A Legendre multiwavelet based method is developed in this paper to solve second kind hypersingular integral equation by converting it into a Cauchy singular integro-differential equation. Multiscale representation of the singular and differential operators is obtained by employing Legendre multiwavelet basis. An estimate of the error of the approximate solution of the integral equation is obtai...

In this paper we study the Cauchy problem of doubly singular parabolic equations ut = div ∇u σ ∇um + t x u with non-negative initial data. Here −1 < σ ≤ 0, m > max 0 1 − σ − σ + 2 /N satisfying 0 < σ +m ≤ 1, p > 1, and s ≥ 0. We prove that if θ > max − σ + 2 , 1 + s N 1 − σ − m − σ + 2 , then pc = σ +m + σ +m− 1 s + σ + 2 1+ s + θ /N > 1 is the critical exponent; i.e, if 1 < p ≤ pc then every n...

We apply the method of determinants to study the distribution of the largest singular values of large real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the largest singular values are different from the Tracy-Widom law. Among other corollaries of our method we show an interesting connection between the mathematical expectations of the deter...

The article investigates the inverse problem for a complete, inhomogeneous, higher-order Sobolev type equation, together with Cauchy and overdetermination conditions. This was reduced to two equivalent problems in aggregate: regular singular. For these problems, theory of polynomially bounded operator pencils is used. unknown coefficient original equation restored using method successive approx...

Of concern here is the numerical solution of singular integral equations of Cauchy type; i.e., equations involving principal value integrals. The unknown function is expressed as the product of an appropriate weight function and a cubic spline. The problem is reduced to a system of linear algebraic equations which is solved for the approximate values of the function at the knots. An estimate is...

We present new fast algorithms for solving the Toeplitz and the Toeplitz-plus-Hankel least squares problems. These algorithms are based on a new fast algorithm for solving the Cauchy-like least squares problem. We perform an error analysis and provide conditions under which these algorithms are numerically stable. We also develop implementation techniques that signiicantly reduce the execution ...

Abstract. We continue our analysis of the Cauchy problem for viscous system of conservation, under natural assumptions. We examine in which way does the existence time depend upon the viscous tensor B(u). In particular, we consider singular limits, where the rank of the symbol B(u; ξ) drops at the limit. This covers a lot of situations, for instance that of the limit of the Navier-Stokes-Fourie...

A class of nonlinear parabolic systems with quadratic nonlinearities in the gradient (the case of two spatial variables) is considered. It is assumed that the elliptic operator of the system has a variational structure. The behavior of a smooth on a time interval [0, T ) solution to the Cauchy-Neumann problem is studied. For the situation when the “local energies” of the solution are uniformly ...

The existence of classical solutions to the Cauchy problem for the Boltzmann equation without angular cutoff has been extensively studied in the framework when the solution has Maxwellian decay in the velocity variable. cf. [8, 6] and the references therein. In this paper, we prove local existence of solutions with polynomial decay in the velocity variable for the Boltzmann equation with soft p...

We consider the Cauchy problem for a system of balance laws derived from chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove global well-posedness classical solutions to when only order spatial derivatives initial data is sufficiently small, and are shown converge prescribed constant equilibrium states as time goes infinity. Then tha...