Search

In this note we present that the optimal investment decision can be characterized by two boundaries that satisfy a system of coupled integral equations of Fredholm type and propose a numerical algorithm for resolution.

A boundary value problem for the equations of heat and mass transfer in a half-space is considered, which contains in the boundary conditions derivatives of a higher order than the order of the derivative of a system of equations. The solution of the boundary problem is found in the form of a double layer potential. The lemma on finding the limit of the normal derivative of the finite order of the function in the neighborhood of the hyperplane xn = 0 is given. Using the boundary conditions, a system of integro-differential equations (SIDE) is obtained. The characteristic part of the SIDE is solved by the method of Fourier-Laplace integral transforms. The conditions for the correctness and incorrectness of the problem, expressed in terms of the given constants and boundary conditions, are found. Using the regularization method, the SIDE is reduced to a system of Volterra-Fredholm integral equations. A theorem on the solvability of the boundary value problem for a parabolic system of heat and mass transfer equations is presented.

The Boundary Element Method is often used in the theory of plate bending. Considered problem is a coupled problem of the fluid-structure type. A plate is immersed partially or totally in the fluid (water). It is assumed that the fluid is inviscid and incompressible. The boundary and domain integral equations are used to describe the plate motion and the hydrodynamic pressure of the surrounding liquid. The set of constant boundary elements for a plate and internal collocation points associated with lumped masses associated with rectangular sub-surfaces are introduced. Present formulation leads to the fully populated liquid mass matrix, which is adjoined to a diagonal plate mass matrix.

In this paper the path integral method developed for dynamic systems driven by the Poisson impulse process (by Di Paola and Santoro [2]) is generalized to dynamic systems driven by Erlang renewal impulse processes. Rather than the conventional joint probability density of the response, the joint probability density-distribution function of the continuous-jump response process and of the Markov states of the auxiliary pure jump process (pertinent to the underlying Erlang renewal process) is introduced. Accordingly, rather than the usual single Chapman-Kolmogorov-Smoluchowski integral equation, the set of integral equations is formulated. The number of these equations is equal to the number of Markov states of the auxiliary pure jump process.

In this paper we propose new methods of interpretation and verification of interval solutions obtained using interval parametric integral equations systems (interval PIES). These strategies concern the examples with the analytical solutions as well as more sophisticated examples. To obtain errors of precisely defined solutions (for comparison with interval ones), we base on well known methods of errors calculation. All of considered boundary value problems examples are described by Laplace’s equation. For modeling the uncertainty of boundary value problems we propose the modified directed interval arithmetic, which was previously applied in mentioned interval PIES method.In this paper we propose new methods of interpretation and verification of interval solutions obtained using interval parametric integral equations systems (interval PIES). These strategies concern the examples with the analytical solutions as well as more sophisticated examples. To obtain errors of precisely defined solutions (for comparison with interval ones), we base on well known methods of errors calculation. All of considered boundary value problems examples are described by Laplace’s equation. For modeling the uncertainty of boundary value problems we propose the modified directed interval arithmetic, which was previously applied in mentioned interval PIES method.

The Nystrom-Clenshaw-Curtis (NCC) quadrature, which was proposed in [S. Y. Kang, I. Koltracht, and G. Rawitscher, Math. Comp. 72, 729–756 (2003)], is a highly accurate numerical method for solving integral equations with semi-smooth kernel. In this paper, we introduce the basic idea of the NCC quadrature and derive an NCC quadrature for Volterra integral equations with proportional delays. Numerical results are presented to illustrated the high accuracy of the method.

The main purpose of this paper is to increase accuracy of modeling uncertainly defined boundary shape in parametric integral equations systems method (PIES), by inclusion of interval NURBS curves into mathematical formalism of the method. We model the uncertainty of the boundary shape by interval control points and examine such strategy on examples of 2D elasticity problems. Accuracy improvement of the boundary shape modeling affects the solutions accuracy. Additionally using interval NURBS we make the modeling process much easier and ensure the continuity of the shape.The main purpose of this paper is to increase accuracy of modeling uncertainly defined boundary shape in parametric integral equations systems method (PIES), by inclusion of interval NURBS curves into mathematical formalism of the method. We model the uncertainty of the boundary shape by interval control points and examine such strategy on examples of 2D elasticity problems. Accuracy improvement of the boundary shape modeling affects the solutions accuracy. Additionally using interval NURBS we make the modeling process much easier and ensure the continuity of the shape.

