Narrow Results

The problem of optimal recovering high-order mixed derivatives of bivariate functions is studied. For numerical differentiation, a new version of the truncation method is proposed. It is established that this algorithm is order-optimal both in the sense of accuracy and in terms of the amount of involved discrete information.

In this paper, we consider the system of $p$-Laplacian equations with critical growth

We analyze the results published in this journal about two conditionally solvable quantum-mechanical models. We show that the authors failed to derive the spectrum correctly because they did not take into consideration all the roots of the truncation condition used for that aim and did not interpret them correctly.

A numerical method using the truncation technique on the integrand is developed for computing singular minimizers or singular minimizing sequences in variational problems involving the Lavrentiev phenomenon. It is proved that the method can detect absolute minimizers with various singularities whether the Lavrentiev phenomenon is involved or not. It is also proved that, when the absolute infimum is not attainable, the method can produce minimizing sequences. Numerical results on Manià's example and a two-dimensional problem involving the Lavrentiev phenomenon with continuous Sobolev exponent dependence, are given to show the efficiency of the method.

A convergence theory is established for a truncation method in solving polyconvex elasticity problems involving the Lavrentiev phenomenon. Numerical results on a recent example by Foss et al., which has a polyconvex integrand and admits continuous singular minimizers, not only verify our convergence theorems but also provide a sharper estimate on the upper bound of a perturbation parameter for the existence of the Lavrentiev phenomenon in the example.

Here, we argue that the results shown in [L. L. Vitória and K. Bakke, Int. J. Mod. Phys. D 27 (2018) 1850005] are not correct. The truncation of the Frobenius method only yields some particular bound states from which one cannot derive any sound physical conclusion. The existence of allowed model parameters is an artifact of such procedure. Besides, the authors failed to obtain most of the solutions given by the truncation of the Frobenius series as well as their connection with the actual eigenvalues of the problem.

Methods of comparing discrete and continuous cable models of single neurons and dynamical phenomena observed in neurobiology can be described with infinite-coupled systems of semilinear parabolic differential-functional equations of the reaction-diffusion-convection type or infinite systems of ordinary integro-differential equations. It is known that numerous problems in computational neuroscience use finite systems of equations based on the so-called compartmental model. It seems a natural idea to extend the results obtained in the theory of finite systems onto infinite systems. However, this requires stringent assumptions to be adopted to achieve compatibility. In most instances the dynamics of infinite systems behave differently to their finite-dimensional projections. The truncation method applied to infinite systems of equations and presented herein yields a truncated system consisting of the first N equations of the infinite system in N unknown functions. A solution of infinite system is defined as the limit when N → ∞ of the sequence of approximations {z_N}_N=1,2,…, where are defined as solutions of suitable finite truncated systems with corresponding initial-boundary conditions. Geometrically, it may be described as the projection of an infinite system of differential equations considered in a function abstract space of infinite dimension (such as Banach or Hilbert space) onto its finite-dimensional subspaces.