Analytical study of solitons to nonlinear time fractional parabolic equations

The some of the well-known nonlinear time fractional parabolic partial differential equations is studied in this paper. The fractional complex transform and the first integral method are employed to construct one-soliton solutions of these equations. The power of this manageable method is confirmed. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions.     

The existence of periodic solutions of impulsive differential equations of mixed type

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Burnett simulations of gas flow in microchannels

The Burnett equations with slip boundary conditions are used to model the gas flow in microchannels in transition flow regime. As the Navier-Stokes equations are not appropriate to model the gas flow in this regime, the higher-order Burnett equations are adopted in the present study. In earlier studies, convergent solutions of the Burnett equations of microPoiseuille flow could only be obtained when Knudsen number is less than 0.2. By using a relaxation method on the boundary values, convergent solutions of the Burnett equations can be obtained even when Knudsen number reaches 0.4. The solutions of Burnett equations agree very well with experimental data and direct simulation Monte Carlo (DSMC) results. The pressure distributions and velocity profiles are then discussed in detail.     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

Exact solutions of the generalized Navier– Stokes equations for benchmarking

The generalized Navier– Stokes equations for incompressible viscous flows through isotropic granular porous medium are studied. Some analytical classic solutions of the Navier– Stokes equations are generalized to the case of the considered equations. Obtained solutions of generalized equations reduce to classic ones as porosity effect disappears. Average velocity of generalized solutions is calculated and evaluated in two limiting regimes of flow. In the shallow conduit, the generalized flow rate approximates the free (without porous medium) flow rate and in the case of removed boundaries this approaches Darcy's law. The use of the derived exact solutions for benchmarking purposes is described. Copyright © 2002 John Wiley & Sons, Ltd.     

A class of similarity solutions for radial motions of compressible hyperelastic spheres and cylinders

A class of similarity solutions is obtained for radial motions of spherical and cylindrical bodies made of a certain type of compressible hyperelastic materials. The equations satisfied by the infinitesimal generators of the symmetry group of the unified governing first order field equations for spheres and cylinders are found. It is shown that these equations admit a special class of solutions which generate a five-parameter group of transformations. The form of the strain energy function corresponding to the resulting symmetry group is evaluated. The similarity variable is determined and ordinary differential equations satisfied by similarity solutions are obtained. Numerical solutions are given for a Ko material which falls into the class of admissible materials.     

Boundary integral equations of unique solutions in elasticity

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Conservation integrals and determination of HRR singularity fields

The angular distribution functions of HRR singularity fields are analyzed via conservation integrals. Two functional equations are proved for the angular distribution functions and can be used for their solutions. The detailed forms of the functional equations and the final governing equations for solutions are given for the cases of plane strain and plane stress. Accurate numerical results are also given for some typical parameters and the equivalence of different governing equations is proved.     

Applications of the group of equations of motion of a polytropic gas

The machinery of Lie theory (groups and algebras) is applied to the system of equations governing the unsteady flow of a polytropic gas. The action on solutions of transformation groups which leave the equations invariant is considered. Using the invariants of the transformation groups, various symmetry reductions are achieved in both the steady state and the unsteady cases. These reduce the system of partial differential equations to systems of ordinary differential equations for which some closed-form solutions are obtained. It is then illustrated how each solution in the steady case gives rise to time-dependent solutions.     

A unified compatibility method for exact solutions of non-linear flow models of Newtonian and non-Newtonian fluids

Criteria are established for higher order ordinary differential equations to be compatible with lower order ordinary differential equations. Necessary and sufficient compatibility conditions are derived which can be used to construct exact solutions of higher order ordinary differential equations subject to lower order equations. We provide the connection to generalized groups through conditional symmetries. Using this approach of compatibility and generalized groups, new exact solutions of non-linear flow problems arising in the study of Newtonian and non-Newtonian fluids are derived. The ansatz approach for obtaining exact solutions for non-linear flow models of Newtonian and non-Newtonian fluids is unified with the application of the compatibility and generalized group criteria.     

Representation of solutions to equations of hyperbolic type

The general solutions to hyperbolic equations of fourth and sixth order are obtained using Vekua’s method for the representation of the general solutions to elliptic equations of order 2n with the aid of n analytic functions. It is assumed that the right-hand sides of the hyperbolic equations can be expanded in time series of sines. The systems of equations of various approximations for a prismatic thin body in terms of moments with respect to the system of Legendre polynomials can be reduced to these equations and to some hyperbolic-type equations of higher order.     

Bending of thin circular rings

The basic equations for the bending of circular rings are deduced from a set of accurate equations for circular cylindrical shells. The advantages in using these differential equations as compared with the customary energy method are shown through examples. It turns out that solutions of these equations can be as easily obtained as solutions of the well-known differential equation for straight beams. It is also shown that the center line of the ring is essentially inextensible, which is assumed ab initio in the classical ring theory.     

