Nonlinear Singular Integro-differential Equations Research Articles

On One Point Singular Nonlinear Initial Boundary Value Problem for a Fractional Integro-Differential Equation via Fixed Point Theory

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order θ∈[0,1]. The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented.

Longitudinal Shear of a Composite Material in Which the Binder and Inclusions are Weakened by Cohesive Cracks

Background: An elastic medium is considered that is weakened by a doubly periodic system of round holes filled with washers made of a homogeneous elastic material and has cohesive cracks, the surface of which is uniformly covered with a cylindrical non-metallic film. The binder (medium) is weakened by two doubly periodic systems of rectilinear cohesive cracks, directed parallel to the abscissa and ordinate axes, and the length of the crack and their dimensions are not the same (different). General concepts are constructed that describe a class of problems with doubly periodic stress distribution outside circular holes and cohesive cracks under longitudinal shear. The analysis of the limit equilibrium of cracks within the framework of the end zone model is performed on the basis of a nonlocal failure criterion with a force condition for the advancement of the crack tip and a deformation condition for determining the advance of the edge of the end zone of the crack. In solving the problem, the main resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations obtained in each approximation were solved by the Gaussian method with the choice of the main element for different order values depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing crack growth. Objective: In the considered problem, the problem of longitudinal slip weakened by cracks parallel to the x axis in the filling medium with two pairs of two periodic cracks weakened by 2 periodic circular holes is considered. In the problem, the boundary problem along the contour of the circular holes is determined, and at the same time, the surface problem is established for the cracks parallel to the x and y axes. The solution of the problem is sought analytically, and taking into account the boundary condition, a system of linear algebraic equations for circular holes and 3 singular integral equations for cracks is obtained. Theoretical Framework: The problem is solved analytically, as the system of linear algebraic equations obtained for circular holes and the system of linear integral equations are jointly solved, and the stress intensity factor at the end of the cracks is determined. Method: The problem was solved using "Kolosova Muskhalashvili" functions and "Collondia's method of solving singular integral equations". Results and Discussion: The main essence of the considered issue is that the stress intensity factor at the end of the cracks is determined according to Fig. 1. Research Implications: For the first time, the stress intensity factor was calculated in composite materials with two pairs of biperiodic cracks of different lengths weakened by biperiodic round holes. Originality/Value: For the first time, the stress intensity coefficient was determined in composite materials with two periodic two pairs of cohesive cracks.

Thermal Stressed State of a Bimaterial with Interface Crack Filled with a Compressible Liquid

We formulate the problem of thermoelasticity for a bimaterial with interface crack filled with a liquid with regard for the pressure and heat conductivity of the liquid. We consider the case of partial crack closure on the edges under the action of a compressive load and a heat flux. The problem is reduced to a system of nonlinear singular integrodifferential equations for a temperature jump between the crack faces and its opening displacement. The length of the contact zone is determined from the condition of smooth closure of crack faces at the points separating the open and closed parts of the crack. We analyze the changes in the crack length and height and in the pressure of liquid depending on the compressive loads, density of the heat flux, and its direction. We also determine the dependence of the mode-II stress intensity factor on the level of pressure and the thermal conductivity coefficient of liquid.

Transverse shear of an isotropic elastic medium in the case when the binder and inclusions are weakened by crack initiation

An elastic medium is considered, weakened by a doubly periodic system of round holes, filled with absolutely rigid inclusions, soldered along the bypass and has a crack initiation. The medium (binder) is weakened by two periodic systems of rectilinear crack initiation directed collinear to the abscissa and ordinate axes, and their sizes are not the same. General representations are constructed that describe a class of problems with a doubly periodic stress distribution outside circular holes and cracks under transverse shear. The analysis of the limiting equilibrium of cracks in the framework of the end zone model is carried out on the basis of a nonlocal fracture criterion with a force condition for the propagation of the crack tip and a deformation condition for determining the advancement of the edge of the end zone of the crack. Basic resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations in each approximation were solved by the Gaussian method with the choice of the principal element for different values of the order of M, depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing the growth of cracks.

Investigation approach for a nonlinear singular Fredholm integro-differential equation

In this paper, we examine the existence and uniqueness of the solution of nonlinear integro-differential Fredholm equation with a weakly singular kernel. Then, we develop an iterative scheme to approach this solution using the product integration method. Finally, we conclude with a numerical tests to show the effectiveness of the proposed method.

Contact Problems for Elastic Plates with Finite-Length Nonlinearly Deformable Stringers Glued to Their Boundaries

The problem to determine the mechanical field in a homogeneous half-plane supported by a finite homogeneous stringer, material of which obeys the nonlinear Hooke’s law, is considered. The contact between the plate and stringer is realized by a thin glue layer. The posed problem is reduced to a nonlinear singular integrodifferential equation. Using the Schauder fixed-point principle, we prove the existence of a solution to this equation. The uniqueness of the solution of the problem is proved. Using the small-parameter method, we obtain a system of recurrence linear singular integral equations of the first kind.

