According to the literature [31], there is lim_{eta to 0} chi^{varpi +theta -1}E_{vartheta,varpi+theta } bigl({eta} { chi ^{vartheta}} bigr)=frac{chi^{varpi +theta -1}}{Gamma ( {varpi+theta } )}, which is the integral kernel of the Liouville-Caputo FDO.
The point is that the integral kernel in (15.16) is Hilbert Schmidt for (text {Im}(k) ge 0) if V is Rollnik.
Let K j, k, α be the integral operator with the kernel K j, k, α ( x, y, t, τ ) = | x ′ | β − 2 + 2 k + | α | | y ′ | − β ∂ x α ∂ t k K j ( x, y, t, τ ), where | α | ≤ 2. As was shown in the proof of Lemma 2.1, this operator is bounded in L p ( D × R ).
In other words, (phantom {dot {i}!}X_{2_{mathsf {Orth}}}in mathcal {T}_{22setminus 12}) is zero everywhere the integral kernel H 12 τ,t) is nonzero.
This can be seen by analyzing the integral kernel for (G_0 x,y kappa ^2)) which is a modified Bessel function of the second kind (see [612, discussion following (6.9.35)]) which is how Jensen [286] does it or by looking at (15.19).
end{aligned}To calculate this integral, we recall (16) and stress again that the diagonal of the integral kernel (Pi _{y_3,k}(x_1,x_2,y_1,y_2)) is is known to be the constant (frac{|y_3|}{2pi }).
We note that, formally, the identity (2.8) can be rewritten as relation (1.6) where A is the "integral operator" with kernel v b v.
This norm can be calculated explicitly by using the integral kernel of the operator (P_{Omega ^mathsf c }Pi _{B,k}).
(1) Here, (x t)) is the population density at t, a, b, c are positive constants, a expresses the innate capacity of increase, b, c express the density restriction coefficient, the integral kernel k is continuous and meets (int_{0}^{infty} k(s), ds = 1) and (int_{0}^{infty} sk(s),ds < + infty).
These FDOs involving the MLFs in the integral kernel have been applied to model many physical phenomena, such as the anomalous relaxation, heat-transfer problems, viscoelastic problems, Euler-Lagrange equation and the boundary value problem, extensively (see [18, 28 36]).