A numerical solution of the Fredholm integral equations can be obtained by many methods. Most of them lead to a solution of a system of linear equations with fully populated matrices. In the case of collocation or product integration methods, each element of the matrix is an integral, which needs to be calculated. It causes high computing time in multidimensional problems. Computing time can be reduced by the Nystrom method. It is based on substitution of the integral by a numerical integration rule. It has the advantage that only diagonal elements of the matrix are integrals. When the kernel function is singular, a singularity subtraction is needed. However it can not be used for every kernel function and every integration rule. The main point of this paper is the convergence conditions of the Nystrom method as applied to a special multidimensional integral equation. The paper includes an illustrative example.

In this paper, we deal with the application of Successive Over-Relaxation (SOR) iterative method for solving fuzzy Fredholm integral equations of the second kind (FFIE-2). In addition to that, we apply the trapezoidal rule to derive the approximate solution of FFIE-2 which consists of a system of integral equations. Next, the approximate equation is used to develop a system of linear equations. Then, we consider SOR iterative method to solve the generated system of linear equations. Next, SOR iterative method is implemented on some numerical examples. Finally, the numerical results is discussed in details by comparing the number of iterations, the computational time, and the Hausdorff distance to analyze the performance of proposed method. Based on the numerical results obtained from all the numerical examples by using Gauss-Seidel (GS) and SOR methods, it can be pointed out that SOR method is more efficient than the GS method.

The goal of this paper is to give a useful method for solving problems formulated by two dimensional stochastic integral equations driven by space-time white noise. Here, we consider two dimensional block pulse functions (2-D BPFs) and their operational matrix and stochastic operational matrix of integration. Since solution of parabolic partial differential equations driven by space-time white noise leads to a stochastic integral equations, we introduced a new method by 2-D BPFs and their operational integration matrix to transform a stochastic mixed Volterra-Fredholm integral equation to a system of algebraic equations. The benefit of this method is lower cost of setting up the system of equations without any integration. So, the computational cost of operations is low. The method is applied to two test examples to illustrate the accuracy and implementation of the method.

The Szegӧ kernel plays an important role in conformal mapping and satisfies a boundary Kerzman-Stein integral equation. The series representation for the Szegӧ kernel for an annulus region is well known. In this paper, the Adomian decomposition method is applied to solve the boundary Kerzman-Stein integral equation. The method provides another series form for the Szegӧ kernel for the annulus region. The two series are shown to be equivalent. Some numerical examples are also presented to compare the speed of convergence of the two series.

The main goal of this study is to provide an operator-pencil framework for the investigation of many-interval boundary-value-transmission problems (BVTP) with eigenparameter appearing in the boundary-transmission conditions. By applying an our own approaches the considered problem is transformed into an eigenvalue problem for suitable integral equation in terms of which it is defined a concept of generalized eigenfunctions. We introduce some self-adjoint compact operators in suitable Sobolev spaces such a way that the considered problem can be reduced to an operator-pencil equation. Finally, it is shown that the spectrum is discrete and the set of generalized eigenfunctions form a Riesz basis of the suitable Hilbert space.

The paper presents a numerical method for processing a large amount of data in detection of anomalies of the chemical elements distribution in polymetallic deposits. Mathematically, the problem reduces to solving the Fredholm integral equation of the first kind for large number of different right-hand parts, while the kernel remains unchanged. The algorithm consists of two stages: at the first stage a number of problems are solved, for which knowing the right part is not required, in the second stage, the previously obtained data are used to find solutions of the integral equation for different right-hand sides. This approach allows the data processing “in-situ” and to define prospects of the established abnormal areas, allocate the overriding areas for geological exploration. The algorithm is tested on model and real data.

The axisymmetric contact problem with fretting wear for an elastic foundation with a longitudinally nonuniform (surface nonuniform) coating and a rigid punch with a rough foundation has been solved for the first time. The case of linear wear is considered. The nonuniformity of the coating and punch roughness are described by a different rapidly changing functions. This strong nonuniformity arises when coatings are deposited using modern additive manufacturing technologies.The problem is reduced the solution of an integral equation with two different integral operators: a compact self-adjoint positively defined operator with respect to the coordinate and the non-self-adjoint integral Volterra operator with respect to time. The solution is obtained in series using the projection method of the authors. The efficiency of the proposed approach for constructing a high-accuracy approximate solution to the problem (with only a few expansion terms retained) is demonstrated.

