The Price of a lookback option as the solution of a boundary-value problem for the heat equation

A lookback option is priced by solving the third boundary-value problem for the heat equation. The application of the Laplace transform makes it possible to represent the option price as a certain integral expressible in terms of the distribution of the first arrival time of a Brownian motion at a given level. Translated from Prikladnaya Matematika i Informatika, No. 28, pp. 66 – 72, 2008.     

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.     

Impression of a semiinfinite stamp in an elastic strip with regard for friction and adhesion

We consider the problem of contact interaction between a semiinfinite stamp with rectilinear base and an elastic strip with one rigid side. Friction forces in the contact region are taken into account. These forces lead to the division of the contact region into slipping and adhesion zones. With the use of the Wiener–Hopf method, a system of integral equations is reduced to an infinite system of algebraic equations. The computational results of stresses and strains at the boundary and at inner points of the elastic strip are presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 138–149, January–March, 2008.     

Determination of the temperatures and heat flows for a conical slide support taking account of frictional heating

We consider an axisymmetric problem of heat conduction taking account of frictional heating in a conetorus pair that models the functioning of a conical support. The bodies are pressed together and are rotating about a common axis. Heat is generated in the region of contact of the bodies due to frictional forces. Outside the region of contact there is heat exchange with the surrounding medium. The thermal contact between the two bodies is nonideal. The problem is reduced to a system of integral equations whose solution is constructed by the method of successive approximations. We give the results of numerical studies of the temperature distribution and heat flows from the geometric and thermophysical parameters of the body. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 19–27.     

Finite abelian group methods in iterative algorithms for solving boundary-value problems with a boundary surface of complex structure

We consider external boundary-value problems for the Laplace equation on surfaces of complex structure. Various iterative computational schemes are constructed for numerical solution of the corresponding integral equations using set-theoretical group methods. Simulation results of electron-optical images are reported. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 16–27.     

Contact with break-off under bending of an elastic strip with a rigid disk

We consider the problem of the theory of elasticity of the contact interaction of a rigid circular disk and an elastic strip, which rests upon two supports with disturbance of contact in the middle part of the contact region. On the basis of the Wiener–Hopf method, an integral equation of the problem is reduced to an infinite system of algebraic equations. The size of the zone of break-off of the boundary of the strip from the disk and the distribution of contact stresses are determined.     

An asymptotic method of solving transient dynamic contact problems of the theory of elasticity for a strip

An asymptotic method is proposed for solving transient dynamic contact problems of the theory of elasticity for a thin strip. The solution of problems by means of the integral Laplace transformation (with respect to time) and the Fourier transformation (with respect to the longitudinal coordinate) reduces to an integral equation in the form of a convolution of the first kind in the unknown Laplace transform of contact stresses under the punch. The zeroth term of the asymptotic form of the solution of the integral equation for large values of the Laplace parameter is constructed in the form of the superposition of solutions of the corresponding Wiener-Hopf integral equations minus the solution of the corresponding integral equation on the entire axis. In solving the Wiener-Hopf integral equations, the symbols of the kernel of the integral equation in the complex plane is presented in special form — in the form of uniform expansion in terms of exponential functions. The latter enables integral equations of the second kind to be obtained for determining the Laplace-Fourier transform of the required contact stresses, which, in turn, is effectively solved by the method of successive approximations. After Laplace inversion of the zeroth term of the asymptotic form of the solution of the integral equations, the asymptotic solution of the transient dynamic contact problem is determined. By way of example, the asymptotic solution of the problem of the penetration of a plane punch into an elastic strip lying without friction on a rigid base is given. Formulae are derived for the active elastic resistance force on the punch of a medium preventing the penetration of the punch, and the law of penetration of the punch into the elastic strip is obtained, taking into account the elastic stress wave reflected from the strip face opposite the punch and passing underneath it.     