ON THE DIRICHLET PROBLEM FOR AN ELLIP-TIC FUNCTIONAL-DIFFERENTIAL EQUATION WITH AFFINE TRANSFORMATION OF THE ARGUMENT

The paper considers the Dirichlet problem for an elliptic functionaldifferential equation containing a combination of shifts and contractions of the argument of an unknown function under the Laplace operator. Sufficient conditions for unique solvability are established. It is also shown that the problem can have an infinite-dimensional variety of solutions. This work is done in the framework of the State Contract in accordance with the Additional Agreement No. 075-03-2020-239/2 from 07.07.2020, registry No. 248, Budgetary Classification Code 01104730290059611, for the project titled ‘Non-linear singular integro-differential equations and boundary value problems’. The authors also acknoowledge support by the Russian Foundation for Basic Research (project no. 18-41-200001p-a).

Runge–Kutta convolution quadrature methods with convergence and stability analysis for nonlinear singular fractional integro-differential equations

The main concern of this paper is to develop and analyze the high-order Runge–Kutta convolution quadrature (RKCQ) method for obtaining the numerical solution of nonlinear fractional integro-differential equations (FIDEs) with weakly singular kernels. We first study the existence and uniqueness of solutions for the original problem. Then, the convergence and stability results of the RKCQ method are obtained. Finally, some numerical experiments are reported to illustrate the effectiveness of the proposed schemes.

Contact Interaction of a Plate with a Nonlinear Elastic Stringer

A problem for determining the mechanical field in a homogeneous half-plane supported by a finite homogeneous stringer, material of which obeys the nonlinear Hooke law, is considered. The posed problem is reduced to a nonlinear singular integro-differential equation. Using the Schauder fixed-point theorem, the existence of a solution for this equation is proved. Using the small parameter method, a system of recurrence linear singular integral equations of the first kind is obtained.

Nonlinear Singular Integro-Differential Equations with an Arbitrary Parameter

The maximally monotone operator method in real weighted Lebesgue spaces is used to study three different classes of nonlinear singular integro-differential equations with an arbitrary positive parameter. Under sufficiently clear constraints on the nonlinearity, we prove existence and uniqueness theorems for the solution covering in particular, the linear case as well. In contrast to the previous papers in which other classes of nonlinear singular integral and integro-differential equations were studied, our study is based on the inversion of the superposition operator generating the nonlinearities of the equations under consideration and the establishment of the coercitivity of the inverse operator, as well as a generalization of the well-known Schleiff inequality.

Non-local dynamics governing the self-induced motion of a planar vortex filament

While the Hasimoto planar vortex filament is one of the few exact solutions to the local induction approximation (LIA) approximating the self-induced motion of a vortex filament, it is natural to wonder whether such a vortex filament solution would exist for the non-local Biot-Savart dynamics exactly governing the filament motion, and if so, whether the non-local effects would drastically modify the solution properties. Both helical vortex filaments and vortex rings are known to exist under both the LIA and non-local Biot-Savart dynamics; however, the planar filament is a bit more complicated. In the present paper, we demonstrate that a planar vortex filament solution does exist for the non-local Biot-Savart formulation, provided that a specific non-linear integral equation (governing the spatial structure of such a filament) has a non-trivial solution. By using the Poincaré–Lindstedt method, we are able to obtain an accurate analytical approximation to the solution of this integral equation under physically reasonable assumptions. To obtain these solutions, we approximate local effects near the singularity of the integral equation using the LIA and non-local effects using the Biot-Savart formulation. Mathematically, the results constitute an analytical solution to an interesting nonlinear singular integro-differential equation in space and time variables. Physically, these results show that planar vortex filaments exist and maintain their forms under the non-local Biot-Savart formulation, as one would hope. Due to the regularization approach utilized, we are able to compare the structure of the planar filaments obtained under both LIA and Biot-Savart formulations in a rather straightforward manner, in order to determine the role of the non-locality on the structure of the planar filament.

Thermoelastic Interaction of the Bodies with Regular Surface Textures in the Presence of Heat-Permeable Media in the Intercontact Gaps

We study the thermoelastic contact of two isotropic semiinfinite bodies with periodic system of grooves on the surface of one of these bodies in the presence of heat-permeable media in the intercontact gaps. We reduce the formulated contact problem to a system of nonlinear singular integrodifferential equations for the height of gaps and the jump of temperature between their surfaces, which is solved numerically. With regard for the directional effect of the heat flow, we analyze the influence of the thermomechanical load and thermal conductivity coefficient of the filler on the contact parameters of the analyzed system.

