Moreover, (i) ( λ, u ) ∈ C 1 + with ∥u∥∞ = ξ2j- 1 for some j ∈ ℕ* implies that λ ≥ 2; (ii) ( λ, u ) ∈ C 1 + with ∥u∥∞ = ξ2j for some j ∈ ℕ* implies that λ ≤ 1 2. Actually, such continua C 1 + can be obtained as upper limits in the sense of Kura-towski of sequence of solution continua from associated continuous problems.
As an example, the equation x ′ = f ( x ) = x ⋅ χ ( 0, + ∞ ) ( x ) has an infinity of solutions (a continuum of solutions [8], p.15]) x T ( t ) = ( t − T ) 2 4 ⋅ χ ( T, + ∞ ) ( t ) defined on the nonnegative half-line which start from x ( 0 ) = 0. Here, by χ we denote the characteristic function of a Lebesgue-measurable set.
We present a set of restrictions on the fractional differential equation x ( t ) = g ( x ( t ) ), t ≥ 0, where α ∈ ( 0, 1 ) and g ( 0 ) = 0, that leads to the existence of an infinity of solutions (a continuum of solutions) starting from x ( 0 ) = 0.
Previous research has shown that the application of residual Monte Carlo methods to the solution of continuum equations, such as the radiation transport equation, is problematic for all but the simplest of cases.
For simulations of tissue mechanics, we have used a nonlinear Finite Element Method which provides an approximate solution of continuum elasticity problems on domains with complex geometry (Zienkiewicz et al., 2005).
The velocity profiles converge to the classical benchmark solutions in continuum regime with the mesh size being much larger than the local particle mean free path.
Good agreement on structural eigenfrequencies between the effective continuum solutions and the exact solutions shows that the proposed continuum model can accurately simulate the dynamic behavior of the cellular solids.
Stress equilibrium and traction free boundary conditions constrain the solution at the continuum scale.
Checking against full atom molecular mechanics calculations show that the continuum solution has high accuracy.
It is difficult to integrate the analytical solution of the continuum model because its coefficients are functions of spatial coordinates.
