We consider a second-order periodic operator with two distant perturbations on the real axis. Perturbations are real finite continuous potentials. The objective is to investigate a behavior of the eigenvalues of the perturbed operator when the distance between the potentials tends to infinity. The study issue is an existence of perturbed eigenvalues in the case of a double limiting eigenvalue (the simple and isolated eigenvalue of a periodic operator with first potential + the simple and isolated eigenvalue of a periodic operator with а second potential).The paper aim is to construct the first terms of the asymptotic expansions of the perturbed eigenvalues and the corresponding eigenfunctions in the case of a double limiting eigenvalue.The technique to obtain the results can also find application when constructing complete asymptotic expansions of perturbed eigenvalues and their corresponding eigenfunctions. The finiteness of the distant potentials allowed us to reveal the complex exponential structure of the asymptotics obtained.The main results include the following:the first terms of the asymptotic expansions of the perturbed eigenvalues and their corresponding eigenfunctions;symmetry of the first corrections of the asymptotics of the perturbed eigenvalues with respect to zero;exponential structure of the asymptotics of perturbed eigenvalues and their corresponding eigenfunctions.
Qualitative analysis of dynamic systems is a powerful research tool used to solve a great deal of application problems. In mathematical models described by ordinary differential equations, the study of equilibrium positions has actually become the standard: their number and location depending on the parameters, as well as issues of local stability. In mathematical models in various fields of chemistry and biology, it is also important that the natural requirements for the possible values of phase variables are not violated in the process of dynamics (as a rule, this is the non-negativity condition). More sophisticated study orientation is global sustainability.The need for a qualitative analysis of a dynamic system arises because, as a rule, differential equations are not integrated and it is impossible to define the general properties of the system trajectories by integration. However, even if the system of differential equations is integrated, the formulas can be so cumbersome that their use in the analysis of the system dynamics turns out to be a challenge.In this paper, we study an 8-dimensional system of ordinary differential equations that describes the dynamics of cancer during chemotherapy and immunotherapy. The phase space of the system is a non-negative orthant.The paper studies the equilibrium positions of the system. There is always one equilibrium in a system, but there may be two more equilibria. There are conditions for the system parameters found, under which the equilibrium position, being always present is asymptotically stable. Moreover, there is a provement that the asymptotic stability is global.
The problems concerning the physical process studies that lead to mathematical models based on a parabolic type equation are of great practical importance.When solving the parabolic equation, a transition from the one-dimensional equation to the multidimensional one gives a rise to some difficulties. The trouble is that amount of computations significantly increases in the transition from the one-dimensional problems to the multidimensional ones In this regard a task to build the economical difference schemes for a numerical solution of multidimensional problems is a challenge.A difference scheme that approximates the problem over time is economical, if it is unconditionally stable and the desirable number of arithmetic operations in the transition from layer to layer is proportional to the number of nodes in each time layer.The paper dwells on the construction of a locally one-dimensional (economical) difference scheme for the approximate solution of a parabolic type equation of a general form in a multidimensional domain, the main idea of which is to reduce a complex problem to a sequential solution of boundary value problems of a simpler structure. At the same time, there is a construction of the economical, unconditionally stable difference scheme for each of the intermediate problems. To have a numerical solution of the problem, there is a construction of the locally one-dimensional difference scheme of A.A. Samarsky. Using a method of energy inequalities allows us to obtain a priori estimates in the differential and difference interpretations, whence it follow that there are the uniqueness, stability, as well as the convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem. For a bi-dimensional problem, a numerical solution algorithm is constructed.
We consider the Laplacian with two distant perturbations on the plane. Perturbations are a real finite continuous potentials. The investigation is aimed at a behavior of the perturbed operator eigenvalues when the distance between the potentials tends to infinity. The study concerns an existence of the perturbed eigenvalues in the case of a double limiting eigenvalue (a double eigenvalue of the Laplacian with the first finite potential).The paper aim is to construct the first terms of the asymptotic expansions of the perturbed eigenvalues and the corresponding eigenfunctions in the case of a double limiting eigenvalue.The technique used to obtain the results is also applicable to the construction of complete asymptotic expansions of perturbed eigenvalues аnd their corresponding eigenfunctions. The finiteness of the distant potentials made it possible to reveal the complex exponential-power structure of the asymptotics obtained.The main study results include the following:the first terms of the asymptotic expansions of the perturbed eigenvalues and their corresponding eigenfunctions;the first corrections of the asymptotics of the perturbed eigenvalues being equal to zero;exponential-power structure of the asymptotics of perturbed eigenvalues and their corresponding eigenfunctions.
Reducing pneumatic valves, which are part of modern pneumatic control systems, are widely used in the automation of production and technological processes in various fields of technology. The efficiency of using pressure reducing valves depends on their parameters and characteristics. To select the optimal parameters of pressure reducing valves, various methods are used based on the search for the minimum of the target (target or criteria) function.The object of study is a three-line direct-acting pressure reducing valve, which ensures the determination of its optimal parameters. To calculate the optimal parameters of the pressure reducing valve, the method of orthogonal experimental design was applied. The main advantage of this method is that in this case all variables are varied simultaneously.The article presents the design features of a three-line direct-acting pressure reducing valve.The developed mathematical model of a pressure reducing valve is presented, which is a system of equations for a three-line direct-acting pressure reducing valve, which should be used to select the optimal parameters of the pressure reducing valve.To calculate the optimal parameters of the pressure reducing valve, the method of orthogonal experimental design was used. As an optimality criterion, the integral criterion of the squared error of the transient process, the change in the pressure level in the outlet line of the valve, was chosen.The optimal parameters of the pressure reducing valve are found, and transient processes of pressure change in the outlet line of the pressure reducing valve are calculated. The main requirements for optimization and selection of the optimal parameters of pressure reducing valves are formulated. A comparison of the transient processes of the outlet pressure before and after optimization is given.Based on the results of the work, the following conclusions were drawn:From the obtained comparative results of the transient processes of the level of outlet pressure in the outlet of the pressure reducing valve before and after optimization, it was found that after optimization the pressure reducing valve has a faster response.The speed has been increased from 0.35 s to 0.27 s, that is, the speed has increased by 1.30 times.The transient overshoot is less than 5%, and the static error is reduced by 1.9% and is completely eliminated.