Effect of Partial Closure of an Interface Crack with Heat-conducting Filler and Surface Films in the Case of Thermal Loading of a Bimaterial

We formulate the problem of thermoelasticity for a bimaterial body with interface crack with regard for the contact of its faces in the central part under the action of thermal strains caused by a heat flow perpendicular to the interface. The components of the bimaterial have different heat-conduction coefficients and linear coefficients of thermal expansion. The crack is filled with a heat-conducting medium and its faces have thermal resistance caused by the presence of thin surface films. The problem is reduced to a system of nonlinear singular integrodifferential equations for the jump of temperature between the crack faces and the crack opening displacements. For the solution of this problem, we develop an iterative algorithm based on the method of successive approximations. We analyze the influence of heat flow and thermal resistance of the films on the size of the contact region of the crack faces, crack opening displacements, jump of temperature between the crack faces, and the stress intensity factors for interface stresses.

Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations

This paper presents a computational method for solving a class of system of nonlinear singular fractional Volterra integro-differential equations. First, existences of a unique solution for under studying problem is proved. Then, shifted Chebyshev polynomials and their properties are employed to derive a general procedure for forming the operational matrix of fractional derivative for Chebyshev wavelets. The application of this operational matrix for solving mentioned problem is explained. In the next step, the error analysis of the proposed method is investigated. Finally, some examples are included for demonstrating the efficiency of the proposed method.

Periodic contact problem of thermoelasticity for bodies with rough surfaces in local regions

We study the thermoelastic interaction of two isotropic half planes with periodically located rough regions on the boundary of one of these planes. The influence of asperities on the heat exchange between the bodies is taken into account by the contact thermal resistance, which is inversely proportional to the contact pressure on the boundaries of the bodies. We reduce the contact problem to a nonlinear singular integrodifferential equation and propose an iterative algorithm for its solution. The influence of the density and direction of heat flow on the contact parameters of the analyzed structures is investigated.

Nucleation of cracks in perforated solids under longitudinal shear

The problem of fracture mechanics on nucleation of cracks emanating from circular contours of holes of perforated isotropic solids under longitudinal shear is under consideration. The solution of the equilibrium problem of a perforated solid under longitudinal shear with prefracture zones is reduced to solving an infinite algebraic system and a nonlinear singular integrodifferential equation with a kernel of the Cauchy type. Forces in crack nucleation zones are found by solving these equations. The condition of crack emergence is formulated taking into account the criterion of the limiting discontinuity of material displacements.

Cracks with interfacial bonds in the hub of a friction pair

The contact deformation of the hub of a plunger pair is considered. It is assumed that, during the repeated reciprocating motion of the plunger, fracture of the materials of the elements of the contact pair occurs. The friction surface of the bushing contains rectilinear cracks with the end zones characterized by the presence of interaction between the cracks faces. The boundary-value problem of equilibrium of the hub of a friction pair with a crack reduced to solving system of nonlinear singular integro-differential equations with a Cauchy type kernel.

Interaction of prefracture zones and crack-visible cavity in a burning solid with mixed boundary conditions

We consider local failure of a medium weakened by a crack-visible cavity, whose surfaces burns. It is assumed that in the vicinity of the technological defect (crack-visible cavity), there are areas of weakened interparticle bonds of the material, and mixed boundary conditions are given on the boundary of the fuel blasting cartridge. The mathematical description of the calculated model on the interaction of prefracture zones for opening and development of a crack-visible cavity in burning fuel is given. The determination of unknown parameters characterizing opening and growth of the crack-visible cavity is reduced to the solution of a system of nonlinear singular integrodifferential equations. The normal and shear forces in the prefracture zones and end zones of the crack-visible cavity are found from the solution of this system of equations. The condition of crack-visible cavity growth is formulated taking account of the criterion of the limit traction of the bonds in the material.

Effect of heat-conductive filler of interface gap on thermoelastic contact of solids

The thermoelastic contact of two isotropic solids separated by an interface gap is considered. The gap is formed due to an initial sloping smooth groove on the boundary of one of solids and filled with heat-conductive substance (gas or liquid). The heat is supposed to flow from the material with the higher thermal distortivity into the material with the smaller one. The gap filler influence on heat transfer between two solids is modeled by thermal resistance linearly dependent on the height of the gap. The conditions of perfect thermal contact and nonfrictional mechanical contact are assumed to be satisfied at the interface outside the gap. The contact problem is reduced to a set of two nonlinear singular integrodifferential equations, which is solved numerically. The effect of the imposed heat flow and heat conduction of the gap filler on the length and height of the gap, temperature jump at the interface, distribution of heat flows along and through the interface and distribution of contact stresses is analyzed.

Numerical model of a body containing a crack “healed” by injection

We construct a computational model of a body damaged by a crack and strengthened by using the injection technologies. As a distinctive feature of the model, we can mention the possibility of taking into account nonlinearity in the course of deformation of the material of the filler. The mathematical problem is reduced to the solution of a nonlinear singular integrodifferential equation. Under the assumption that the contour of the defect is elliptic, we obtain the exact solution of the integral equation one constant of which is found from the solution of a transcendental equation. We find the stresses acting in the material of the filler, the stress intensity factors, and the ultimate loads for bodies with cracks healed by injection materials whose stress–strain diagrams change from linear to nonlinear.