[en] Linear axisymmetric Stokes flow over a porous spherical particle is investigated. An exact analytical solution for the fluid velocity components and the pressure inside and outside the porous particle is obtained. The solution is generalized to include the cases of arbitrary three-dimensional linear shear flaw as well as translational-shear Stokes flow. As the permeability of the particle tends to zero, the solutions obtained go over into the corresponding solutions for an impermeable particle. The problem of translational Stokes flow around a spherical drop (in the limit a gas bubble or an impermeable sphere) was considered. A solution of the problem of translational Stokes flow over a porous spherical particle was given. Linear shear-strain Stokes flow over a porous spherical drop was investigated

[en] The existing modifications of the method of functional separation of variables are overviewed. A new more general approach is proposed to the construction of exact solutions of nonlinear equations of mathematical physics, which is based on an implicit transformation and the splitting principle. The effectiveness of the approach is illustrated by nonlinear diffusion equations that contain reaction and convective terms as well as variable coefficients. A substantial number of new exact solutions with functional separation of variables and generalized traveling wave solutions are described

[en] We describe a new method for the construction of exact solutions of nonlinear reaction-diffusion equations with delay. All equations under consideration contain one or two arbitrary functions of one variable. A number of new exact solutions with the generalized separation of variables have been described and some more complex solutions have been obtained in the form of linear combinations of solutions with generalized separation of variables and traveling-wave solutions. The presented solutions contain free parameters (the number of such parameters can be arbitrary in some cases) and can be used to solve some problems and to test approximate analytical and numerical methods for solving similar and more complex nonlinear differential difference equations

[en] New classes of exact solutions of nonstationary equations for heat and mass transfer with a nonlinear sources are described. Different dependences of the transfer factors on temperature (concentration) are considered. Main attention is paid to general equations containing arbitrary functions. The method of functional separation of variables is used for deriving the exact solutions

[en] Various cases where the construction of exact solutions of nonlinear equations of mathematical physics leads to overdetermined systems of ordinary differential equations with parameters that are not included in the original partial differential equation are described. An appropriate nonclassical problem for ordinary differential equations with parameters is formulated and the new concept of the conditional cost of the exact solution is introduced

[en] Authors present periodic and antiperiodic solutions, composite solutions resulting from a nonlinear superposition of generalized separable and traveling wave solutions, and others. Some results are extended to nonlinear delay reaction-diffusion equations with time-varying delay

[en] We consider nonlinear delay reaction-diffusion equations. We present a number of exact traveling wave solutions, which can be represented in terms of elementary functions. We consider equations with quadratic, power, exponential, and logarithmic nonlinearities, as well as more complex equations with the kinetic function depending on one or two arbitrary functions of a single argument. All of the solutions obtained involve several free parameters and can be used to test approximate analytical and numerical methods for solving similar and more complex nonlinear differential-difference equations

[en] The asymmetric and symmetric decomposition methods for various classes of three-dimensional linear (and model nonlinear) systems of equations arising in the theory of elasticity, thermoelasticity, and thermoviscoelasticity, in the mechanics of viscous and viscoelastic incompressible and compressible barotropic gases have been described. It has been shown that the solution of the considered system of four three-dimensional stationary and nonstationary equations in the absence of body forces can be expressed in terms of solutions of three independent equations

[en] The unsteady Navier-Stokes equations with two space variables have been studied in the paper. The results obtained can be used to solve certain model problems and make it possible to effectively estimate the domain of applicability and accuracy of numerical, asymptotic, and approximate analytical methods in hydrodynamics

[en] We consider one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients. We describe a few new methods for constructing exact solutions of such equations; these methods are based on using invariant subspaces for the corresponding nonlinear differential operators. A number of new exact generalized and functional separable solutions have been obtained. All of the equations and solutions involve several free parameters. The exact solutions obtained can be used to test approximate analytical and numerical methods for solving nonlinear delay reaction-diffusion equations