Asymptotic methods in the nonlinear theory of viscoelasticity

Dynamic and quasistatic problems of the nonlinear theory of viscoelasticity are described by nonlinear integrodifferential and integral equations. Methods of averaging various classes of nonlinear integrodifferential and integral equations are described and asymptotic expansions of the solutions of these equations are given.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 221–229, March–April, 1974.     

Numerical methods in the theory of viscoelasticity

Formulas are given for calculating the coefficients of differential operators of defining equations on the basis of given approximations of the relaxation kernels in the form of the sum of exponential curves. As the defining equations it is suggested to use quadrature formulas into which are substituted the relaxation kernels found experimentally without preliminary analytic approximation. A three-dimensional difference problem of the linear isotropic theory of viscoelasticity is formulated. The direct and inverse -transformation establishing the correspondence between the viscoelastic and elastic difference problems is introduced. The specific characteristics of the use of the net, Ritz, finite-element, and variation-difference methods in solving problems of viscoelasticity theory are examined. A method facilitating the arrangement of the information on relaxation kernels in a computer memory is indicated.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 417–428, May–June, 1973.     

On the approximate solution of weakly singular integro-differential equations of Volterra type

Recently, the convergence rate of the collocation method for integral and integro-differential equations with weakly singular kernels has been studied in a series of papers [1–7]. The present paper belongs to the same series. We analyze the possibility of constructing approximate solutions of high-order accuracy on a uniform or almost uniform grid for weakly singular integro-differential equations of Volterra type.Translated from Differentsialnye Uravneniya, Vol. 40, No. 9, 2004, pp. 1271–1279.Original Russian Text Copyright © 2004 by Pedas.     

On the rate of convergence of methods of projection-iterative type for fredholm equations with periodic analytic kernels

Optimal rates of convergence of projection-iterative methods and methods of Sokolov type are found for a certain class of Fredholm equations with analytic kernels that appear within the framework of the method of boundary integral equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1155–1161, September, 1995.     

Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels

We propose iterated fast multiscale Galerkin methods for the second kind Fredholm integral equations with mildly weakly singular kernel by combining the advantages of fast methods and iteration post-processing methods. To study the super-convergence of these methods, we develop a theoretical framework for iterated fast multiscale schemes, and apply the scheme to integral equations with weakly singular kernels. We show theoretically that even the computational complexity is almost optimal, our schemes improve the accuracy of numerical solutions greatly, and exhibit the global super-convergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the methods.     

Contribution to the nonlinear theory of viscoelasticity

The author presents a generalization of certain results obtained in [3] relating to the region of rigid behavior of plastics for which the nonlinearity of the viscoelastic properties is important even in the area of small strains. Certain questions connected with the derivation of the basic equations of viscoelasticity are considered for small strains. Formulas are obtained for the resolvents of kernels of arbitrary order. The general equations of the "principal quadratic theory of viscoelasticity" are derived.Mekhanika Polimerov, Vol. 2, No. 4, pp. 498–507, 1966     

Averaging of electrical properties of anisotropic fiber-metal composites

The electrical properties of fiber-metal composites with anisotropic components were averaged within the framework of a regular structure model. The corresponding current problem was reduced to a system of two integral equations with elliptical kernels. The effective electrical conductivity tensor of such materials was defined as several functionals based on integral equation solutions for the boundary problem. The calculation results are given.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 263–268, March–April, 1997.     

Boundary-value problems for Helmholtz equations in an angular domain. I

The boundary-value problems are investigated that arise when studying the diffraction of acoustic waves on an infinite cylinder with cross-section of an arbitrary shape situated inside a wedge so that the axis of the cylinder is parallel to the edge of the wedge. The potential theory is worked out which enables one to reduce these boundary-value problems to integral equations on a one-dimensional contour — the boundary of the cross-section of this cylinder. The theorems on existence and uniqueness of solutions to the boundary-value problems and the corresponding integral equations are proved. For this case, a principle of limit absorption is established. Effective algorithms for calculating the kernels of the integral operators are constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 403–418, March, 1993.     

A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations

This article considers a stabilized finite element approximation for the branch of nonsingular solutions of the stationary Navier–Stokes equations based on local polynomial pressure projection by using the lowest equal-order elements. The proposed stabilized method has a number of attractive computational properties. Firstly, it is free from stabilization parameters. Secondly, it only requires the simple and efficient calculation of Gauss integral residual terms. Thirdly, it can be implemented at the element level. The optimal error estimate is obtained by the standard finite element technique. Finally, comparison with other methods, through a series of numerical experiments, shows that this method has better stability and accuracy.     

