Boundary integral equations for a body with inhomogeneous inclusions

On the basis of the partially singular differential equations of the stationary problem of heat conduction and the quasi-static problem of thermoelasticity, written taking account of conditions of nonideal thermomechanical contact, we derive boundary integral equations for a body with inhomogeneous inclusions. We propose a method of solving these equations taking account of the order of the principal term of the asymptotics of the solution in neighborhoods of the corners of the contact surfaces. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 37–41.     

Reduction of elasticity theory boundary problems for inhomogeneous media to sets of integral equations

The boundary problem of elasticity theory in stresses or displacements for materials which are continuously inhomogeneous along one coordinate is reduced by means of Laplace and Helmholtz equations to a set of four integro-differential equations, two of which are singular. Each of the equations contains integrals for the contour of the transverse section of a body which is assumed to be piecewise-smooth, and integrals for a region coincident with the section of the body.Sumy. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 20–23, 1990.     

Numerical solution of the three-dimensional Dirichlet problem for inhomogeneous media by the method of integral equations

We consider the three-dimensional Dirichlet problem for equations of elliptic type in inhomogeneous media. The problem can be reduced to a system of loaded Fredholm integral equations of the second kind over the volume. We prove the uniqueness of a classical solution of the problem. We suggest a numerical solution algorithm of iterative type. An example of the numerical solution of the problem is considered, and the convergence of the iterative procedure is demonstrated numerically.     

An exact methodology for solving nonlinear diffusion equations based on integral transforms

This paper develops a new methodology for the solution of nonlinear diffusion equations. The solution technique is based on integral transforms and leads to exact numerical results. We apply the formal methodology to the problem of one-dimensional transient heat conduction. A new form of the heat equation is developed using a generalized expression for temperature-dependent thermal conductivity, based on a power-series expansion, for the three standard orthogonal coordinate systems. The resulting form of the heat equation suggests that the finite integral transform technique may reduce the dimensionality of the heat equation prior to the initiation of any numerical procedure. An example in a slab with linearly varying thermal conductivity is shown to produce exact results for the temperature distribution.     

A new boundary integral equation formulation for plane orthotropic elastic media

A new boundary integral equation formulation for solving plane elasticity problems involving orthotropic media is presented in this paper. Based on the real variable fundamental solutions of the considered problems, a limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) and a novel decomposition technique to the fundamental solutions, the regularized BIEs with indirect unknowns, which do not involve the direct calculation of CPV and HFP integrals, are established. The limiting process is done in global coordinates and no separate numerical treatment for strong and weak singular integrals was necessary. The current method does not need to transform the considered problems into isotropic ones as is normally done in the existing literature, so no inverse transform is required. The numerical implementation is carried out using both discontinuous quadratic elements and exact elements, which is developed to model its boundary with negligible error. The validity of the proposed scheme is demonstrated by three numerical examples. Excellent agreement between the numerical results and exact solutions was obtained even with using small amounts of element.     

Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method.     

Semilinear fractional differential equations based on a new integral operator approach

Semilinear fractional differential equations (SFDEs) often arise in some dynamical systems. In most of the existing literature, the fixed point theorems are used to prove the existence and uniqueness of the solutions of SFDEs. In this paper, we give a new way to prove the existence and uniqueness of the solutions by introducing a new integral operator associated with the Mittag–Leffler function.     

Two-sided estimates for the solution of a nonlinear integral equation in a diffusion problem

For a particular nonlinear integral equation, its solution is constructed using sharp two-sided estimates and their convergence to the solution is proved.     

Solutions through nonclassical potential symmetries for a generalized inhomogeneous nonlinear diffusion equation

In this paper, we consider a class of generalized diffusion equations which are of great interest in mathematical physics. For some of these equations that model fast diffusion, nonclassical and nonclassical potential symmetries are derived. These symmetries allow us to increase the number of solutions. These solutions are unobtainable neither from classical nor from classical potential symmetries. Copyright © 2007 John Wiley & Sons, Ltd.     

System of nonlinear integral equations for the unknown functions in a functional-differential hyperbolic equation

We consider the inverse problem for a functional-differential equation in which the delay function and a function occurring in the source are unknown. The values of the solution and its derivative at x = 0 are given as additional information. We derive a system of nonlinear integral equations for the unknown functions. This system is used to prove a uniqueness theorem for the inverse problem.     

A new reliable algorithm based on the sinc function for the time fractional diffusion equation

This paper deals with the numerical solution of time fractional diffusion equation. In this work, we consider the fractional derivative in the sense of Riemann-Liouville. At first, the time fractional derivative is discretized by integrating both sides of the equation with respect to the time variable and we arrive at a semi–discrete scheme. The stability and convergence of time discretized scheme are proven by using the energy method. Also we show that the convergence order of this scheme is O(τ2?α). Then we use the sinc collocation method to approximate the solution of semi–discrete scheme and show that the problem is reduced to a Sylvester matrix equation. Besides by performing some theorems, the exponential convergence rate of sinc method is illustrated. The numerical experiments are presented to show the excellent behavior and high accuracy of the proposed hybrid method in comparison with some other well known methods.     

Inverse problem with a time‐integral condition for a fractional diffusion equation

We find the conditions for the unique solvability of the inverse problem for a time‐fractional diffusion equation with Schwarz‐type distributions in the right‐hand sides. This problem is to find a generalized solution of the Cauchy problem and an unknown space‐dependent part of an equation's right‐hand side under a time‐integral overdetermination condition.     

On the approximation method of solving integral equations in diffusion problems

In this paper we consider the one-dimensional problem of heat or mass transport in the system with moving ends. We show that without solving the heat transfer equation, the heat flux flowing out from the system can be found when temperature on the boundary of this system is known. We make use of the Banach contraction theorem for appropriate integral equations. Our method also enables us to find the distribution of temperature in the whole domain that forms the physical system.     

Nystrm interpolants based on the zeros of Legendre polynomials for a non-compact integral operator equation

We consider two different Nystrm interpolants for the numericalsolution fo the following singular integral equation arising from a problem of determining the distribution of stressin a thin elastic plate in the vicinity of a cruciform crack.These interpolants originate from the discretization of theintegral by two different quadrature formulas of interpolatorytype based on the zeros of Legendre orthogonal polynomials.The first quadrature is of product type and integrates exactlythe kernel; the second one is the well-known Gauss-Legendreformula. First we derive uniform convergence estimates for the two basicquadrature rules. Then by properly modifying the interpolantassociated with the Gauss-Legendre rule we prove its stabilityand derive for it a uniform error estimate of the type O(n^–4+),>0 as small as we like. We also show that if we had beenable to prove the stability of the first (modified) interpolantwe would have obtained a similar convergence estimate. Finally,for the Gauss-Legendre interpolant we prove that in any closedsubinterval [ 1] (0, 1] the rate of convergence is at leastO(n^–6+). Some numerical results which show the accuracy of our approximantsare also presented.     

Solution of the spatial Tricomi problem for a singular mixed-type equation by the method of integral equations

We consider a mixed-type singular differential equation in a bounded spatial domain in a specific form and prove the unique solvability of the Tricomi problem for the mentioned equation.