Inverse Problems of a Fractional Differential Equation with Bessel Operator

Inverse initial and inverse source problems of a time-fractional differential equation with Bessel operator are considered. Results on existence and uniqueness of solutions to these problems are presented. The solution method is based on series expansions using a set of Bessel functions of order zero. Convergence of the obtained series solutions is also discussedComment: 11 pages, No figure

(2E)-4-tert-Butyl-2-(4-meth­oxy­benzyl­idene)-3,4-dihydro­naphthalen-1(2H)-one

In the title compound C22H24O2, the exocyclic C=C double bond is in an E configuration and the tert-butyl group is in an axial position on the cyclo­hexa­none ring. The cyclo­hexa­none ring in the dihydro­naphthalene fused-ring system adopts a half-chair conformation in both independent two mol­ecules in the asymetric unit. The benzene ring system is oriented angles of 43.97 (12) and 39.24 (12)° with respect to the naphthyl ring system in the two independent mol­ecules. In the crystal, mol­ecules are linked via C—H⋯O hydrogen bonds and C—H⋯π inter­actions

(2E)-2-Benzyl­idene-4-ethyl-3,4-dihydro­naphthalen-1(2H)-one

In the title compound, C19H18O, the exocyclic C=C double bond has an E configuration. The ethyl substituent on the cyclo­hexa­none ring is in an axial position. The cyclo­hexa­none ring adopts a half-chair conformation, presumably due to conjugation in the benzene ring

Diethyl 2-{(dibenzyl­amino)[4-(trifluoro­meth­yl)phen­yl]meth­yl}malonate

The asymmetric unit of the title compound, C29H30F3NO4, contains two independent mol­ecules. In each independent mol­ecule, one of two terminal ethyl groups is disordered over two conformations: the occupancies of major components were fixed at 0.53 and 0.64 in the two mol­ecules. In the crystal structure, weak inter­molecular C—H⋯O hydrogen bonds link mol­ecules into chains propagating along [10]

Ethyl 2-amino-4-phenyl-4H-1-benzo­thieno[3,2-b]pyran-3-carboxyl­ate

The title heterocyclic compound, C20H17NO3S, was synthesized by condensation of ethyl cyano­acetate with (Z)-2-benzyl­idenebenzo[b]thio­phen-3(2H)-one in the presence of a basic catalyst in ethanol. The phenyl and ester groups make dihedral angles of 77.67 (6) and 8.52 (6)°, respectively, with the benzothienopyran ring system [maximum r.m.s. deviation = 0.1177 (13) Å]. In the crystal, centrosymmetric dimers are formed through pairs of N—H⋯O hydrogen bonds between the amine and ester groups. Inter­molecular C—H⋯N hydrogen bonds and C—H⋯π inter­actions involving the thio­phene ring are also observed

3,4-Bis(4-nitro­phen­yl)-1,2,5-oxadiazole 2-oxide

The title compound, C14H8N4O6, a new 1,2,5-oxadiazole N-oxide derivative, was formed by dimerization of 4-nitro­benz­al­de­hyde oxime. The compound crystallizes with two independent mol­ecules per asymmetric unit. The N-oxide O atom is disordered over two sites in each mol­ecule; site occupancy factors are 0.57/0.43 and 0.5/0.5. The mean planes through the two benzene rings are inclined to the planar 1,2,3-oxadiazole ring by 25.03 (11) and 41.64 (11)° in one mol­ecule, and 22.58 (11) and 42.66 (11)° in the other mol­ecule, the smaller angle being for the ring on the oxide side of the oxadiazole ring in each case. In the crystal structure, the individual mol­ecules form centrosymmetric dimers linked via C—H⋯O hydrogen bonds. The dimers of one mol­ecule are then linked to those of the other mol­ecule via C—H⋯O hydrogen bonds, forming a three-dimensional network

4-tert-Butyl-4′-(4-meth­oxy­phen­yl)-3′-(4-methyl­phen­yl)-1,2,3,4-tetra­hydro­spiro­[naphthalene-2,5′(4′H)-1,2-oxazol]-1-one

In the title compound, C30H31NO3, the tolyl ring is almost coplanar with the isoxazole ring [dihedral angle = 12.51 (7)°], whereas the meth­oxy­phenyl ring is almost perpendicular to the isoxazole ring [dihedral angle = 89.77 (5)°]. In the crystal, mol­ecules are connected through C—H⋯O hydrogen bonds, forming chains running along the a axis

Optimal boundary control of distributed systems involving dynamic boundary conditions

In this paper we consider Lagrange type control problem for systems involving dynamic boundary conditions that is, with boundary operators containing time derivatives. Assuming the existence of optimal controls, B-evolutions theory is used to present necessary conditions of optimality. The result is illustrated by an example from heat transfer problem and also an algorithm for computing optimal controls is presented