Convergence estimates for abstract second order differential equations with two small parameters and Lipschitzian nonlinearities

In a real Hilbert space $H$ we consider the following singularly perturbed Cauchy problem ... We study the behavior of solutions $u_{\varepsilon\delta}$ in two different cases: $\varepsilon\to 0$ and $\delta \geq \delta_0>0;$ $\varepsilon\to 0$ and $\delta \to 0,$ relative to solution to the corresponding unperturbed problem.We obtain some {\it a priori} estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of $t=0.$

Two parameter singular perturbation problems for sine-Gordon type equations

In the real Sobolev space $H_0^1(\Omega)$ we consider the Cauchy-Dirichlet problem for sine-Gordon type equation with strongly elliptic operators and two small parameters. Using some {\it a priori} estimates of solutions to the perturbed problem and a relationship between solutions in the linear case, we establish convergence estimates for the difference of solutions to the perturbed and corresponding unperturbed problems. We obtain that the solution to the perturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of $t=0.$

Statistical Convergence Estimates for (p, q)-Baskakov-Durrmeyer Type Operators

Available with pdf

Convergence Estimates for Gupta-Srivastava Operators

The Grüss-Voronovskaya-type approximation results for the modified Gupta-Srivastava operators are considered. Moreover, the magnitude of differences of two linear positive operators defined on an unbounded interval has been estimated. Quantitative type results are established as we initially obtain the moments of generalized discrete operators and then estimate the difference of these operators with the Gupta-Srivastava operators.

Analysis of the Orbital Approach Dynamics of the Space Debris Collector to the Fragment of Debris by the Method of Thrust Reversal with Interruption

The debris collector and a debris fragment move along random noncoplanar orbits in the altitude range of 400--2000 km. The thrust of the promising engine is 5000--25 000 N, the specific impulse of the promising fuel is not lower than 20 000 m/s. The remaining fuel after approach is not less than the specified. The debris collector undocks from the base station, transfers from its orbital plane to the debris fragment orbital plane, performs phasing, approaches the fragment, grabs it and returns to the base station. The paper considers only the stage of orbital approach. The duration of the entire flight mission is limited to one day. The phasing time is insufficient, therefore, at the start time of the orbital approach, the distance to the target is ~ 100 km, the relative velocity is ~ 1 km/s. On the other hand, for reliable and safe grabbing of a debris fragment, it is necessary to provide a distance of ~ 1 m and a relative velocity of ~ 1 m/s. It is shown that this can be achieved by approach using the method of thrust reversal with interruption. An effective algorithm of approach with target is proposed. An analysis of the orbital approach dynamics was performed by joint numerical integration of the orbital motion equations of the debris collector and the debris fragment by the 4th-order Runge --- Kutta method. Approach is performed in 6 cycles. In each cycle, the engine turns on three times. Two cycles are performed by sustainer engines, four cycles are performed by auxiliary engines of lower thrust. The fuel depletion and the non-sphericity of the Earth's gravitational field according to the 2nd zonal harmonic are taken into account. Calculation example is considered. Convergence estimates of the integration procedure by the resultant distance to the target and the resultant relative velocity are given. Resultant orbital approach is oscillation process with heavy damping. Damping is ensured by multiple firings of the sustainer (auxiliary) engine

An Inverse Problem for a Two-Dimensional Time-Fractional Sideways Heat Equation

In this paper, we consider a two-dimensional (2D) time-fractional inverse diffusion problem which is severely ill-posed; i.e., the solution (if it exists) does not depend continuously on the data. A modified kernel method is presented for approximating the solution of this problem, and the convergence estimates are obtained based on both a priori choice and a posteriori choice of regularization parameters. The numerical examples illustrate the behavior of the proposed method.

A modified Tikhonov regularization method for a class of inverse parabolic problems

This paper deals with the problem of determining an unknown source and an unknown initial condition in a abstract final value parabolic problem. This problem is ill-posed in the sense that the solutions do not depend continuously on the data. To solve the considered problem a modified Tikhonov regularization method is proposed. Using this method regularized solutions are constructed and under boundary conditions assumptions, convergence estimates between the exact solutions and their regularized approximations are obtained. Moreover numerical results are presented to illustrate the accuracy and efficiency of the proposed method.