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More and more researchers have found that fractional differential equations play an important role in mathematical physics and can be used to describe some physical phenomenon more exactly than integral differential equations.
Moreover, for any such fixed (N > N_r), the number of times that H attains a value (w in S_N) with (|w|exactly the integral (frac{1}{2 pi i} int frac{H' z)}{H z -w} text{d}z) around tH z -w}dary edges of (S_N).
During the whole connection time ΔT i,S, vehicle i ((iin mathbb {N})) can not exactly download integral pieces since it is out of the communication range of S, resulting in the failed connection L i between i and S while the nth piece is transferring.
In general, it is more difficult to compute the previous integral exactly, so we use the Legendre-Gauss quadrature formula for this purpose.
This type of quadrature methods exhibits an optimality property with respect to the number of required nodes to calculate the integral exactly for all multivariate polynomials up to a total degree R. For example, it holds that (K=2Q) for (R=3) and (K=2Q^{2}+1) for (R=5).
In the inset in Fig. 7A we plot the results of the same simulations after manually scaling the elongation speed so that its integral exactly matches the change in muscle length.
By using these natural coordinates as the finite element basis functions we can integrate the corresponding integrals exactly thereby achieving a high level of accuracy and efficiency for modeling physical problems on the sphere.
Thus, the obtained differentiator is exactly given by an integral formula which depends on a set of design parameters.
In this paper, we develop high order interpolants, from any basis that constitutes a partition of unity, which satisfy these integral relations exactly.