The Ward-Tordai integral equation governs the diffusion-controlled surfactant adsorption at air/liquid interfaces. In this paper the Ward-Tordai equation is generalized in two directions. First, the adsorption is assumed to take place at a liquid/liquid interface, where the surfactant is soluble in both liquid phases. Second, the diffusion in the bulk phases is anomalous and is governed by time-fractional diffusion equations. For the computation of the change of adsorption with time two numerical techniques are proposed and compared. Numerical results are presented.

Let 0 0, f″(u) > 0 for u > 0. By using Green’s function, the problem is converted into an integral equation. It is shown that there exists tb such that for 0 ≤ t 0, f″(u) > 0 for u > 0. By using Green’s function, the problem is converted into an integral equation. It is shown that there exists tb such that for 0 ≤ t < tb, the integral equation has a unique nonnegative continuous solution u; if tb is finite, then u is unbounded in [0, tb). Then, u is proved to be the solution of the original problem.

The extraction of negative ions request conditions such as electrons are magnetized, so that a collisionless model is clearly overoptimistic, and collisions must be included to model electron co-extraction (an undesired effect). Collision dependence from speed is fully treated in this paper, addressing also the difficult case of Coulomb collisions. The kinetic effects are important in the sheath and presheath region, giving integral equations for electron density, current and collision variation along extraction axis z. With a careful description of the closed orbit variables and of the recirculation effect, integral kernels are reduced to expression suitable for computation. Numerical results for the plasma conductivity and average emission angle in Coulomb collision dominated regime are given and compared with similar results of constant collision frequency models.The extraction of negative ions request conditions such as electrons are magnetized, so that a collisionless model is clearly overoptimistic, and collisions must be included to model electron co-extraction (an undesired effect). Collision dependence from speed is fully treated in this paper, addressing also the difficult case of Coulomb collisions. The kinetic effects are important in the sheath and presheath region, giving integral equations for electron density, current and collision variation along extraction axis z. With a careful description of the closed orbit variables and of the recirculation effect, integral kernels are reduced to expression suitable for computation. Numerical results for the plasma conductivity and average emission angle in Coulomb collision dominated regime are given and compared with similar results of constant collision frequency models.

The 2-D problem of an infinite elastic solid with a nanoinclusion of a different from circular shape is solved. The interfacial stresses are acting at the interface. Contact of the inclusion with the matrix satisfies the ideal conditions of cohesion. The generalized Laplace – Young law defines conditions at the interface. To solve the problem, Gurtin – Murdoch surface elasticity model, Goursat – Kolosov complex potentials and the boundary perturbation method are used. The problem is reduced to the solution of two independent Riemann – Hilbert’s boundary problems. For the circular inclusion, hypersingular integral equation in an unknown interfacial stress is derived. The algorithm of solving this equation is constructed. The influence of the interfacial stress and the dimension of the circular inclusion on the stress distribution and stress concentration at the interface are analyzed.The 2-D problem of an infinite elastic solid with a nanoinclusion of a different from circular shape is solved. The interfacial stresses are acting at the interface. Contact of the inclusion with the matrix satisfies the ideal conditions of cohesion. The generalized Laplace – Young law defines conditions at the interface. To solve the problem, Gurtin – Murdoch surface elasticity model, Goursat – Kolosov complex potentials and the boundary perturbation method are used. The problem is reduced to the solution of two independent Riemann – Hilbert’s boundary problems. For the circular inclusion, hypersingular integral equation in an unknown interfacial stress is derived. The algorithm of solving this equation is constructed. The influence of the interfacial stress and the dimension of the circular inclusion on the stress distribution and stress concentration at the interface are analyzed.

In this study, the Two-point Explicit Group Successive Over Relaxation (2-EGSOR) iterative method is used to solve a system of linear equations generated from the discretization of system of second kind Fredholm integral equations. The discretization process is done by using quadrature scheme. The formulation and implementation of proposed iterative methods are also presented. Then numerical experiments are carried out onto three test problems to verify the effectiveness of the Gauss-Seidel (GS), Successive Over Relaxation (SOR) and 2-EGSOR methods. The numerical results shown that the 2-EGSOR iterative method with quadrature scheme is more efficient compared to GS and SOR methods.

