Boundary Value Problems weak solution

Examining these solutions, we see that the temperature becomes nonuniform and the velocity profile nonlinear. These results are to be expected from a qualitative point of view. In a sense, the most important conclusion is that the regular asymptotic expansion in terms of the small parameter Br provides a method to obtain an approximate solution of the highly nonlinear boundary-value problem to evaluate the influence of weak dissipation, which can clearly be applied to other problems. 

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). 

In this study we have discussed the definition of the boundary value problems in linear, isotropic, homogeneous, nonlocal elasticity proposed by Eringen. The discussion is based on the concept of weak solution which is widely used in the theory of partial differential equations. An Inequality which is analogous to Korn s First Inequality has been given and the results on the existence and uniqueness of the boundary value problems in nonlocal elasticity are summarized. Finally an alternative definition of the boundary value problems in nonlocal elasticity is criticized. 

Weak formulation of the problem of elasticity. With a view to using the finite element method to obtain solutions to the problem for elastic bodies, it is necessary to convert the boundary value problem (1.7) to what is known as a weak formulation. To this end, let w be an arbitrary displacement which satisfies the homogeneous essential boundary condition, i.e. 

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