Two-dimensional Integral Equations Research Articles

Mixed Fourier Legendre spectral Galerkin methods for two-dimensional Fredholm integral equations of the second kind

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations (fies) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems (evps) associated with the two-dimensional fies. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fies in both L2 and L∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L2 and L∞ norms. The numerical illustrations are introduced for the error of these methods.

A numerical method for two-dimensional Hammerstein integral equations

"In this paper we investigate a collocation method for the approximate solution of Hammerstein integral equations in two dimensions. As in [8], col- location is applied to a reformulation of the equation in a new unknown, thus reducing the computational cost and simplifying the implementation. We start with a special type of piecewise linear interpolation over triangles for a refor- mulation of the equation. This leads to a numerical integration scheme that can then be extended to any bounded domain in R2, which is used in collocation. We analyze and prove the convergence of the method and give error estimates. As the quadrature formula has a higher degree of precision than expected with linear interpolation, the resulting collocation method is superconvergent, thus requiring fewer iterations for a desired accuracy. We show the applicability of the proposed scheme on numerical examples and discuss future research ideas in this area."

Integral Models Based on Volterra Equations with Prehistory and Their Applications in Energy

The paper addresses the application of Volterra integral equations of the first kind for modeling dynamic power systems. We study the problem of forecasting the commissioning of capacities of the electric power system, taking into account various hypotheses about the dynamics of equipment aging, and the known prehistory. The numerical results of the application of two models to the problem of the development of a large electric power system using the example of the Unified Energy System of Russia are presented. Theoretical results were formulated for a two-dimensional Volterra integral equation of the first kind with variable limits of integration. This class of equations arises when solving the actual problem of identifying variable characteristics of a nonlinear dynamic system of the “input-output” type.

Chebyshev Spectral Projection Methods for Two-Dimensional Fredholm Integral Equations of Second Kind

In this paper, we approximate the two-dimensional linear Fredholm integral equations (fies) with smooth kernels using Chebyshev spectral Galerkin and collocation methods. The existence and convergence analysis for the problem have been discussed. The errors between approximated solutions with exact solution have been evaluated in $$L^2_\omega $$ norm applying both these methods. We also solve the associated eigenvalue problems (evps) by using above methods and obtain the errors between approximated eigenelements with the exact eigenelements in $$L^2_\omega $$ and $$L^\infty _\omega $$ norms. Numerical examples are presented to illustrate the theoretical results.

Solving systems of fractional two-dimensional nonlinear partial Volterra integral equations by using Haar wavelets

Abstract The main aim of this paper is to use the operational matrices of fractional integration of Haar wavelets to find the numerical solution for a nonlinear system of two-dimensional fractional partial Volterra integral equations. To do this, first we present the operational matrices of fractional integration of Haar wavelets. Then we apply these matrices to solve systems of two-dimensional fractional partial Volterra integral equations (2DFPVIE). Also, we present the error analysis and convergence as well. At the end, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

Numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types)

Purpose This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types). Design/methodology/approach The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations. Findings Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods. Originality/value The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

Nyström Method to Solve Two-Dimensional Volterra Integral Equation with Discontinuous Kernel

In this paper, a linear two-dimensional Volterra integral equation of the second kind with the discontinuous kernel is considered. The conditions for ensuring the existence of a unique continuous solution are mentioned. The product Nystrom method, as a well-known method of solving singular integral equations, is presented. Therefore, the Nystrom method is applied to the linear Volterra integral equation with the discontinuous kernel to convert it to a linear algebraic system. Some formulas are expanded in two dimensions. Weights’ functions of the Nystrom method are obtained for kernels of logarithmic and Carleman types. Some numerical applications are presented to show the efficiency and accuracy of the proposed method. Maple18 is used to compute numerical solutions. The estimated error is calculated in each case. The Nystrom method is useful and effective in treating the two-dimensional singular Volterra integral equation. Finally, we conclude that the time factor and the parameter v have a clear effect on the results.

Solving Fractional Two-Dimensional Nonlinear Partial Volterra Integral Equation by Using Bernoulli Wavelet

Solving Fractional Two-Dimensional Nonlinear Partial Volterra Integral Equation by Using Bernoulli Wavelet

Numerical solutions of two-dimensional nonlinear integral equations via Laguerre Wavelet method with convergence analysis

In this paper, the approximate solutions for two different type of two-dimensional nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis. Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.

Solutions of Fractional Differential Type Equations by Fixed Point Techniques for Multivalued Contractions

This paper involves extended b − metric versions of a fractional differential equation, a system of fractional differential equations and two-dimensional (2D) linear Fredholm integral equations. By various given hypotheses, exciting results are established in the setting of an extended b − metric space. Thereafter, by making consequent use of the fixed point technique, short and simple proofs are obtained for solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional linear Fredholm integral equation.

Two-dimensional Euler polynomials solutions of two-dimensional Volterra integral equations of fractional order

This paper proposes a method based on two-dimensional Euler polynomials combined with Gauss-Jacobi quadrature formula. The method is used to solve two-dimensional Volterra integral equations with fractional order weakly singular kernels. Firstly, we prove the existence and uniqueness of the original equation by Gronwall inequality and mathematical induction method. Secondly, we use two-dimensional Euler polynomials to approximate the unknown function of the original equation, and the Gauss-Jacobi quadrature formula is used to approximate the integrals in the original equation. Thirdly, we prove the existence and uniqueness of the solution of approximate equation, and the error analysis of the proposed method is given. Finally, some numerical examples illustrate the efficiency of the method.

