A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel

This work is concerned with the numerical solution of a nonlinear weakly singular Volterra integral equation. Owing to the singular behavior of the solution near the origin, the global convergence order of product integration and collocation methods is not optimal. In order to recover the optimal orders a hybrid collocation method is used which combines a non-polynomial approximation on the first subinterval followed by piecewise polynomial collocation on a graded mesh. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. A comparison is made with the standard graded collocation method.     

An integral equation with a difference kernel

In this note a new method of solving a class of integral equations with difference kernels is given. It is based on establishing a connection between the solution of the given equation and that of the corresponding equation on the half-axis. This method allows us to reduce the given equation to a new integral equation with the kernel of a simple structure.Translated from Matematicheskie Zametki, Vol. 19, No. 6, pp. 927–932, June, 1976.     

The heat equation with nonlinear generalized Robin boundary conditions

Let Ω⊂RN be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H¹-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup SB=(SB(t))_t?0 on L²(Ω). We also obtain that this semigroup is ultracontractive in the sense that for every u,v∈Lp(Ω), p?2 and every t>0, one has

     

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem.     

Fredholm integral equation with singular kernel

In this paper, we solve the Fredholm integral equation of the first and second kind when the kernel takes a singular form. Also, some important relations for Chebyshev polynomial of integration are established.     

Fredholm-Volterra integral equation with singular kernel

The purpose of this paper is to obtain the solution of Fredholm-Volterra integral equation with singular kernel in the space L₂(?1, 1) × C(0,T), 0 ≤t ≤T < ∞, under certain conditions. The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernel using the Toeplitz matrices. Also, the error estimate is computed and some numerical examples are computed using the MathCad package.     

Approximate collocation method for an integral equation with a logarithmic kernel

An integral operator is defined on the set of functions expandable in a Fourier-Chebyshev series. The expansion is used to prove convergence of the proposed method and an error bound is derived.Consider the integral operator L, (1) $$L\varphi = \frac{1}{\pi }\int\limits_{ - 1}^1 {\ln \left| {x - t} \right|\frac{{\varphi (t)}}{{\sqrt {1 - t^2 } }}dt} ,\left| x \right| \leqslant 1.$$      

Asymptotics for the nonlinear dissipative wave equation

We are interested in the asymptotic behaviour of global classical solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like , , 0)$"> and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.

     

On a generalized Hammerstein-type integral equation

Theorems on the existence, uniqueness and continuous dependence of the solution of a generalized Hammerstein-type integral equation are given. The well-known Banach fixed-point theorem is employed to establish the results.     

On a generalized Hammerstein-type integral equation

Theorems on the existence, uniqueness and continuous dependence of the solution of a generalized Hammerstein-type integral equation are given. The well-known Banach fixed-point theorem is employed to establish the results.     

Numerical solution of an integral equation with a logarithmic kernel

When formulating boundary value problems within different branches of mathematical physics, one encounters an integral equation whose kernel is equal to the logarithm of the distance between two points on a plane, closed, smooth, and simple curve. This equation can be replaced by a system of linear algebraic equations which can be solved numerically.In the present paper we investigate two methods by which this replacement can be performed. Several examples are given in the literature where one of the methods is used. In contrast to this we here put forward a second method, which gives a higher accuracy without requiring more computational effort.     

Singular integral equation with Cauchy kernel on a complicated contour

We present a reasonably comprehensive exposition of the theory of a singular integral equation with Cauchy kernel for the case in which the integration contour is a set of disjoint smooth open arcs. We construct numerical schemes for this equation and give an order estimate for the accuracy of the approximate solutions.     

Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions

This paper presents a new boundary integral method for the solution of Laplace’s equation on both bounded and unbounded multiply connected regions, with either the Dirichlet boundary condition or the Neumann boundary condition. The method is based on two uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. Numerical results are presented to illustrate the efficiency of the proposed method.     

Asymptotics of the heat kernel for nonminimal differential operators

A method of calculating the coefficients in the asymptotic expansion of the heat-conduction kernel is generalized to nonminimal differential operators. The first nontrivial coefficients of the expansion for second-order nonminimal operators on Riemannian manifolds of arbitrary dimension are calculated.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 43, No. 11, pp. 1541–1551, November, 1991.