The Method of Abel-Type Integral Equations for Identification of Nonlinear Systems of Structural Seismodynamics

The problem on determination of the nonlinear dissipative and elastic characteristics of some vibrating systems that are encountered in structural seismodynamics is considered. Systems of integral Volterra equations of the first kind (Abel-type equations) are constructed on the basis of approximate analytical solutions to problems on the forced vibration of quasilinear vibrating systems. Such equations relate the nonlinear stiffness and dissipation characteristics with the characteristics of motion, which can be obtained experimentally. The solutions of the integral equations derived are represented in the form of quadrature Stieltjes-integral formulas     

Series solutions of annular axisymmetric stagnation flow and heat transfer on moving cylinder

The steady, laminar, incompressible flow and heat transfer of a viscous fluid between two circular cylinders for two different types of thermal boundary conditions are investigated. The governing Navier-Stokes and thermal equations of the flow are reduced to a nonlinear system of ordinary differential equations. The equations are solved analyt- ically using the homotopy analysis method （HAM）. Convergence of the HAM solutions is discussed in detail. These solutions are then compared with recently obtained numericM and perturbative solutions. Plots of the velocity and temperature profiles are provided for various values of the relevant parameters.     

Application of group analysis to stochastic equations of fluid dynamics

Group analysis is used to study stochastic equations of fluid dynamics. Determining equations for admitted Lie groups of transformation involving independent and dependent variables and Wiener processes are obtained. It is shown that, as in the case of deterministic differential equations, admitted groups make it possible to reduce invariant solutions of stochastic differential equations to solutions with a smaller number of independent variables.     

Energy version of the kinetic equations of isothermal creep and long-term strength

Energy-type kinetic equations of inelastic rheological deformation are proposed in which the elastic, plastic, and creep strains are the additive components of the total strain, and the damage parameter is taken into account. A model of viscoelastic material with a creep kernel of exponential type is considered. The Lyapunov stability of solutions under constant stress is studied. The stability range of the solutions of the differential equations of the mathematical model corresponding to asymptotically bounded creep is established. It is shown that the instability range of the solutions corresponds to the onset of the third stage of creep. The relationship is determined between the Lyapunov stability of the solutions and the stability of the computational algorithm for the numerical solution of the system of equations. The proposed model is experimentally verified. It is shown that the calculated and experimental data are in good agreement.     

Some properties of three-dimensional Biltrami flows

The investigation of Beltrami flows is important for the research on the mechanism of turbulent structure. In this paper the general solutions of the Beltrami flows are given, which depend explicitly on the solutions of three independent Helmholtz equations with scalar unknowns. Velocity fields of Beltrami flows can then be obtained explicitly after the application of some curl operations on the solutions of Helmholtz equations. On the basis of the exact solutions of Euler equations given above, we obtain one kind of exact solutions of non-steady Navier-Stokes equations which are also the Beltrami flows. Some interesting examples of Beltrami flows other than “ABC flows”, “Kolmogolov flows”, “Rayleigh-Bernard flows”, “Q-flows” are given. The detailed analytic results of these examples will be published in the near future.     

Exact solutions of the kinetic-moment equations of a mixture of monatomic gases

An extension is given of the class of exact solutions of the kinetic-moment equations for a monatomic gas in the absence of external forces [1] to the case of a mixture of monatomic Maxwellian gases with account for external forces. Very simple solutions of this class are obtained which are examples of the normal solutions of the Boltzmann equations in the Chapman-Enskog sense [2]. Conclusions are summarized concerning the applicability of the various methods of solving the Boltzmann equations and their properties, obtained on the basis of an analysis of the indicated exact solutions.The author wishes to thank M. N. Kogan and A. A. Nikol'skii for their interest in the study.     

Fundamental Solutions for Zero-Coupon Bond Pricing Models

The transformation approach is employed to reduce one-factor bond-pricing equations into the heat equation for which the fundamental solution is wellknown. These transformations are subsequently used to construct the fundamental solutions of two zero-coupon bond-pricing equations. The closed-form analytical solutions of the Cauchy initial value problems of the two bond-pricing model equations are then obtained.     

Method of constructing estimates of the solution of equations of filtration of an aerated liquid

The exact solutions of the nonlinear equations of filtration of an aerated liquid have been obtained in [1–3]. In [4] the system of equations of an aerated liquid have been reduced to the heat-conduction equation under certain assumptions. An approximate method of computing the nonsteady flow of an aerated liquid is given in [5], where the real flow pattern is replaced by a computational scheme of successive change of stationary states. In [6] the same problem is solved by the method of averaging. In the present article estimates of the solution of the equations for nonstationary filtration of an aerated liquid in one-dimensional layer are constructed under certain conditions imposed on the desired functions. These estimates can be used as approximate solutions with known error or for the verification of the accuracy of different approximate methods. We note that the use of comparison theorem for the estimate of solutions of equations of nonlinear filtration is discussed in [7–9]. The methods of constructing estimates of solutions of various problems of heat conduction are given in [10, 11]