A Pizzetti-type formula for the heat operator

We provide an asymptotic expansion of the integral mean of a smooth function over the Heat ball. Namely we generalize to the Heat operator the so-called Pizzetti’s Formula, which expresses the integral mean of a smooth function over an Euclidean ball in terms of a power series with respect to the radius of the ball having the iterated of the ordinary Laplace operator as coefficients. Similarly here, we express the heat integral mean as a power series with respect to the radius of the heat ball, whose coefficients are powers of a distorted heat operator. We also discuss sufficient conditions to have a finite sum. Received: 27 May 2005     

The elastoplastic stressed state of a half-plane under local nonstationary heating

We solve the thermoplastic problem for a semi-infinite plate under local nonstationary heating by heat sources. The physical equations are taken to be the relations of the nonisothermic theory of plastic flow associated with the Mises fluidity condition. The solution of the problem is constructed by the method of integral equations and the self-correcting method of sequential loading, where time is taken as the loading parameter. We carry out numerical computations of the stresses in the case of heating a plate with heat output by normal-circular heat sources. We study the problem of optimization of heating regimes in order to introduce favorable residual compressive stresses (from the point of view of hardness) in a given region of a half-plane. Two figures.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 29–34.     

Spectral gaps in the dirichlet and neumann problems on the plane perforated by a doubleperiodic family of circular holes

We show that the spectrum of the Dirichlet and Neumann problems for the Laplace operator in the plane perforated by a double–periodic family of circular holes contains gaps (even any a priori given number of gaps) of certain radii of holes. The result is obtained by asymptotic analysis of the cell spectral problem, interpreted as a problem in a domain with thin bridges. Some open questions are stated.     

A system of boundary integral equations for orthotropic shells of positive curvature with slits and holes

We propose a method of constructing a system of boundary integral equations for the problem of the stress state of an orthotropic shell with slits and holes. Using the theory of distributions and the two-dimensional Fourier transform, we reduce the problem to a system of boundary integral equations. In the solution obtained the kernels of the system of integral equations do not contain the direction cosines of the unit outward normal vector explicitly. There are no extra-integral terms. The matrix of the kernels is symmetric. The kernels are regular or have a logarithmic singularity. Two figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 59–69.     

Poincaré-Steklov integral equations and the Riemann monodromy problem

We consider the Poincaré-Steklov singular integral equation obtained by reducing a boundary value problem for the Laplace operator with a spectral parameter in the boundary condition to the boundary. It is shown that this equation can be restated equivalently in terms of the classical Riemann monodromy problem. Several equations of this type are solved in elliptic functions. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow Institute of Physical Engineering (Physical-Technical). Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 9–22, April–June, 2000. Translated by V. E. Nazaikinskii     

On a Certain Class of Integral Equations Associated with Hankel Transforms

This paper deals with a new class of Fredholm integral equations of the first kind associated with Hankel transforms of integer order. Analysis of the equations is based on operators transforming Bessel functions of the first kind into kernels of Weber–Orr integral transforms. Their inverse operators are established by means of new inversion theorems for the Hankel and Weber–Orr integral transforms of functions belonging to L ¹ and L ². These operators together with the proven Paley–Wiener’s theorem for the Weber–Orr transform enable to regularize the equations and, in special cases, to derive explicit solutions. The integral equations analyzed in this paper can be employed instead of dual integral equations usually treated with the Cooke–Lebedev method. An example manifests that it may be preferable because of the possibility to control norms of operators in the regularized equations.      

On the determination of stress concentration in a stretched plate with two holes

We present a short survey of studies of the elastic interaction of two holes in a stretched plate. Special attention is paid to the study of the concentration of stresses on the contours of closely positioned holes. For two identical elliptic holes, numerical results are obtained by the method of singular integral equations. With the help of the limit transition, we determined the stress intensity factors at the vertices of semi-infinite parabolic notches. A comparison of the numerical data with known analytic solutions for two circular holes and collinear cracks is performed.     

