Class Of Dual Integral Equations Research Articles

The solution of a certain class of dual integral equations with the right-hand side in the form of a Fourier series and its application to the solution of contact problems for inhomogeneous media

Abstract Using the bilateral asymptotic method, a semi-analytical solution of a dual integral equation with its right-hand side in the form of a Fourier series is constructed. This equation arises in the solution of a number of contact problems of elasticity theory for bodies with inhomogeneous coatings. The efficiency of the method is illustrated in the example of the solution of the plane contact problem on bending of a beam lying on a functionally graded strip with arbitrary variation of the elastic moduli with depth. It is assumed that the strip is perfectly bonded to an elastic half-plane. Numerical results are presented for a strip whose Young's modulus varies harmonically with depth. In this case, Young's modulus of the substrate is 100 times greater than at the lower boundary of the coating.

Solution of a dual integral equation arising in the contact problems of elasticity theory with the full Fourier series as the right-hand side

A class of dual integral equations is analyzed which arises in solution of a wide range of plane and antiplane contact problems of elasticity theory for a half-plane with functionally graded coating. In particular, a similar equation arises in solution of the contact problem on indentation in the presence of tangential stresses on a surface. The solution of the dual integral equation is sought in the form of a sum of even and odd functions. It makes possible to reduce the problem to independent solution of two dual integral equations over odd and even functions. Kernel transform of these equations is approximated by a product of fractional quadratic functions. The solution of dual integral equations is constructed in approximated analytical form by the bilateral asymptotic method. The expressions obtained are asymptotically exact for small and large values of a characteristic geometrical parameter.

Some Criteria for Coating Effectiveness in Heavily Loaded Line Elastohydrodynamically Lubricated Contacts—Part II: Lubricated Contacts

Contacts of indentors with functionally graded elastic solids may produce pressures significantly different from the results obtained for homogeneous elastic materials (Hertzian results). It is even more so for heavily loaded line elastohydrodynamically lubricated (EHL) contacts. The goal of the paper is to indicate two distinct ways the functionally graded elastic materials may alter the classic results for the heavily loaded line EHL contacts. Namely, besides pressure, the other two main characteristics which are influenced by the nonuniformity of the elastic properties of the contact materials are lubrication film thickness and frictional stress/friction force produced by lubricant flow. The approach used for analyzing the influence of functionally graded elastic materials on parameters of heavily loaded line EHL contacts is based on the asymptotic methods developed earlier by the authors such as Kudish (2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman & Hall/CRC Press, Boca Raton, FL), Kudish and Covitch (2010, Modeling and Analytical Methods in Tribology, Chapman & Hall/CRC Press, Boca Raton, FL), Aizikovich et al. (2002, “Analytical Solution of the Spherical Indentation Problem for a Half-Space With Gradients With the Depth Elastic Properties,” Int. J. Solids Struct., 39(10), pp. 2745–2772), Aizikovich et al. (2009, “Bilateral Asymptotic Solution of One Class of Dual Integral Equations of the Static Contact Problems for the Foundations Inhomogeneous in Depth,” Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, p. 317), Aizikovich and Vasiliev (2013, “A Bilateral Asymptotic Method of Solving the Integral Equation of the Contact Problem for the Torsion of an Elastic Halfspace Inhomogeneous in Depth,” J. Appl. Math. Mech., 77(1), pp. 91–97), Volkov et al. (2013, “Analytical Solution of Axisymmetric Contact Problem About Indentation of a Circular Indenter Into a Soft Functionally Graded Elastic Layer,” Acta Mech. Sin., 29(2), pp. 196–201), Vasiliev et al. (2014, “Axisymmetric Contact Problems of the Theory of Elasticity for Inhomogeneous Layers,” Z. Angew. Math. Mech., 94(9), pp. 705–712), Aizikovich et al. (2008, “The Deformation of a Half-Space With a Gradient Elastic Coating Under Arbitrary Axisymmetric Loading,” J. Appl. Math. Mech., 72(4), pp. 461–467), and Aizikovich et al. (2010, “Inverse Analysis for Evaluation of the Shear Modulus of Inhomogeneous Media in Torsion Experiments,” Int. J. Eng. Sci., 48(10), pp. 936–942). More specifically, it is based on the analysis of contact problems for dry contacts of functionally graded elastic solids and the lubrication mechanisms in the inlet and exit zones as well as in the central region of heavily loaded lubricated contacts. The way the solution of the EHL problem for coated/functionally graded materials is obtained provides a very clear structure of the solution. The solution of the EHL problem in the Hertzian region is very close to the solution of the dry contact problem while in the inlet and exit zones the solutions of the EHL problem with the right asymptotes coming from the solution of the dry contact problem can be related to the solutions of the classic EHL problem for homogeneous materials.

