We study a continuous analog of strictly non-Volterra quadratic stochastic operators with parameters on a two-dimensional simplex. The explicit form of the fixed point and its type are found. For particular parameter values, an analytic solution of the Cauchy problem is obtained and a comparative analysis with the numerical solution of the system is conducted. We prove that the trajectories of the system on the invariant manifold converge to the equilibrium position exponentially rapidly. As an example, an analytic solution of the Cauchy problem for the linearized system in a neighborhood of the fixed point is presented, which demonstrates the vanishing on the invariant manifold of the coefficient of the exponential term with a positive eigenvalue in the exponent. An analytic solution of the system for general parameters is constructed as well. The solutions are represented in the form of special power series whose coefficients are determined from recurrence relations. It is shown that the representation of the solution is uniquely determined by the set of free coefficients. The uniform convergence of the constructed series on an infinite interval starting from a certain time is proved. The efficiency of this method is demonstrated by comparison with the numerical solution.
A system of acoustic equations in which the Lamé parameter has a memory (i.e., depends on the history) is considered. The dependence has a specific form and is described by an integral convolution operator whose kernel is the product of two functions, one depending on the space coordinates and the other, on time. For this system of equations, we study an acoustic tomography problem in which three functions, namely, the speed of acoustic waves, the medium density, and the component of the memory kernel depending on the space variables, should be reconstructed. To solve this problem, we use information on a series of solutions of direct problems for the acoustic equations with point sources. This information is given for a finite time interval at the points of some sphere inside which the coefficients to be recovered are assumed to be unknown. We show that the problem in question can be reduced to the inverse kinematic problem for reconstructing the speed of sound and two X-ray tomography problems to be solved consecutively.
We consider an ordinary differential operator of order 2m with integral conditions containing derivatives of the unknown function. We obtain an a priori estimate for the solutions of the problem for sufficiently large values of the spectral parameter. We prove the discreteness of the spectrum and its sectorial structure as well as the Fredholm property of the corresponding problem.
We consider various classes of abstract singular equations generalizing the Euler–Poisson–Darboux equation and containing the squared generator of an exponentially decaying semigroup. Sufficient conditions for the unique solvability of the corresponding incomplete problems are obtained.
This paper investigates the step-like contrast structure in a class of higher-order quasilinear singularly perturbed boundary value problems with the same type of singularity. To overcome the difficulty that the conditions for determining the transition point are overdetermined in the real domain, we extend the problem to the complex domain and employ complex-analytic methods. By utilizing the boundary layer method and the smooth matching technique, we not only construct the asymptotic expansion of the step-like solution with an interior layer but also rigorously prove the uniform validity of this smooth solution together with remainder estimates. The results show that, under appropriate analyticity and sign conditions, such quasilinear systems can achieve a balance between the number of equations and the number of unknowns for determining the transition point in the complex domain, thus yielding a solution with a step-like structure in the real domain. This work provides new theoretical tools and analytical approaches for the asymptotic analysis of higher-dimensional quasilinear singular perturbation systems.