Pressure of an elastic half space on a rigid base with rectangular hole in the case of a liquid bridge between them

A model of contact between an elastic half space and a rigid base with a shallow surface rectangular hole is proposed. The hole contains an incompressible liquid and gas. The liquid occupies the middle part of the hole and forms a capillary bridge between the opposite surfaces. The remaining volume of the hole is filled with gas under a constant pressure. The liquid completely wets the surfaces of the bodies. The pressure drop at the liquid–gas interface caused by the surface tension is defined by the Laplace formula. The corresponding plane contact problem for the elastic half space is essentially nonlinear because the pressure of the liquid and the length of the capillary in the contact-boundary conditions are not known in advance and depend on the external load. The problem is reduced to a system of three equations (a singular integral equation for the function of height of the hole and two transcendental equations for the length of the capillary and the height of the meniscus). An analytic-numerical procedure for the solution of these equations is proposed. Dependences of the length of the capillary and the pressure drop at the liquid–gas interface on the external load, volume of liquid, and its surface tension are analyzed. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 150–156, January–March, 2008.     

A heat transfer problem with exponential memory and the associated thermal stresses

In this study, a heat transfer problem defined by the Caputo–Fabrizio derivative, which is known to behave by the exponential decaying law, is addressed in an axially symmetric cylindrical region. Thus, the fundamental solutions of the heat diffusion process and the associated thermal stresses are aimed to find. For this purpose, Laplace and finite Hankel integral transforms are applied according to the geometry of the region. To obtain the thermal stresses, constitutive relations of the classical thermoelasticity theory are used. The effects of fractional orders on the diffusion process are illustrated graphically using MATLAB.     

Integration of equations of one-dimensional problems of elasticity and thermoelasticity for inhomogeneous cylindrical bodies

We propose a method for direct integration of differential equations of equilibrium and continuity in terms of stresses in the case of one-dimensional quasistatic problems of elasticity and thermoelasticity for inhomogeneous and thermosensitive isotropic cylindrical bodies. The solution of each of the one-dimensional problems is reduced to a Volterra integral equation of the second kind, which makes it possible to propose a rapidly convergent iteration method of computations. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, vol. 41, No. 2, pp. 124–131, April–June, 1998.     

A problem of generalized magneto-thermoelastic thin slim strip subjected to a moving heat source

The generalized thermoelastic theory with thermal relaxation, in the context of Lord and Shulman theory, is used to investigate the magneto-thermoelastic problem of a thin slim strip placed in a magnetic field and subjected to a moving plane of heat source. The generalized magneto-thermoelastic coupled governing equations are formulated. By means of the Laplace transform and numerical Laplace inversion, the governing equations are solved. Numerical calculations for the considered variables are performed and the obtained results are presented graphically. The effects of moving heat source speed and applied magnetic field on temperature, stress and displacement are studied. It is found from the graphs that the temperature, thermally induced displacement and stress in the strip are found to decrease at large heat source speed, and the magnetic field significantly influences the variations of non-dimensional displacement and stress. However, it has no effect on the non-dimensional temperature.     

An error analysis of Runge–Kutta convolution quadrature

An error analysis is given for convolution quadratures based on strongly A-stable Runge–Kutta methods, for the non-sectorial case of a convolution kernel with a Laplace transform that is polynomially bounded in a half-plane. The order of approximation depends on the classical order and stage order of the Runge–Kutta method and on the growth exponent of the Laplace transform. Numerical experiments with convolution quadratures based on the Radau IIA methods are given on an example of a time-domain boundary integral operator.     

The heat conduction problem for orthotropic shells with a system of cuts

Using the Fourier integral method we have solved the heat conduction problem for an orthotropic shell of arbitrary Gaussian curvature with a system of thermally insulated cuts. In the process we have taken account of heat exchange on the lateral surfaces of the shells. We have studied the influence of the anistropy properties of the material on the distribution of the perturbed temperature field. Using the example of a system consisting of two cuts we have studied the dependence of jumps in the integral characteristics of the temperature on the relative locations of the cuts. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 50–54.