On the Application of the Collocation Method in Spaces of p-Summable Functions to the Fredholm Integral Equation of the Second Kind

This work is devoted to the study of the numerical solution by the spline collocation method of the Fredholm equation of the second kind. For numerical solutions of such problems, the classical collocation method using polynomials is not always realizable in spaces of p-summable functions for numerical solutions of such problems. It is not always possible to obtain characteristics and estimates of errors of such approximations even in the case of its implementation. In this regard, in recent years, in practice, approximations are built using finite-difference methods. The purpose of this study is to obtain estimates of the error of the obtained approximate solution in the spaces indicated above. In addition, several statements were obtained about a pointwise estimate of this error at collocation nodes in terms of the kernel norm in specially constructed spaces of functions summable over the second variable. To obtain the main results, third-order collocation splines are used, as well as integral and averaged modules of smoothness. In this case, the results obtained can become a starting point for working with collocation splines of higher orders. In the case of the third order, the exact constants involved in the estimates are obtained. These results can be extended to the case of linear, parabolic collocation splines, as well as splines of order higher than the third.

Asymptotic analysis of a mathematical model of the atherosclerosis development

A mathematical model, which takes into account new experimental results about diverse roles of macrophages in the atherosclerosis development, is proposed. Using technic of upper and lower solutions, the existence and uniqueness of its positive solution are justified. After the nondimensionalization, small parameters are found and the multiscale analysis of the corresponding perturbed problem is performed when those parameters tend to zero. In particular, the limit two-dimensional problem, which is a coupled system of reaction–diffusion equations and ordinary differential equations, is derived; the asymptotic approximation is constructed; the uniform pointwise estimate for the difference between the solution of the original problem and the solution of the limit problem as well as the respective [Formula: see text]-estimates for the fluxes are proved.

POINTWISE ESTIMATE OF DEVIATION OF KRIAKIN POLYNOMIAL FROM A FUNCTION, CONTINUOUS ON A SEGMENT

New estimates for the algebraic polynomials that approximate a function continuous on a segment involving moduli of continuous of high orders are obtained, namely the pointwise estimates.

Sharp estimates for the gradient of the generalized Poisson integral for a half-space

A representation of the sharp coefficient in a pointwise estimate for the gradient of the generalized Poisson integral of a function f on {{\mathbb{R}}^{n}} is obtained under the assumption that f belongs to {L^{p}} . The explicit value of the coefficient is found for the cases {p=1} and {p=2} .

Generalized Poisson integral and sharp estimates for harmonic and biharmonic functions in the half-space

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on ℝn−1 is obtained under the assumption that f belongs to Lp. It is assumed that the kernel of the integral depends on the parameters α and β. The explicit formulas for the sharp coefficients are found for the cases p = 1, p = 2 and for some values of α, β in the case p = ∞. Conditions ensuring the validity of some analogues of the Khavinson’s conjecture for the generalized Poisson integral are obtained. The sharp estimates are applied to harmonic and biharmonic functions in the half-space.

Decay property of the Timoshenko–Cattaneo system

We study the Timoshenko system with Cattaneo’s type heat conduction in the one-dimensional whole space. We investigate the dissipative structure of the system and derive the optimal [Formula: see text] decay estimate of the solution in a general situation. Our decay estimate is based on the detailed pointwise estimate of the solution in the Fourier space. We observe that the decay property of our Timoshenko–Cattaneo system is of the regularity-loss type. This decay property is a little different from that of the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]) in the low frequency region. However, in the high frequency region, it is just the same as that of the Timoshenko–Fourier system (see [N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier’s type heat conduction, J. Hyperbolic Differential Equations 11 (2014) 135–157]) or the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]), although the stability number is different. Finally, we study the decay property of the Timoshenko system with the thermal effect of memory-type by reducing it to the Timoshenko–Cattaneo system.