A numerical method for solving two-dimensional nonlinear Volterra fuzzy integral equation (2D-NVFIE) of the second kind will is introduced. We convert a nonlinear Volterra fuzzy integral equation to a nonlinear system of Volterra integral equation in crisp case. We use Adomian Decomposition Method (ADM) to find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the nonlinear Volterra fuzzy integral equation. Also, the existence and uniqueness of the solution and convergence of the proposed method are proved. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

In this paper, we study some properties of a class of integral operators that depends on the cosine and sine Fourier transforms. In particular, we will exhibit properties related with their invertibility, the spectrum, Parseval type identities and involutions. Moreover, a new convolution will be proposed and consequent integral equations will be also studied in detail. Namely, we will characterize the solvability of two integral equations which are associated with the integral operator under study. Moreover, under appropriate conditions, the unique solutions of those two equations are also obtained in a constructive way. published

In this paper, the systematization of numerical (implemented on a computer) randomized functional algorithms for approximation of a solution of Fredholm integral equation of the second kind is carried out. Wherein, three types of such algorithms are distinguished: the projection, the mesh and the projection-mesh methods. The possibilities for usage of these algorithms for solution of practically important problems is investigated in detail. The disadvantages of the mesh algorithms, related to the necessity of calculation values of the kernels of integral equations in fixed points, are identified. On practice, these kernels have integrated singularities, and calculation of their values is impossible. Thus, for applied problems, related to solving Fredholm integral equation of the second kind, it is expedient to use not mesh, but the projection and the projection-mesh randomized algorithms.

We study a numerical approach for solving integral equation with adjoint generalized Neumann kernel related to conformal mapping. Previously, computation of conformal mapping of M + 1 connected regions require solving at least M + 1 integral equations with adjoint generalized Neumann kernel separately. We apply global simpler GMRES which solve nonsymmetric system with multiple right-hand sides to solve M + 1 integral equations simultaneously. We also apply fast multipole method for several matrix vector products in every iteration of global simpler GMRES. Numerical example is given to illustrate the effectiveness of the proposed method.

Cauchy type singular integral equations with index zero naturally occur in the field of aerodynamics. Literature is very much developed for these equations and Chebyshevs polynomials are most frequently used to solve these integral equations. In this paper, a residual based Galerkins method has been proposed by using Legendre polynomial as basis functions to solve Cauchy singular integral equation of index zero. It converts the Cauchy singular integral equation into system of equations which can be easily solved. The test examples are given for illustration of proposed numerical method. Error bounds are derived as well as implemented in all the test examples.

Based fitting parametrization and transformations of the exact non-linear integral equation for the unified function E of radiation exit from a homogeneous slab of an arbitrary optical thickness, approximate analytical expressions have been obtained for the corresponding slab brightness coefficients in the case of strongly elongated phase functions. The accuracy of the obtained fitting analytical approximations has been evaluated by using appropriate calculations in the visible spectrum region (λ = 400-800 nm) for Heyney-Greenstein phase functions in dependence of the single scattering albedo, vision angles, solar zenith distance and azimuths.

In recent studies of parametric integral equations system (PIES), the input data, necessary to define the shape of boundary, was defined in precise way. However, it is just assumption for further calculations. In practice even the most accurate measurement instruments generate errors. Therefore, in this paper we decide to propose the method for modelling and solving the boundary value problems with uncertainly defined shape of boundary. In view of advantages in precisely defined problems, we decide to generalize PIES method. To define the uncertainty of the input data we propose the modification of directed interval arithmetic.

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed.

This paper presents a boundary integral equation method with the adjoint generalized Neumann kernel for conformal mapping of unbounded multiply connected regions. The canonical region is the entire complex plane bounded by an infinite straight slit on the line Im ω = 0 and finite logarithmic spiral slits. Some linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on a multiply connected region. These integral equations are uniquely solvable. The kernel involved in these integral equations is the adjoint generalized Neumann kernel.

Mathematical models of boundary value problems of elasticity incorporating surface stress are reviewed. In the case of 2-D problems, the basic ideas and the complete set of equations are presented. It is shown singular and hypersingular integral equations which have been derived by solving some 2-D problems using complex potentials and boundary perturbation technique.

Ionic liquids are very important entry to industry and technology. Because of their unique properties they may classified as a new class of materials. IL usually classified as a high temperature ionic liquids (HTIL) and room temperature ionic liquids (RTIL). HTIL are molten salts. There are many research studies on molten salts such as recycling, new energy sources, rare elements mining. RTIL recently become very important in daily life industry because of their “green chemistry” properties. As a simple view ionic liquids consist of one positively charged and one negatively charged components. Because of their Coulombic or dispersive interactions the local structure of ionic liquids emerges. In this presentation the local structural properties of the HTIL are discussed via correlation functions and integral equation theories. RTIL are much more difficult to do modeling, but still general consideration for the modeling of the HTIL is valid also for the RTIL.