On the rate of convergence of projection-iterative methods for classes of weakly singular integral equations

For classes of weakly singular integral equations of the second kind whose kernels have a power singularity, we find the optimal order of the rate of convergence of projection-iterative methods. Moreover, iterative methods of the Sokolov type are considered and, for weakly singular equations with differentiable coefficients, we present estimates of the rate of convergence of such methods.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 498–505, April, 1995.     

On Optimization of Direct Methods of Solving Weakly Singular Integral Equations

We find the exact order of optimal accuracy of adaptive direct methods for the approximate solution of integral equations with potential-type kernels and for Peierls integral equations arising in transport theory. Moreover, for these equations we indicate the adaptive direct method of optimal order.     

On the order of the error in discretization methods for weakly singular second kind non-smooth solutions

In general, second kind Volterra integral equations with weakly singular kernels of the formk(t,s)(t –s)^– posses solutions which have discontinuous derivatives att=0. A discrete Gronwall inequality is employed to prove that, away from the origin, the error in product integration and collocation schemes for these equations is of order 2-.     

Quasilinear theory of viscoelasticity and the small parameter method

A closed quasilinear quadratic theory of viscoelasticity is proposed for bodies with physical nonlincarity. All the basic unique relations between the stress and strain tensors and time are established. Time relations between the invariant characteristics of combined loading processes are given. Integral equations relating the secondary kernels of creep and relaxation are obtained, and their connection with the corresponding kernels of the linear theory is determined. The singular properties of the kernels are elucidated, and their principal parts are isolated. Examples of possible expressions for the secondary kernels are given, and their general properties are examined.Mekhanika Polimerov, Vol. 2, No. 2, pp. 170–189, 1966Read at the Riga Conference on Polymer Mechanics, 11 November 1965.     

Averaging method applicable to the dynamics of not fully elastic bodies

Using well-known methods, various problems of dynamics are reduced to a standard system of integral differential equations, accounting for heredity effects in the form of multiple integrals according to the theory of Volterra. An averaging scheme is applied to this standard system, converting it to a system of differential equations which are much simpler that the initial equations. A theorem is proved which establishes the similarity of the solutions of these systems.V. I. Lenin Tashkent State University. Translated from Mekhanika Polimerov, No. 5, pp. 940–942, September–October, 1972.     

Integral equations of the third kind with unbounded operators

We consider linear functional equations of the third kind in L ₂ with arbitrary measurable coefficients and unbounded integral operators with kernels satisfying broad conditions. We propose methods for reducing these equations by linear continuous invertible transformations either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. To the integral equations obtained after the reduction, one can apply various exact and approximate methods of solution; in particular, the two approximate methods developed in this article.     

Optimization of algorithms for the approximate solution of the Volterra equations with infinitely differentiable kernels

For the Volterra equations with analytic kernels, we establish the exact power order of complexity of their approximate solutions and show that the optimal power order is realized by the method of simple iterations based on the use of information in the form of the values of kernels and free terms at certain points. In addition, for the Volterra equations with infinitely differentiable kernels, we determine the minimal order of the error of direct methods and construct a method which realizes this order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1534–1545, November, 1994.The work was supported by the Foundation for Fundamental Researches of the Ukrainian State Committee on Science and Technology.     

On the exact degree of complexity of a class of operator equations of the second kind in a Hilbert space

The exact exponent of complexity is found for approximate solutions of a certain class of operator equations in a Hilbert space. A method for information setup and the algorithm for realization of this optimal degree are presented. As a consequence, we find the exact exponent of complexity for approximate solutions of Fredholm integral equations of the second kind whose kernels and free terms include square integrable -derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 893–903, July, 1994.     

The study of dual integral equations with generalized Legendre functions

Closed form solutions for dual integral equations involving generalized Legendre functions as kernels are obtained. Connected to these dual integral equations an exact solution for dual integral equations involving sine functions as kernels is also obtained. Properties of generalized Legendre functions and the inversion theorem for the generalized Mehler-Fock transforms are used to obtain the solution of dual integral equations     

Variational Principle for General Diffusion Problems

We employ the Monge–Kantorovich mass transfer theory to study the existence of solutions for a large class of parabolic partial differential equations. We deal with nonhomogeneous nonlinear diffusion problems (of Fokker–Planck type) with time-dependent coefficients. This work greatly extends the applicability of known techniques based on constructing weak solutions by approximation with time-interpolants of minimizers arising from Wasserstein-type implicit schemes. It also generalizes previous results of the authors, where proofs of convergence in the case of a right-hand side in the equation is given by these methods. To prove the existence of weak solutions we establish an interesting maximum principle for such equations. This involves comparison with the solution for the corresponding homogeneous, time-independent equation.     

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.