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We consider the mean value theorem for the wave equation in the Euclidean space to be a starting point to produce other mean value theorems for the wave equations on a sphere and in a space. We use transformations connecting the wave equation in the Euclidean space and the wave equation in non-Euclidean spaces. Also, we prove two-point mean value formulas for elliptic equations in the Lobachevskii space and on a sphere.

In this paper we consider optimal recovery of the B-elliptic operator with its powers of a smooth function from the Fourier–Bessel transform given on some convex subset. We construct a series of optimal recovery methods. It turns out that information about the Fourier–Bessel transform outside of some ball with the center at zero is superfluous, optimal methods do not use it. The methods themselves differ in the way they process available information.

5-Acetyl-4-hydroxy-2H-1,3-thiazine-2,6(3H)-dione reacted with cyclohexyl and benzyl (4-aminophenyl)carbamates in boiling ethanol in the presence of a catalytic amount of glacial acetic acid to give the corresponding Schiff bases, cyclohexyl and benzyl (4-{[(E)-1-(4-hydroxy-2,6-dioxo-3,6-dihydro-2H-1,3-thiazin-5-yl)ethylidene]amino}phenyl)carbamates. Heating of the latter in dimethylformamide for 2 h was accompanied by evolution of COS with the formation of cyclohexyl and benzyl {4-[6-methyl-2,4-dioxo-1,2,3,4-tetrahydropyrimidin-1-yl]phenyl}carbamates. The condensation of the title compound with benzene-1,2-diamine in propan-2-ol, followed by addition of 5-arylfuran-2-carbaldehyde and trifluoroacetic acid afforded 5-(2-{5-[4-bromo(nitro)phenyl]furan-2-yl}-2,3-dihydro-1H-1,5-benzodiazepin-4-yl)-4-hydroxy-2H-1,3-thiazine-2,6(3H)-diones. 5-((E)-3-{5-[4-Bromo(nitro)phenyl]furan-2-yl}prop-2-enoyl)-4-hydroxy-2H-1,3-thiazine-2,6(3H)-diones were synthesized by condensation of 5-acetyl-4-hydroxy-2H-1,3-thiazine-2,6(3H)-dione with 5-[4-bromo(nitro)phenyl]furan-2-carbaldehydes in chloroform in the presence of catalytic amounts of pyridine and piperidine.

The reaction of methyl N-{4-[2-(2-oxo-2,3-dihydro-1H-indol-3-ylidene)acetyl]phenyl}carbamate with tosylmethyl isocyanide in tetrahydrofuran in the presence of potassium tert-butoxide led to the formation of methyl N-{4-[(4-oxo-4,5-dihydro-3H-pyrrolo[2,3-c]quinolin-1-yl)carbonyl]phenyl}carbamate. Three-component condensation of methyl N-{4-[(E)-3-R-prop-2-enoyl]phenyl}carbamates and methyl N-{4-[2-(2-oxo-2,3-dihydro-1H-indol-3-ylidene)acetyl]phenyl}carbamate with sarcosine and paraformaldehyde in boiling toluene with simultaneous removal of water by azeotropic distillation afforded methyl N-(4-{4-[4-R-1-methyl-pyrrolidin-3-yl]carbonyl}phenyl)carbamates and methyl N-[4-(1′-methyl-2-oxo-1,2-dihydrospiro[indole-3,3′-pyrrolidin]-4′-ylcarbonyl)phenyl]carbamate, respectively.

We prove a sufficient condition for the stability of a stationary solution to a system of nonlinear partial differential equations of the diffusion model describing the growth of malignant tumors. We also numerically simulate stable and unstable scenarios involving the interaction between tumor and immune cells.

This paper describes an accompanying distributions technique that allows to obtain mean value formulas for linear homogeneous partial differential equations. One of these formulas can be interpreted as a generalization of the Asgeirsson principle for the string vibration equation into the case of an arbitrary natural order. In addition, this mean value formula is an exact difference scheme for a two-dimensional linear homogeneous equation with a symbol factorized up to linear factors.

The brief presentation of the method of fast expansions is given to solve nonlinear differential equations. Application rules of the operator of fast expansions are specified for solving differential equations. According to the method of fast expansions, an unknown function can be represented as the sum of the boundary function and Fourier series sines and cosines for one variable. The special construction of the boundary functions leads to reasonably fast convergence of the Fourier series, so that for engineering calculations, it is sufficient to consider only the first three members. The method is applicable both to linear and nonlinear integro-differential systems. By means of applying the method of fast expansions to nonlinear Navier-Stokes equations the problem is reduced to a closed system of ordinary differential equations, which solution doesn't represent special difficulties. We can reapply the method of fast expansions to the resulting system of differential equations and reduce the original problem to a system of algebraic equations. If the problem is n -dimensional, then after n -fold application of the method of fast expansions the problem will be reduced to a closed algebraic system. Finally, we obtain an analytic-form solution of complicated boundary value problem in partial derivatives. The flow of an incompressible viscous fluid of Navier–Stokes is considered in a curvilinear pipe. The problem is reduced to solving a closed system of ordinary differential equations with boundary conditions by the method of fast expansions. The article considers peculiarities of finding the coefficients of boundary functions and Fourier coefficients for the zero-order and first-order operators of fast expansions. Obtaining the analytic-form solution is of great interest, because it allows to analyze and to investigate the influence of various factors on the properties of the viscous fluid in specific cases.

In this paper, we describe hypothetical possibilities for applying nonlinear dynamics methods to linguistic research. We suggest to refer to a large text corpus (meta-book of the writer, national language, language of the branch of knowledge) as a fractal object and to measure its fractal dimension (by Hausdorff). We consider the question of practical calculation of fractal dimension, as well as the question of motivation for applying the fractal concept to the language. In addition, we propose to apply some methods of the qualitative theory of differential equations to modelling the growth of the natural language vocabulary.

We note from a general point of view that adding diffusion terms to ordinary differential equations, for example, to logistic ones, can in some cases improve sufficient conditions for the stability of a stationary solution. We give examples of models in which the addition of diffusion terms to ordinary differential equations changes the stability conditions of a stationary solution.

To study properties of solutions to differential equations by using the Fourier transform, we generalize the notion of accompanying distribution of a differential operator to the case of the Fourier-Bessel transform. Bibliography: 2 titles.

We consider the macroeconomic model of T. Puu, describing the fluctuations of gross income in a given region. With a special combination of savings and investment rates, income deviations will take place only for a finite period of time, after which the income will return to a stationary state. This e˙ect is known in mathematical physics as the Huygens principle. We investigated this model using statistical methods of data analysis. The obtained results suggest that the Huygens effect hypothesis in certain historical periods of the Russian economy is plausible. We have also considered the structure of the set of stationary zeros for nontrivial solutions to a stationary equation.