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For a mixed elliptic-hyperbolic type equation in a mixed domain, when the elliptic part is a sector of the circle, and the hyperbolic part consists of two characteristic triangles, the Bitsadze–Samarskii type nonlocal problem is investigated. A feature of such problems is that the boundary conditions in the hyperbolic parts of the boundary are determined by a first-order differential operator and pointwise link the values of the partial derivatives of the desired solution on the characteristics with the partial derivatives on arbitrary monotone curves lying inside the characteristic triangles of the equation. To solve the problem, the methods of the theory of partial differential equations and the theory of singular integral equations as well as the methods of energy integrals and complex analysis were used, with the help of which the existence and uniqueness theorem for the solution of the investigated problem is proved. Also, a method of reducing the investigated problem to an equivalent system of singular integral equations is shown, also, a method for solving this system is proposed, which allows to obtain a solution to the problem in explicit form.

The Cauchy problem in Rd,d≥1, for a non-local in time p-Laplacian equations is considered. The nonexistence of nontrivial global weak solutions by using the test function method is obtained.

A new strategy has been proposed for the synthesis of 9-arylhexahydroacridine-1,8-diones by three-component condensation of aromatic aldehydes with 5,5-dimethylcyclohexane-1,3-dione and ammonium acetate in ethanol, using nontoxic, available, and inexpensive phosphate fertilizers [monoammonium phosphate (MAP), diammonium phosphate (DAP), and triple super phosphate (TSP)] as heterogeneous catalysts. Notable advantages of the proposed method include operational simplicity, low cost of the reactants, favorable environmental profile, good yield, short reaction time, recovery and reuse of the catalyst, and valorization of Moroccan natural phosphate.

A Timoshenko type beam subject to a viscoelastic damping in the rotational displacement component is considered. Taking into account a neutral type delay, we prove a fast stability result despite the previously observed destabilizing effect due to delays in such systems. The proof relies on the introduction of nine different functionals with which we modify the energy of the system. These functionals are carefully selected and adapted to cope with both the viscoelasticity and the neutral delay.

The novelty of this work is to propose a new process consisting to the modification of triple superphosphate (TSP) by sodium nitrate (NaNO3) for the preparation of a heterogeneous catalytic support Na2Ca(HPO4)2 (calcium hydrogen disodium phosphate). This product has been characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM) and infrared spectroscopy (IR). The catalytic activity of Na2Ca(HPO4)2 was examined in Knoevenagel condensation between 4-chlorobenzaldehyde and malononit rile as a model reaction. The optimization of the reaction operating conditions was carried out based on a study of the effect of various parameters, namely reaction time, the catalyst charge, the nature, and volume of solvent on the reaction yield. From these considerations, this catalyst proves its efficiency since it allows to obtain the product with excellent yields in a short reaction time. Moreover, it has been reused several times without any significant loss of its activity. The process developed herein can be considered as an ecological and economical method because the catalyst used is non-toxic, cost-effective and easy to prepare.