Sentence examples for has singularities from inspiring English sources

One can see from Equation 5 that the effective dielectric function has singularities at ω = 0 and ω = ΩTO.

The powerful side of our method is that it can easily compute solutions even if the equation has singularities.

The behaviour of Vandermonde matrices is different when the density of ω has singularities and depends on the density growth rates near the singularities points.

(3.35) As in the one-population model, the Perron eigenvalue has singularities, reflecting the possible breakdown of the Perron–Frobenius theorem in the thermodynamic limit.

The Liouville-Caputo representation has the disadvantage that its kernel has singularities, meaning this kernel includes memory effects and therefore this definition cannot accurately describe the full effect of the memory.

Remark that we do not need to subtract the boundary value of functions on (|x|=eR) in the integrand on the left-hand side of (1.6) in spite of the fact that the integrand on the right-hand side has singularities on (|x|=R), (|x|=eR), and (|x|=e^{2}R).

The fractional differential equation with Riemann-Stieltjes integral conditions considered in Theorem 3.1 is singular, that is, (f t,u,v)) has singularity at (t=0) or (t=1) and (v=0), and (g t,v)) has singularity at (t=0) or (t=1) and (v=0), which generalizes and improves the known results for continuous functions in [16, 27 29].

In BVP (4.21), note that (a(t)) is singular at (t=0) and (t=1), that is to say, the nonlinearity has singularity at (t=0) and (t=1).

In particular, we consider that (f t,u,v)) has singularity at (t=0) or 1 and (v=0), (g t,v)) has singularity at (t=0) or 1 and (v=0).

In Example 4.1, the nonlinearity term f has singularity at (t=0) and (t=frac{1}{2}).

However, contrary to the Caputo-Fabrizio derivative in this paper, the Caputo-Katugampola fractional derivative has singularity for (t =tau ).