We consider the uniform attractors for the two dimensional nonautonomous g-Navier-Stokes equations in bounded domain $\Omega$. Assuming $f=f(x,t)\in L_{\text{loc}}^2$, we establish the existence of the uniform attractor in $L^2(\Omega )$ and $D\left(A^{1/2}\right)$. The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

We consider the application of a new analytic method based on homotopy analysis to the solution of the steady flow of a viscous incompressible fluid past a fixed circular cylinder. The solutions obtained using this method produce some interesting results. For instance, an analytic verification of the critical Reynolds number $R_d$ for which a standing vortex first appears behind the cylinder is given for the first time and found to be $R_d\preccurlyeq 2.4$. Since these values of the critical Reynolds number are beyond the range of validity of Oseen theory, no analytic verification of this value had previously been given. As a check on the accuracy of the solutions, the calculated drag coefficients at 6th-order approximation are found to agree reasonably well with experimental measurements for $R_d\simeq 30$ which is considerably larger than the $R_d\simeq 1$ results currently available using other analytic techniques. This buttresses the usefulness of the homotopy analysis method (HAM) as an important tool in solving highly nonlinear problems.

This paper presents an analytical solution of the hyperbolic heat conduction equation for moving semi-infinite medium under the effect of time dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by $g\left(x,t\right)=I\left(t\right)\left(1-R\right)\mu e^{-\mu x}$ while the semi-infinite body has insulated boundary. The solution is obtained by Laplace transforms method, and the discussion of solutions for different time characteristics of heat sources capacity (constant, instantaneous, and exponential) is presented. The effect of absorption coefficients on the temperature profiles is examined in detail. It is found that the closed form solution derived from the present study reduces to the previously obtained analytical solution when the medium velocity is set to zero in the closed form solution.

A boundary value problem is posed for an integro-differential beam equation. An approximate solution is found using the Galerkin method and the Jacobi nonlinear iteration process. A theorem on the algorithm error is proved.

We deal with an application of the fixed point theorem for nonexpansive mappings to a class of control systems. We study closed-loop and open-loop controllable dynamical systems governed by ordinary differential equations (ODEs) and establish convexity of the set of trajectories. Solutions to the above ODEs are considered as fixed points of the associated system-operator. If convexity of the set of trajectories is established, this can be used to estimate and approximate the reachable set of dynamical systems under consideration. The estimations/approximations of the above type are important in various engineering applications as, for example, the verification of safety properties.

We introduce the concept of generalized $S\left(C,A,B\right)$-pairs which is related to generalized $S\left(A,B\right)$-invariant subspaces and generalized $S\left(C,A\right)$-invariant subspaces for infinite-dimensional systems. As an application the parameter-insensitive disturbance-rejection problem with dynamic compensator is formulated and its solvability conditions are presented. Further, an illustrative example is also examined.

The problem of concentric pervious spheres carrying a fluid source at their centre and rotating slowly with different uniform angular velocities ${\Omega }_1$, ${\Omega }_2$ about a diameter has been studied. The analysis reveals that only azimuthal component of velocity exists, and the couple, rate of dissipated energy is found analytically in the present situation. The expression of couple on inner sphere rotating slowly with uniform angular velocity ${\Omega }_1$, while outer sphere also rotates slowly with uniform angular velocity ${\Omega }_2$, is evaluated. The special cases, like (i) inner sphere is fixed (i.e., ${\Omega }_1=0$), while outer sphere rotates with uniform angular velocity ${\Omega }_2$, (ii) outer sphere is fixed (i.e., ${\Omega }_2=0$), while inner sphere rotates with uniform angular velocity ${\Omega }_1$, and (iii) inner sphere rotates with uniform angular velocity ${\Omega }_1$, while outer sphere rotates at infinity with angular velocity ${\Omega }_2$, have been deduced.

We discuss and derive the analytical solution for three basic problems of the so-called time-fractional telegraph equation. The Cauchy and Signaling problems are solved by means of juxtaposition of transforms of the Laplace and Fourier transforms in variable t and x, respectively. the appropriate structures and negative prosperities for their Green functions are provided. The boundary problem in a bounded space domain is also solved by the spatial Sine transform and temporal Laplace transform, whose solution is given in the form of a series.

We study Hopf bifurcation solutions to the Monodomain model equipped with FitzHugh-Nagumo cell dynamics. This reaction-diffusion system plays an important role in the field of electrocardiology as a tractable mathematical model of the electrical activity in the human heart. In our setting the (bounded) spatial domain consists of two subdomains: a collection of automatic cells surrounded by collections of normal cells. Thus, the cell model features a discontinuous coefficient. Analytical techniques are applied to approximate the time-periodic solution that arises at the Hopf bifurcation point. Accurate numerical experiments are employed to complement our findings.

This paper considers the problem of controlling the solution of an initial boundary-value problem for a wave equation with time-dependent sound speed. The control problem is to determine the optimal sound speed function which damps the vibration of the system by minimizing a given energy performance measure. The minimization of the energy performance measure over sound speed is subjected to the equation of motion of the system with imposed initial and boundary conditions. Using the modal space technique, the optimal control of distributed parameter systems is simplified into the optimal control of bilinear time-invariant lumped-parameter systems. A wavelet-based method for evaluating the modal optimal control and trajectory of the bilinear system is proposed. The method employs finite CAS wavelets to approximate modal control and state variables. Numerical examples are presented to demonstrate the effectiveness of the method in reducing the energy of the system.

A theoretical model developed by Stone describing a three-level trophic system in the Ocean is analysed. The system consists of two distinct predator-prey networks, linked by competition for nutrients at the lowest level. There is also an interaction at the level of the two preys, in the sense that the presence of one is advantageous to the other when nutrients are low. It is shown that spontaneous oscillations in population numbers are possible, and that they result from a Hopf bifurcation. The limit cycles are analysed using Floquet theory and are found to change from stable to unstable as a solution branch is traversed.

The ejectors are used commonly to extract gases in the petroleum industry where it is not possible to use an electric bomb due the explosion risk because the gases are flammable. The steam ejector is important in creating and holding a vacuum system. The goal of this job is to develop an object oriented parallel numerical code to investigate the unsteady behavior of the supersonic flow in the ejector diffuser to have an efficient computational tool that allows modeling different diffuser designs. The first step is the construction of a proper transformation of the solution space to generate a computational regular space to apply an explicit scheme. The second step, consists in developing the numerical code with an-object-oriented parallel methodology. Finally, the results obtained about the flux are satisfactory compared with the physical sensors, and the parallel paradigm used not only reduces the computational time but also shows a better maintainability, reusability, and extensibility accuracy of the code.