On a class of dual integral equations involving trigonometrical functions

In this paper we consider certain dual integral equations involving trigonometrical functions whose closed form solutions are obtained. Solutions are obtained by using the properties of Mehler-Fock transforms and the inversion theorem of the generalized Mehler-Fock transforms. The solutions of these dual integral equations have applications in engineering.

A note on the polynomial solution of a class of dual integral equations arising in mixed boundary value problems of elasticity

The present paper is concerned with finding an effective polynomial solution to a class of dual integral equations which arise in many mixed boundary value problems in the theory of elasticity. The dual integral equations are first transformed into a Fredholm integration equation of the second kind via an auxiliary function, which is next reduced to an infinite system of linear algebraic equations by representing the unknown auxiliary function in the form of an infinite series of Jacobi polynomials. The approximate solution of this infinite system of equations can be obtained by a suitable truncation. It is shown that the unknown function involving the dual integral equations can also be expressed in the form of an infinite series of Jacobi polynomials with the same expansion coefficients with no numerical integration involved. The main advantage of the present approach is that the solution of the dual integral equations thus obtained is numerically more stable than that obtained by reducing themdirectly into an infinite system of equations, insofar as the expansion coefficients are determined essentially by solving asecond kind integral equation.

On the solutions of a class of dual integral equations occurring in diffraction problems

It is shown that there exists a category of two-dimensional diffraction problems, which can be put into a ‘standard form‘ of dual integral equations. These diffraction problems include: diffraction of electromagnetic waves by a finite strip, a finite slit, the diffraction of scalar or vector elastic waves by a rigid strip or crack, etc. A general method for solving such dual integral equations is given by the artifice of constructing a set of functions of compact support biorthogonal to another given set of functions. The sufficient conditions for a given dual integral equations to be solvable in this manner are also determined. Hence, the method forms a complement to the Weiner-Hopf method. To illustrate the method solutions are obtained for a bench-mark problem : the diffraction of light by a finite perfectly conducting strip (or equivalently the diffraction of SH waves by a crack). Comparison with results obtained by others for low, intermediate and high frequencies show the utility and accuracy of the method for the entire range of frequencies. Both the near field and the far field are obtained, the latter is shown to correspond to the Fraunhoffer diffraction pattern for high frequency. It is also shown that for the equivalent crack problem the stress intensity factor (SIF) fluctuates rapidly with changes in the angle of incidence for high frequencies, thus making the SIF especially sensitive to angle of incidence at high frequencies.

Asymptotic properties of the approximate solution of a class of dual integral equations

An investigation is presented of dual integral equations generated by various plane contact problems: a strip resting without friction on a rigid base (Problem 1), a strip clamped along the base (Problem 2), a wedge with a clamped face (Problem 3), and also be axisymmetric problems relating to the action of a ring-shaped stamp on a half-space (Problem 4), and the interaction of an elastic bandage with an elastic cylinder /1/ (Problem 5). The strip, wedge, half-space and cylinder may be uniform, laminar of continuously inhomogeneous. Analogous equations in terms of Laplace transforms are obtained in problems of coupled thermo-elasticity and consolidation theory of water-saturated media for the bodies listed here /2/. The method described in /3/ is generalized to construct solutions of the above problems. Well-posedness and solvability classes are established for the equations, proving that the approximate method proposed here is asymptotic in both directions with respect to a characteristic geometric parameter λ = H a ( H is the thickness of the strip and a is half the thickness of the stamp) in Problems 1 and 2, or λ = 2 ln ( b a ) ( a and b are the distances from the nearest and farthest points at which the stamp touches the boundary of the wedge to its vertex) in Problem 3, λ = 2 ln ( b a ) ( a is the inner radius of the stamp and b its outer radius) in Problem 4, λ = R a ( R is the radius of the cylinder and a half the thickness of the bandage) in Problem 5. In problems of coupled thermo-elasticity and consolidation theory λ also involves the parameter p of the Laplace transform with respect to the time coordinate /2/. The method is illustrated in relation to a contact problem for a strip continuously inhomogeneous in depth.

