Integro-Differential Equations over a Closed Circuit with Gaussian Function in the Kernel

Integro-differential equations with kernels including hypergeometric Gaussian function that depends on the arguments ratio are studied over a closed curve in the complex plane. Special cases of the equations considered are the special integro-differential equation with Cauchy kernel, equations with power and logarithmic kernels. By means of the curvilinear convolution operator with the kernel of special kind, the equations with derivatives are reduced to the equations without derivatives. We find out the connection between special cases of the above-mentioned convolution operator and the known integral representations of piecewise analytical functions applied in the study of boundary value problems of the Riemann type. The correct statement of Noetherian property for the investigated class of equations is given. In this case, the operators corresponding to the equations are considered acting from the space of summable functions into the space of fractional integrals of the curvilinear convolution type. Examples of integro-differential equations solvable in a closed form are given.

     

The antiplane problem of electroelasticity for a cylinder with a tunnel crack excited by an arbitrary system of surface electrodes

The antiplane mixed boundary-value problem of electroelasticity of the oscillations of an infinite piezoceramic cylinder, weakened by a curvilinear tunnel crack, is considered. Using special integral representations of the solution, the boundary-value problem is reduced to a system of singular integro-differential equations of the second kind with discontinuous kernels. The results of a numerical realization of the algorithm, characterizing the amplitude-frequency characteristics of a piecewise-uniform cylinder and the behaviour of the components of the electroelastic field in the region and on the boundary of the cylinder under conditions of the inverse piezoelectric effect, are presented.     

Block boundary value methods for solving Volterra integral and integro-differential equations

Reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind Volterra integral equations and Volterra integro-differential equations. These extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for Volterra-type equations. Numerical experiments are presented and the efficiency, accuracy and stability of the schemes are confirmed.     

Integro-differential equations for the characteristic functional that are generated by a system of random integral equations

We consider a boundary value problem for a special system of integro-differential equations with variational derivatives. We establish the relationship between this problem and a system of integral equations with a power-law nonlinearity whose kernels and right-hand sides are random functions. We study the solvability of the boundary value problem. Special cases and examples are considered.     

Banach空间积-微分方程两点边值问题

本文利用一些新的微分和积分不等式研究了Ｂａｎａｃｈ空间中积－微分方程的 两点边值问题解的存在唯一性,给出了解的隐式迭代序列和误差估计式．     

Banach空间积-微分方程两点边值问题

本文利用一些新的微分和积分不等式研究了Ｂａｎａｃｈ空间中积－微分方程的 两点边值问题解的存在唯一性,给出了解的隐式迭代序列和误差估计式．     

INTEGRAL BOUNDARY VALUE PROBLEM FOR FIRST ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS

By developing a comparison result and using the monotone iterative technique, we obtain the existence of the minimal and the maximal solutions to an integral boundary value problem for first order impulsive integro-differential equations.     

Non-classical interface problems for piecewise homogeneous anisotropic elastic bodies

Two non-classical model interface problems for piecewise homogeneous anisotropic bodies are studied. In both problems on the contact surface jumps of the normal components of displacement and stress vectors are given. In addition, in the first problem (Problem H) the tangent components of the displacement vectors are given from both sides of the contact surface, while in the second one (Problem G) the tangent components of the stress vectors are prescribed on the same surface. The existence and uniqueness theorems are proved by means of the boundary integral equation method, and representations of solutions by single layer potentials are established. In the investigation the general approach of regularization of the first kind of integral equations is worked out for the case of two-dimensional closed smooth manifolds. An equivalent global regularizer operator is constructed explicitly in the form of a singular integro-differential operator.     

具积分边界条件的积—微分型参数方程的正解

本文讨论了一类以平板模型迁移方程为背景的具积分型边界条件的控制临界本征方程.藉助L_2空间上的线性算子理论,我们得到了这类方程的控制参数在实轴上的分布情况以及存在符合物理意义的正解的条件.     

