The paper is concerned with the Riesz basis property of a boundary value problem associated in L20,1⊗C2 with the following 2×2 Dirac type equation(0.1)Ly=−iB−1y′+Q(x)y=λy,B=(b100b2),y=col(y1,y2), ...with a summable potential matrix Q∈L10,1⊗C2×2 and b1<0<b2. If b2=−b1=1 this equation is equivalent to one dimensional Dirac equation. It is proved that the system of root functions of a linear boundary value problem constitutes a Riesz basis in L20,1⊗C2 provided that the boundary conditions are strictly regular. By analogy with the case of ordinary differential equations, boundary conditions are called strictly regular if the eigenvalues of the corresponding unperturbed (Q=0) operator are asymptotically simple and separated. In opposite to the Dirac case there is no simple algebraic criterion of the strict regularity whenever b1+b2≠0. However under certain restrictions on coefficients of the boundary linear forms we present certain algebraic criteria of the strict regularity in the latter case. In particular, it is shown that regular separated boundary conditions are always strictly regular while antiperiodic boundary conditions are strictly regular if b1, b2 are coprime integers of different parity. The proof of the main result is based on existence of triangular transformation operators for system (0.1). Their existence is also established here in the case of a summable Q. In the case of regular (but not strictly regular) boundary conditions we prove the Riesz basis property with parentheses. The main results are applied to establish the Riesz basis property of the dynamic generator of spatially non-homogeneous damped Timoshenko beam model.
This work aims to solve large-scale variational inequalities (VIs), which are equivalent to high-dimensional systems of ordinary differential equations (ODEs). The existing physics-informed neural ...network (PINN) approach (Wu and Lisser, 2023) shows superior performance for VIs with less than 1000 variables, but fails for VIs of larger size, due to the increasing number of equations and the requirement of an extensive time interval. To overcome this limitation, we present two algorithms that dynamically adjust the initial condition for the PINN. The first algorithm uses multiple PINNs sequentially to decompose the task, where the best prediction from the current PINN serves as the initial condition for the next PINN. The second algorithm uses a single PINN throughout the solution process, immediately taking any improved prediction as an initial condition and refining the PINN to achieve a better prediction. Finally, we demonstrate the effectiveness of the proposed algorithms on a number of large-scale VI problems with up to 100,000 variables.
•Two PINNs algorithms are proposed for large-scale variational inequalities.•We introduce dynamical adjustment of initial conditions in PINNs.•The first algorithm deals with a sequential multi-network approach.•The second algorithm performs single network with refined initial conditions.•Experiments are conducted on instances with up to 100k variables.
The Riccati equation method is used to establish oscillation and non-oscillation criteria for nonhomogeneous linear systems of two first-order ordinary differential equations. It is shown that the ...obtained oscillation criterion is a generalization of J.S.W. Wong's oscillation criterion.
The differential quadrature method is a well-known numerical approach for solving ordinary and partial differential equations. This work introduces an explicit form for the approximate solution using ...differential quadrature rules. Analogies with Taylor's expansion are presented. Some properties are formally discussed. An interpretation of the approach from the neural networks perspective is also offered. For a fair comparison, we selected from the literature relevant examples numerically solved by approaches mainly in the realm of Taylor formalism, including a kind of neural network. Compared to the known numerical solutions, the obtained results show the good performance of the method.
We consider the problem of recovery of unknown right-hand sides of the first-order linear systems of ordinary differential equations with periodic coefficients from indirect observations of their ...solutions on a finite system of points and intervals. Under the assumption that right-hand sides and deterministic errors in observations are subjected to some quadratic restrictions, we obtain a posteriori estimates of right-hand sides which are compatible with observations data.
In this paper, we study the properties of trajectories of systems of ordinary differential equations generated by the velocity field of a moving incompressible viscoelastic fluid with memory along ...the trajectories in a domain with multiple boundary components. The case of a velocity field from a Sobolev space with inhomogeneous boundary conditions is considered. The properties of the maximal intervals of existence of solutions to the Cauchy problem corresponding to a given velocity field are investigated. The study assumes the approximation of a velocity field by a sequence of smooth fields followed by a passage to the limit. The theory of regular Lagrangian flows is used.
In this paper, applying the theory of fixed points in complete gauge spaces, we establish some conditions for the existence and uniqueness of monotonic and positive solutions for nonlinear systems of ...ordinary differential equations. Moreover, the paper contains an application of the theoretical results to the study of a class of systems of nonlinear ordinary differential equations.
The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. We introduce and investigate the ...class of weakly regular boundary conditions. We show that this class is much broader than the class of regular boundary conditions introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness. Also we investigate the completeness for 2×2 Dirac type equations subject to irregular boundary conditions. Emphasize that our results are the first results on the completeness for general first order systems even in the case of regular boundary conditions.
Some global solvability criteria for the scalar Riccati equations are used to establish new reducibility criteria for systems of two linear first-order ordinary differential equations. Some examples ...are presented.
In this paper, we will consider three deterministic models for the study of the interaction between the human immune system and a virus: the logistic model, the Gompertz model, and the generalized ...logistic model (or Richards model). A qualitative analysis of these three models based on dynamical systems theory will be performed by studying the local behavior of the equilibrium points and obtaining the local dynamics properties from the linear stability point of view. Additionally, we will compare these models in order to understand which is more appropriate to model the interaction between the human immune system and a virus. Some natural medical interpretations will be obtained, which are available for all three models and can be useful to the medical community.
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