Явные решения двумерного интегрального уравнения типа Вольтерра с граничной особой и сильно-особой линиями, когда корни характеристических уравнений вещественные, разные и равные

The paper studies a two-dimensional integral equation of Volterra type with singular and strongly singular boundary lines. The solution of a two-dimensional integral equation of the Volterra type with singular kernels is sought in the class of continuous functions that vanish on the boundary lines. In the case when the roots of the characteristic equations are real, different and equal, the parameters of the equations are related to each other in a certain way, depending on the roots of the characteristic equations and the sign of the parameters of the integral equation, explicit solutions of the integral equation are found. It is proved that the solutions of a two-dimensional integral equation, depending on the sign of the parameters, can contain from one to four arbitrary continuous functions. The cases are determined when the solution to the integral equation is unique.

Представление решения задачи Гурса для линейных стохастических гиперболических дифференциальных уравнений второго порядка

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

A new generalized definition of singular kernel for two-dimensional Fredholm integral equation

A new generalized definition of singular kernel for two-dimensional Fredholm integral equation

Two queues with time-limited polling and workload-dependent service speeds

In this paper, we study a single-server polling model with two queues. Customers arrive at the queues according to two independent Poisson processes. The server spends random amounts of time in each queue, regardless of the amounts of work present at the queues. The service speed is not constant; it is assumed that the server works at speed at queue i when its current workload equals xi , i = 1, 2. We first focus on the case that all visit times are constant. In the two-queue case, we then compute the Laplace–Stieltjes transform (LST) of the steady-state joint workload distribution. Using a different method in which we exploit the independence of the workloads at visit endings, we compute the joint LST of workloads in the case of an arbitrary number of queues with constant visit times. Next, we consider a two-queue polling model with constant visit times at the first queue and general visit times at the second, and we derive the marginal workload distribution at the first queue. We also investigate the case of a two-queue polling model with exponentially distributed visit times. We determine the steady-state marginal workload distributions, and we formulate a two-dimensional Volterra integral equation for the LST of the steady-state joint workload distribution. We finally show that this equation can be solved by a fixed-point iteration.

A Highly Efficient and Accurate Finite Iterative Method for Solving Linear Two-Dimensional Fredholm Fuzzy Integral Equations of the Second Kind Using Triangular Functions

This work introduces a computational method for solving the linear two-dimensional fuzzy Fredholm integral equation of the second form (2D-FFIE-2) based on triangular basis functions. We have used the parametric form of fuzzy functions and transformed a 2D-FFIE-2 with three variables in crisp case to a linear Fredholm integral equation of the second kind. First, a method based on the use of two m-sets of orthogonal functions of triangular form is implemented on the integral equation under study to be changed to coupled algebraic equation system. In order to solve these two schemes, a finite iterative algorithm is then applied to evaluate the coefficients that provided the approximate solution of the integral problems. Three examples are given to clarify the efficiency and accuracy of the method. The obtained numerical results are compared with other direct and exact solutions.

Numerical solution of two-dimensional nonlinear fuzzy delay integral equations via iterative method and trapezoidal quadrature rule

In recent years, fuzzy delay integral equation has received a considerable amount of attention. From the viewpoint of engineering, time delays in controllers may induce complex dynamic behavior and will limit the performance of the active control. Therefore, time delays play an important role in the dynamic behavior of active control systems. In our paper, we introduce an iterative method using trapezoidal quadrature formula for solving two-dimensional nonlinear fuzzy delay integral equations. In addition, the existence and uniqueness of the solution of these equations are proved. Also, the convergence analysis and a criterion to stop of the presented method are investigated. Moreover, we prove numerical stability analysis of the method through choosing the perturbation in the first iteration. Eventually, some numerical examples are provided to demonstrate the applicability of this procedure.

Solving two-dimensional fuzzy Fredholm integral equations via sinc collocation method

In this paper, we present a numerical method to solve two-dimensional fuzzy Fredholm integral equations (2D-FFIE) by combing the sinc method with double exponential (DE) transformation. Using this method, the fuzzy Fredholm integral equations are converted into dual fuzzy linear systems. Convergence analysis is performed in terms of the modulus of continuity. Numerical experiments demonstrate the efficiency of the proposed method.

Numerical reconstruction of two-dimensional particle size distributions from laser diffraction data

In this paper, we consider the two-dimensional Fredholm integral equation of the first kind. The kernel of this equation models the scattering of a laser beam by spheroids having the same fixed orientation. The unknown function under the integral describes the distribution of spheroids along two semi-axes. The input data are the diffraction pattern corresponding to the scattering of the laser beam by the particles. We show that the Tikhonov regularization method allows one to reconstruct two-dimensional distributions in the case when the diffraction pattern is modulated by white noise with relative amplitude up to 1%. In applications, this means novel possibility of obtaining two-dimensional particle size distributions, rather than one-dimensional ones, as in the classical version of the method. This significantly expands the capabilities of particle sizing via static laser diffraction technique. We show that the solution of the corresponding equation is unique in space and exists when the right-hand-side function is in a set dense in . We provide test experimental results in the framework of laser ektacytometry of red blood cells, which serves as the main application of the proposed approach presently.