Existence and uniqueness theorems in electromagnetic diffraction on systems of lossless dielectrics and perfectly conducting screens

New vector problem of electromagnetic wave diffraction by a system of non-intersecting three-dimensional inhomogeneous dielectric bodies and infinitely thin screens is considered in a quasiclassical formulation as well as the classical problem of diffraction by a lossless inhomogeneous body. In both cases, the original boundary value problem for Maxwell’s equations is reduced to integro-differential equations in the regions occupied by the bodies (and on the screen surfaces). The integro-differential operator is treated as a pseudodifferential operator in Sobolev spaces and is shown to be zero-index Fredholm operator. Uniqueness of solutions is proved under the realistic hypothesis of discontinuity of the dielectric permittivity the boundary of a volume scatterer. This result allowed to establish invertibility of the integro-differential operator in sufficiently broad spaces. For the problem of diffraction on dielectrics and surface conductors, theorem on smoothness of a solution is proved under assumption of data smoothness. The latter implies equivalence between the differential and integral formulations of the scattering problem. The matrix integro-differential operator is proved to be a Fredholm invertible operator. Thus, the existence of a unique solution to both problems is established.     

Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces

This paper investigates the existence of solutions for a class of second-order boundary-value problems with integral boundary conditions of nonlinear impulsive integro-differential equations in Banach spaces. The arguments are based upon the fixed point theorem of strict set contraction operators. Meanwhile, an example is worked out to demonstrate the main results.     

Integral boundary value problems for first order nonlinear impulsive functional integro-differential differential equations

In this paper, we consider the existence of solutions for first order nonlinear impulsive functional integro-differential equations with integral boundary value conditions. We firstly build a new comparison theorem. Then by utilizing the monotone iterative technique and the method of lower and upper solutions, we obtain the existence of extremal solutions or quasi-solutions.     

Some properties of $${\tau }$$-adic expansions on hyperelliptic Koblitz curves

This paper studies the existence of solutions for a six-point boundary value problem of coupled system of nonlinear Caputo (Liouville–Caputo) type sequential fractional integro-differential equations supplemented with coupled nonlocal Riemann–Liouville integral boundary conditions. Our results are based on some classical results of the fixed-point theory. An example is constructed to demonstrate the application of our work. Some interesting observations are also presented.     

BEM with linear complexity for the classical boundary integral operators

Alternative representations of boundary integral operators corresponding to elliptic boundary value problems are developed as a starting point for numerical approximations as, e.g., Galerkin boundary elements including numerical quadrature and panel-clustering. These representations have the advantage that the integrands of the integral operators have a reduced singular behaviour allowing one to choose the order of the numerical approximations much lower than for the classical formulations. Low-order discretisations for the single layer integral equations as well as for the classical double layer potential and the hypersingular integral equation are considered. We will present fully discrete Galerkin boundary element methods where the storage amount and the CPU time grow only linearly with respect to the number of unknowns.

     

Integral boundary value problems for first order integro-differential equations with deviating arguments

This paper deals with first order integro-differential equations of mixed type with deviating arguments. We investigate the existence of solutions of such problems with integral boundary conditions by establishing a comparison result and applying the monotone iterative technique. To obtain corresponding results, we also discuss first order differential inequalities with deviating arguments. Two examples demonstrate the application of our results.     

Boundary value methods for Volterra integral and integro-differential equations

New and effective quadrature rules generated by boundary value methods are introduced. We employ the introduced quadrature rules to construct quadrature methods for the second kind Volterra integral equations and Volterra integro-differential equations. These methods are shown to be effective and possess excellent convergence properties. The nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme. Numerical simulations are presented and confirm the efficiency and accuracy of the methods.     

Boundary-value problem for a parabolic system of integro-differential equations with integral conditions

Using operators of fractional integration and differentiation, we prove a theorem on the wellposedness of a general parabolic boundary-value problem for a system of integro-differential equations with integral operators in boundary conditions. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1610 – 1618, December, 2008.     

Boundary Value Problems for Quasielliptic Equations in a Half-Space

We consider general boundary value problems with data on the boundary of a half-space for quasielliptic equations with constant coefficients. We find integral representations for solutions and study some properties of the kernels of the corresponding integral operators. The results are applied to proving some generalization of the Miranda-Agmon maximum principle.     

Integral boundary value problem for impulsive integro-differential equations in Banach spaces

In this paper, by using the cone theory and monotone iterative technique, we investigate the existence of extremal solutions and unique solution of the integral boundary value problem for a class of first-order impulsive integro-differential equations in a real Banach space. An explicit iterative scheme for the unique solution and an error estimate of the approximation sequence are also derived.     

Key Continuity Equations in Stresses for Axisymmetric Problems of Elasticity and Thermoelasticity

Two key integro-differential continuity equations are derived in stresses for axisymmetric quasi-static problems of elasticity and thermoelasticity in unbounded domains. The possibility of equivalent reduction of the integro-differential equations to the corresponding differential equations is demonstrated. Boundary and integral conditions for solving those equations are formulated.