Eddy current testing to evaluate the condition of metallic parts in a sodium cooled fast reactor under standby conditions is challenging due to the presence of liquid sodium at 250 °C. The eddy current test system should be sensitive enough to capture small signal changes and hence an advanced inspection systems is needed. We have developed new hardware and improved numerical models to predict the eddy current probe signal due to cracks in metallic fast reactor parts by using volume integral equation method. The analytical expressions are derived for the quasi-static time-harmonic electromagnetic fields of a circular eddy current coil which interacts with conductive plate. Naturally, the method of moment is used to approximate the integral equation and obtain the discrete approximation of the field in the crack domain. A simple and accurate analytical method for dealing with the hyper-singularity element evaluation is also provided. An accurate controlled experiment is carried out on the ferromagnetic sta...

In this paper, we propose an efficient iterative method to solve two-dimensional nonlinear Hammerstein fuzzy functional integral equations based on fuzzy Haar wavelets. We prove the convergence of the method by Banach’s fixed point theorem and investigate the numerical stability of the presented method with respect to the choice of the first iteration. Finally, some numerical experiment confirm the theoretical results and illustrate the accuracy of the proposed method.

In this paper, the existence and uniqueness of the solution of a nonlinear singular integral equation, which has broad applications in the theory of elementary particles and scattering, have been investigated in Holder space. An iteration method is proposed for the approximate solution of this nonlinear singular integral equation. Furthermore, it is demonstrated that this iteration method converges.

The algorithm of Fredholm first kind integral equation solution which are used in electro-optics and dynamic light scattering is considered. The presented algorithm, unlike other algorithms aimed at solving the same problem, uses the peculiarities of the kernels of integral equations by applying additional constraints to the solution using specifically constructed penalty functions. This makes the precision of the solution higher for functions of one argument and allows to reliably solve integral equations with the solution as a function of two arguments.The algorithm of Fredholm first kind integral equation solution which are used in electro-optics and dynamic light scattering is considered. The presented algorithm, unlike other algorithms aimed at solving the same problem, uses the peculiarities of the kernels of integral equations by applying additional constraints to the solution using specifically constructed penalty functions. This makes the precision of the solution higher for functions of one argument and allows to reliably solve integral equations with the solution as a function of two arguments.

The numerical solution of a class of boundary value problems for nonlinear fractional differential equations is discussed. Using a suitable integral equation reformulation of the boundary value problem and spline collocation techniques on graded grids, high order methods for solving fractional boundary value problems are constructed.

An approximate analytical theory of weak spectral lines formation in a scattering and absorbing aerosol-molecular atmosphere bounded from below by an arbitrary horizontally uniform reflecting bottom has been considered taking into account the polarization effect of a radiation field in an optically thin atmosphere. The appropriate analytical approximations and parametrizations have been carried out for exact integral equations of polarized radiation within the above-mentioned system by using different correlations between the slab’s optical parameters and matrices of atmospheric scattering and reflecting underlying surface.

In this paper, we study the solvability of a singular integral equation arising in the theory of boundary value problems for the heat equation in an infinite angular domain. With the help of thermal potentials, the boundary value problem of heat conduction is reduced to a singular integral Volterra equation of the second kind. The corresponding homogeneous integral equation was investigated by us earlier in the previous paper of M.T. Jenaliyev and M.I. Ramazanov, and it was shown that in some weight class of essentially bounded functions it has, along with a trivial solution, a family of nontrivial solutions up to a constant factor. In this paper we study the nonhomogeneous integral equation, for which a representation of the general solution is found with the help of the resolvent constructed by us. Estimates of the resolvent are established.

Motivated by the fact that bounded variation (often discontinuous) functions frequently appear when studying integral equations that describe physical phenomena, we focus on the existence of bounded variation solutions for Urysohn integral measure driven equations. Due to numerous applications of Urysohn integral equations in various domains, problems of this kind have been extensively studied in literature, under more restrictive assumptions. Our approach concerns the framework of Kurzweil-Stieltjes integration, which allows the occurrence of high oscillatory features on the right hand side of the equation. A discussion about interesting consequences of our main result (given by particular cases of the measure driving the equation) is presented. Finally, we show the generality of our results by investigating two examples of impulsive type problems (from both theoretical and numerical perspective) and giving an application in electronics industry concerning polarization properties of ferroelectric materials.

The Laplace transform method, along with Computer Algebra Systems (CAS) “Maple” v. 13, are extremely successfully applied for solving a class of integral equations with an arbitrary order, including fractional order integral equations. The combining of both powerful approaches allows students more quickly, enjoyable and thoroughly to master the material.