Boundary null-controllability of linear diffusion–reaction equations

This Note deals with the boundary null-controllability of linear diffusion–reaction equations in a 2D bounded domain. We transform the determination of the sought HUM boundary control into the minimization of a continuous and strictly convex functional. In the case of a rectangular domain where the diffusion tensor is represented by a diagonal matrix, we establish a procedure based on the inner product method that uses a complete orthonormal family of Sturm–Liouville's eigenfunctions to express explicitly the sought control.     

Synthesis and X-ray crystal structure of pyrrolo[1,2-a]benzimidazoles

Reaction of nitrilimines 1 with 2-cyanomethylbenzimidazole 2 gave the 3-arylazo-2-methylpyrrolo[1,2-a]benzimidazole 4a rather than the reported 2-arylazo-3-methylpyrrolo[1,2-a]benzimidazole 3a. The correct structure of the product was determined using X-ray crystal structure analysis. The similar reaction of nitrilimines with 2-aminobenzimidazole 5 gave the acyclic nucleophilic addition product 6.     

Existence of positive homoclinic solutions for damped differential equations

This paper is concerned with the existence of positive homoclinic solutions for the second-order differential equation

$$\begin{aligned} u^{\prime \prime }+cu^{\prime }-a(t)u+f(t,u)=0, \end{aligned}$$

where \(c\ge 0\) is a constant and the functions a and f are continuous and not necessarily periodic in t. Under other suitable assumptions on a and f, we obtain the existence of positive homoclinic solutions in both cases sub-quadratic and super-quadratic by using critical point theorems.

     

Reaction of nitrilimines with 2-substituted aza-heterocycles. Synthesis of pyrrolo[1,2-a]pyridine and pyrimido[2,1-d]1,2,3,5- tetrazine

The reaction of nitrilimine 6a with ethyl pyridine-2-acetate (7) gave the corresponding pyrrolo[1,2-a]pyridine 8, while the reaction of 6b containing an ester moiety afforded the acyclic adduct 9. The reaction of 6a with 2-aminopyrimidine (10) gave the novel unexpected pyrimido[2,1-d]1,2,3,5-tetrazine 11. Acyclic adducts 16 and 17 were obtained from the reaction of 6b with 2-cyanomethylbenzimidazole (14) and 2-aminobenzimidazole (15), respectively.     

Computational method for solving space fractional Fisher's nonlinear equation

This paper aims to present a general framework of the quadratic spline functions to develop a numerical method for solving the nonlinear space fractional Fisher's equation. Using Von Neumann method, the proposed method is shown to be conditionally stable. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm. The results reveal that the proposed approach is very effective, convenient, and quite accurate to such considered problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Synthesis of 1,2,4-Triazol-3-ylmethyl-, 1,3,4-Oxa-, and -Thiadiazol-2-ylmethyl-1H-[1,2,3]-triazolo[4,5-d]pyrimidinediones

1-Carbethoxymethyl-4,6-dimethyl-1H-[1,2,3]triazolo[4,5-d]pyrimidine-5,7(4H,6H)-dione was synthesized and treated with hydrazine hydrate to give the corresponding hydrazide. The latter hydrazide was treated either with phenylisothiocyanate or with carbon disulfide/alc. KOH to afford the corresponding thiosemicarbazide and oxadiazole derivatives. Alkylation of 2-mercapto-1,3,4-oxadiazole with dimethyl sulfate or ethyl chloroacetate gave the corresponding 2-methylthio-, and 2-ethylthioglycolate derivatives. Formation of 1,3,4-thiadiazole, 5-mercapto-1,2,4-triazole, and 1,3,4-oxadiazole were carried out by treating of the latter thiosemicarbazide with conc. H₂SO₄, NaOH/HCl, and HgO. Treating of 5-mercapto-1,2,4-triazole with ethyl chloroacetate afforded the thioglycolate ester. Hydrolysis of the latter with hydrazine hydrate afforded the hydrazide derivatives. Condensation of these hydrazides with monosaccharide aldoses gave the corresponding sugar hydrazones. The novel compounds were tested for antiviral activity against hepatitis B virus and showed moderate activities.