Sect. 2.23.2). In contrast to the FDM, the integral expression contains the variable and its first derivative, thus reducing the order of the differential by one with respect to the variable. Domain discretization uses a mesh of intercormectmg elements and the variable approximated using a chosen interpolation function. 

This is known as the weak formulation. Since the boundary condition for the traction (t = a n) is directiy incorporated in the weak formulation, the traction boundary condition is sometimes called the natural boundary condition. 

By integrating Equation 7 over the time interval / e (0, T] and by performing an integration by parts to the first term of the left hand side, yields the following weak formulation of Equation 1 Find... 

It is generally accepted that the derived weak statement of the problem and also its discretized form should possess the same conserved properties as the initial dynamical system. The celebrated theorem of Emmy Noether provides the basis for the systematic examination of the conserved properties of the weak formulation as well as of the discrete system. In the present paper we utilize this theorem in a weak sense, as it is more consistent with the whole setting. 

Sivaselvan and Reinhorn (2004) have shown that weak formulations analogous to equations (17) through (18) can be obtained for continua. The final formulation derived elsewhere (Sivaselvan and Reinhorn, 2004) is presented below. For a three dimensional continuum, the Lagrangian formulation is given by ... 

The evolution of the elastic-plastic structural state in time is provided a weak formulation using Hamilton s principle. It is shown that a certain class of structures called reciprocal structures has a mixed weak formulation in time involving Lagrangian and dissipation functions. The new form of the Lagrangian developed in this work involves... 

The basic idea of FEM is the consideration of the sotest function (x) This function must vanish at the... 

The weak formulation follows by applying the product rule and the divergence theorem... 

J. Majak, M. Pohlak, M. Eerme, and T. Lepikult, Weak formulation based haar wavelet method for solving differential equations, Applied Mathematics and Computation, vol. 211, no. 2, pp. 488-494, 2009. 

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