Класс точных решений уравнений Обербека-Буссинеска с учетом диссипативной функции Рэлея

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method

The work presents the analysis of the free boundary value problem related to the one-dimensional invasion model of new species in biofilm reactors. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of non-linear hyperbolic partial differential equations governing the microbial species growth and a system of semi-linear elliptic partial differential equations describing the substrate trends. The model is completed with a system of elliptic partial differential equations governing the diffusion and reaction of planktonic cells, which are able to switch their mode of growth from planktonic to sessile when specific environmental conditions are found. Two systems of non-linear differential equations for the substrate and planktonic cells mass balance within the bulk liquid are also considered. The free boundary evolution is governed by a differential equation that accounts for detachment. The qualitative analysis is performed and a uniqueness and existence result is presented. Furthermore, two special models of biological and engineering interest are discussed numerically. The invasion of Anammox bacteria in a constituted biofilm inhabiting the deammonification units of the wastewater treatment plants is simulated. Numerical simulations are run to evaluate the influence of the colonization process on biofilm structure and activity.

Differential transform method for solving linear and non-linear systems of partial differential equations

Singular perturbation problems for systems of partial differential equations of elliptic type

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis.

The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE), (2 + 1)-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.

The Sumudu transform is presented in this paper in a modified form which is aimed at improving its performance and employing it along with a modified iteration method in order to determine the solution to a system of nonlinear partial differential equations. This includes a theoretical analysis of the associated modified Sumudu transform. It also includes an explanation of the mathematical method for utilizing the transform in conjunction with the modified iteration technique. The iteration method is employed to determine the nonlinear terms of the equations. The research is valuable in the sense that it allows approximate and exact solution configurations to be determined by combining the modified Sumudu transform with a modified iteration method. As another benefit, the modified Sumudu transform can be developed and enhanced to be applicable to a wide range of equations, making it an effective solution tool. By combining techniques, a final advantage is that the solutions can be derived quickly and easily as a result of the combined approach. Finally, an old transformation which has been modified from the Sumudu transform is combined with the modified iteration method to examine its capability of yielding convergent solutions by incorporating the modified iteration method into it.

The similarity solution for the steady two-dimensional flow of an incompressible viscous and electrically conducting fluid over a non-linearly semi-infinite stretching sheet in the presence of a chemical reaction and under the influence of a magnetic field gives a system of non-linear ordinary differential equations. These non-linear differential equations are analytically solved by applying a newly developed method, namely the Homotopy Analysis Method (HAM). The analytic solutions of the system of non-linear differential equations are constructed in the series form. The convergence of the obtained series solutions is carefully analyzed. Graphical results are presented to investigate the influence of the Schmidt number, magnetic parameter and chemical reaction parameter on the velocity and concentration fields. It is noted that the behavior of the HAM solution for concentration profiles is in good agreement with the numerical solution given in reference [A. Raptis, C. Perdikis, Int. J. Nonlinear Mech. 41, 527 (2006)].

A new family of exact solutions to the unsteady Navier–Stokes equations using canonical transformation with complex coefficients

For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution. Bibliography: 18 titles.

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

The current work investigates the exact analytical solution for 2D mixed convective flow and heat transfer of Casson hybrid nanofluid over stretching/shrinking sheet linear velocity in porous medium. The nonlinear system of partial differential equations (PDEs) is formed for the defined flow which is then converted into nonlinear system of ordinary differential equations (ODEs) via similarity transformations. Then the resultant dimensionless equation is solved analytically to obtain the exact solution for velocity profile. The hybrid nanofluid Cu – Al2O3/H2O is used in the analysis. Further the effect of porous medium, hybrid nanofluids and Brinkman ratio on the velocity profile is examined with the help of graphs. It found many applications in the fields in solar energy, heating and cooling systems, regenerative heat exchange, chemical catalytic reactors and oil extraction etc.

A theoretical analysis of steady and unsteady magnetohydrodynamic (MHD) flow of blood through a stenosed artery with variable viscosity and slip conditions is presented in this work. Slip conditions and variable viscosity dependent on hematocrit for both steady and unsteady blood flow models are taken into consideration in this study. The governing equations have been derived by treating blood as incompressible magnetohydrodynamic (MHD) Newtonian fluid. Galerkin Weighted Residual method, Newton Raphson method and fourth order Runge-Kutta method are used to solve the non-linear differential equations for velocity profiles of the blood flow, flow rate, shear stress and flow resistance for different values of hematocrit parameter, magnetic field parameter and slip velocity. The Galerkin Weighted Residual method is applied to transform the partial differential equations describing the flow into systems of non-linear equations and system of non-linear first order differential equations for the time independent and time dependent MHD flow models respectively. Newton Raphson’s method with the aid of maple software is used to solve the system of non-linear equations while the fourth order Runge-Kutta method is used to solve the system of non-linear first order differential equations. The research analyzed how the flow velocity, flow rate, shear stress and resistance to fluid flow respond to changes in some key parameters such as hematocrit parameter, magnetic field parameter and slip velocity. The results as presented graphically reveal that flow velocity and flow rate decrease while shear stress and resistance to flow increase when the hematocrit parameter increases. Also, flow velocity, flow rate and shear stress increase while resistance to flow decrease when the slip velocity increases. Finally, the increase in magnetic field parameter led to a decrease in flow velocity, flow rate and shear stress but increased the flow resistance. The flow velocity of the unsteady blood flow model is higher when compared to steady blood flow model.

On a system of nonlinear partial differential equations arising in the theory of heat and mass transfer in a nonlinear second-order fluid was published in Nonlinear Analysis Theory and Applications on page 437.