Nonlinear Panel Flutter Analysis at High Supersonic Speed (Dept.M)

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.

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

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 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.

The article presents a class of exact solutions of the Oberbeck-Boussinesq equations for media flows taking into account energy dissipation (transformation of mechanical energy into thermal energy). The construction of a family of exact solutions is based on the use of the Lin-Sidorov-Aristov class. The velocity field is described by linear forms relative to two spatial coordinates. These coordinates are called horizontal or longitudinal. The coefficients of linear forms depend on the third coordinate (vertical or transverse) and time. The pressure field and temperature field are described by quadratic forms with a functional dependence similar to the velocity vector. The article presents a system of nonlinear partial differential equations for determining unknown functions of hydrodynamic fields. Exact solutions are constructed to describe unidirectional flows of the Birikh-Ostroumov type with the dissipation of mechanical energy. The problems for convective flows of the Ekman type of a solid-state rotating fluid with the first Coriolis parameter are formulated. To describe inhomogeneous shear motions of a viscous incompressible fluid, an overdetermined system of nonlinear partial differential equations was investigated for the existence of solutions. It is shown that there is a nontrivial exact solution for flows with spatial acceleration.

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.

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.

A New Multistep Technique Based on the Nonuniform Rational Basis Spline Curves for Nonlinear Transient Heat Transfer Analysis of Functionally Graded Truncated Cone

Sufficient conditions for the existence of one-sided integral manifolds of solutions of systems of nonlinear difference equations are obtained. One-sided nonlinear projections, defining these manifolds, are constructed.

: Complex physical phenomena involving chemically reacting systems or the transport of heat or fluids are often modeled by coupled systems of time-dependent, nonlinear partial differential equations. The difficulties in understanding the stability of the differential equation systems and in designing efficient, accurate numerical methods for their solution are widely recognized and were the focus of this research. We have worked on four general aspects of the analysis and numerical approximation of systems of partial differential equations. These areas of research are: (1) modeling aspects and stability analysis for nonlinear time-dependent partial differential equations; (2) use and analysis of finite element or finite difference methods to discretize coupled systems of nonlinear differential equations; (3) development of adaptive or local grid refinement capabilities to resolve local phenomena in large-scale applications; and (4) development of data structures, preconditioners, and efficient solution algorithms for large-scale problems on new computer architectures. Emphasis has been placed upon multiphase or multicomponent, transport-dominated flow processes with dynamic local phenomena. The research also involved a mix of analysis, algorithm development, and large-scale computation using newer computer architectures.

Robotic and mechatronic systems, autonomous vehicles, electric power systems and smart grids, as well as manufacturing and industrial production systems can exhibit complex nonlinear dynamics or spatio-temporal dynamics which need to be controlled to ensure good functioning and performance. In this comprehensive reference, the authors present new and innovative control and estimation methods and techniques based on dynamical nonlinear and partial differential equation systems. Such results can be classified in five main domains for the control of complex nonlinear dynamical systems using respectively methods of approximate (local) linearization, methods of exact (global) linearization, Lyapunov stability approaches, control and estimation of distributed parameter systems and stochastic estimation and fault diagnosis methods. Control and Estimation of Dynamical Nonlinear and Partial Differential Equation Systems: Theory and applications will be of interest to electrical engineering, physics, computer science, robotics and mechatronics researchers and professionals working on control problems, condition monitoring, estimation and fault diagnosis and isolation problems. It will also be useful to skilled technical personnel working on applications in robotics, energy conversion, transportation and manufacturing.

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.).

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

Homotopy perturbation method (HPM) is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations. This chapter first illustrates the basic idea of the HPM. It uses the present method to solve a nonlinear ordinary differential equation, a nonlinear partial differential equation, and a coupled nonlinear system of partial differential equations. It is worth mentioning that the HPM can also be used for handling linear ordinary and partial differential equations. The test problems demonstrated in the chapter confirm that the HPM is an efficient method for solving linear/nonlinear ordinary, partial, and coupled differential equations. The series usually converges with an increase in the number of terms, but one may not expect the compact form solution always.