Dual integral equations with Fox′s H‐function kernel

The dual integral equations involving Bessel function kernels were first considered by Weber in 1873. The problem comprised of finding potential of an electrified disc which belongs to a general category of mixed boundary value problems. Titchmarsh gave the formal solution using Wiener‐Hopf procedure. We use this direct method as improvised by Busbridge to solve a class of dual integral equations which can be reduced to other known kernels by particularizing the parameters in the Fox′s H‐function.

Explicit solutions for a class of dual integral equations

SynopsisAn explicit solution is derived for the dual integral equationswhen A(ξ) takes the formIt is then shown how the basic technique can be adapted to derive explicit solutions for more general forms of A(ξ) such asandand linear combinations of such sums.

On a class of dual integral equations

Dual integral equations involving generalized Legendre functions and certain trigonometrical functions are considered. Solutions are obtained by using properties of generalized Legendre functions and the inversion theorem for the generalized Mehler — Fock transform.

On a class of dual integral equations

Dual integral equations involving generalized Legendre functions and certain trigonometrical functions are considered. Solutions are obtained by using properties of generalized Legendre functions and the inversion theorem for the generalized Mehler — Fock transform.

On a method of solving dual integral equations: PMM vol. 37, n≗6, 1973, pp. 1087–1097

We expound a method of reducing a class of dual integral equations which find important practical application to infinite algebraic systems of the first kind. The latter system can be reduced to the systems of the second kind by exact inversion of the principal singular part, and the second kind systems can be solved using the method of consecutive approximations [1–6], The dual integral equations generated by the Kontorovich-Lebedev and Mehler-Fock integral transforms are considered as examples as well as the problems of torsion of a truncated elastic sphere by a punch and that of a circular crack in an elastic space.

The solution of one class of dual integral equations connected with the Mehler-Fock transform in the theory of elasticity and mathematical physics: PMM vol. 33, n≗6, 1969, pp. 1061–1068

The solution of one class of dual integral equations connected with the Mehler-Fock transform in the theory of elasticity and mathematical physics: PMM vol. 33, n≗6, 1969, pp. 1061–1068

Function‐Theoretic Solution of a Class of Dual Integral Equations having Application in Supersonic Flow past a Body

Function‐Theoretic Solution of a Class of Dual Integral Equations having Application in Supersonic Flow past a Body

On a class of dual integral equations and their applications to the theory of elasticity

On a class of dual integral equations and their applications to the theory of elasticity

Function-theoretic solution to a class of dual integral equations and an application to diffraction theory

Dual integral equations of the type \[ ∫ 0 ∞ u λ f ( u ) J μ ( r u ) d u = g ( r ) , 0 > r > 1 , ∫ 0 ∞ u ( u 2 + a 2 ) − 1 / 2 f ( u ) J v ( r u ) d u = h ( r ) , 1 > r > ∞ , \int _0^\infty {{u^\lambda }f\left ( u \right ){J_\mu }\left ( {ru} \right )du = g\left ( r \right ),0 > r > 1, \\ \int _0^\infty u {{\left ( {{u^2} + {a^2}} \right )}^{ - 1/2}}f\left ( u \right ){J_v}\left ( {ru} \right )du = h\left ( r \right ),} 1 > r > \infty , \] where g ( r ) g\left ( r \right ) , h ( r ) h\left ( r \right ) are prescribed functions and f ( u ) f\left ( u \right ) is to be found, are solved exactly by the application of function-theoretic methods. As an example, a closed-form solution is obtained for the diffraction of an electromagnetic wave by a plane slit.

The Solution of Certain Dual Integral Equations

SummaryA class of dual integral equations involving Bessel functions is solved by formal application of Mellin transforms.