The matrix a above is AH domain differential operator
, which can be considered as the similar time-domain operator d/dt or frequency-domain operator j[omega] and respective identity matrix I is for 1.
Moreover, the second-order differential equation also can be considered as a product of two first-order differential operators
and the spinor wave function related to the differential equation that is expressed in terms of Rodrigues representations related to the orthogonal polynomials.
where [alpha],[beta], [gamma] [member of] (0,1], [R.sup.i], [N.sup.i], i = 1,2,3, denote linear differential operators
and nonlinear differential operators
, respectively, and [g.sup.i] (x, t) are the source terms.
Let [u.sub.n](x) = [w.sub.n]([x.sub.1])[u.sub.1]([x.sub.2]), where [u.sub.1] is the first eigenfunction of the one-dimensional second-order differential operator
on [L.sup.2]((0, a)) which is identically equal to 1.
We now try to decompose the differential operator
[L.sup.2.sub.[alpha]] into differential operators
[L.sub.A(r)] and [L.sub.B(r)] of first order such that
In this section, we give some theorems that estimate resolvent of an differential operator
on a Hilbert space.
Moreover, let we choose [mathematical expression not reproducible] and A to be differential operator
with generalized Wentzell-Robin boundary condition defined by
Using convolution, we here define the differential operator
[D.sup.(n]) analogue of the operator defined in (8), n [member of] [N.sub.0], by
The external differential operator
in (5) is [[partial derivative].sub.i].
This is an evolutionary algorithm specializing in the mutation process through a differential operator
. After performing the mutation, a cross or recombination process also takes place.
We considered the general form of inhomogeneous nonlinear partial differential equations with initial conditions as given below Equations the remaining linear operator represents a general nonlinear differential operator
and was source term.