Two-dimensional Integral Equations Research Articles

Solvability of Two-Dimensional Integral Equations with Concave Nonlinearity in the Plane

Solvability of Two-Dimensional Integral Equations with Concave Nonlinearity in the Plane

Error Estimates of a Symmetric Spectral Method for a Linear Volterra Integral Equation

A symmetric spectral method is applied to investigate the two-dimensional Volterra integral equation with weakly singular kernels and delays. In this work, the solution of the equation we considered is assumed to be sufficiently smooth so that the spectral method can be applied naturally. Employing three couples of variable transformations, we apply the two-dimensional Gauss quadrature rule to approximate the weakly singular integral with delays and obtain the spectral discretization. Then we derive the convergence results of the proposed approximation scheme. We show that the errors of solution decay exponentially in both the infinity norm and weighted square norm. In the end, we carry out numerical experiments to verify the theoretical results.

Numerical solution of two-dimensional Hammerstein integral equations via quadratic spline collocation

Numerical solution of two-dimensional Hammerstein integral equations via quadratic spline collocation

On the Constructive Solvability of a Two-Dimensional Nonlinear Integral Equation Arising in the Theory of $$p$$-Adic Strings

On the Constructive Solvability of a Two-Dimensional Nonlinear Integral Equation Arising in the Theory of $$p$$-Adic Strings

Methods and effective algorithms for solving multidimensional integral equations

Objectives. Integral equations have long been used in mathematical physics to demonstrate existence and uniqueness theorems for solving boundary value problems for differential equations. However, despite integral equations have a number of advantages in comparison with corresponding boundary value problems where boundary conditions are present in the kernels of equations, they are rarely used for obtaining numerical solutions of problems due to the presence of equations with dense matrices that arise that when discretizing integral equations, as opposed to sparse matrices in the case of differential equations. Recently, due to the development of computer technology and methods of computational mathematics, integral equations have been used for the numerical solution of specific problems. In the present work, two methods for numerical solution of two-dimensional and three-dimensional integral equations are proposed for describing several significant classes of problems in mathematical physics.Methods. The method of collocation on non-uniform and uniform grids is used to discretize integral equations. To obtain a numerical solution of the resulting systems of linear algebraic equations (SLAEs), iterative methods are used. In the case of a uniform grid, an efficient method for multiplying the SLAE matrix by vector is created.Results. Corresponding SLAEs describing the considered classes of problems are set up. Efficient solution algorithms using fast Fourier transforms are proposed for solving systems of equations obtained using a uniform grid.Conclusions. While SLAEs using a non-uniform grid can be used to describe complex domain configurations, there are significant constraints on the dimensionality of described systems. When using a uniform grid, the dimensionality of SLAEs can be several orders of magnitude higher; however, in this case, it may be difficult to describe the complex configuration of the domain. Selection of the particular method depends on the specific problem and available computational resources. Thus, SLAEs on a non-uniform grid may be preferable for many two-dimensional problems, while systems on a uniform grid may be preferable for three-dimensional problems.

Combined Legendre Spectral-Finite Element Methods for Two-Dimensional Fredholm Integral Equations of the Second Kind

In this paper, we will discuss on the Combined Legendre spectral-Finite element methods (CLSFEM) for the two-dimensional Fredholm integral equations with smooth kernel on the Banach spaces and the corresponding eigenvalue problem. In these methods, the approximated finite dimensional space is the cartesian product of spline space and Legendre polynomial space. The problem is approximated by the CLSFEM using orthogonal projection, which projects from the Banach space into the finite dimensional space. The convergence analysis for both Fredholm integral equations and the corresponding eigenvalue problem will be discussed in both L 2 and norms. The numerical results will be shown to validate the theoretical estimate.

A New Four-Step Iterative Procedure for Approximating Fixed Points with Application to 2D Volterra Integral Equations

This work is devoted to presenting a new four-step iterative scheme for approximating fixed points under almost contraction mappings and Reich–Suzuki-type nonexpansive mappings (RSTN mappings, for short). Additionally, we demonstrate that for almost contraction mappings, the proposed algorithm converges faster than a variety of other current iterative schemes. Furthermore, the new iterative scheme’s ω2—stability result is established and a corroborating example is given to clarify the concept of ω2—stability. Moreover, weak as well as a number of strong convergence results are demonstrated for our new iterative approach for fixed points of RSTN mappings. Further, to demonstrate the effectiveness of our new iterative strategy, we also conduct a numerical experiment. Our major finding is applied to demonstrate that the two-dimensional (2D) Volterra integral equation has a solution. Additionally, a comprehensive example for validating the outcome of our application is provided. Our results expand and generalize a number of relevant results in the literature.

A novel method with convergence analysis based on the Jacobi wavelets for solving a system of two-dimensional Volterra integral equations

ABSTRACT The main objective of this paper is to introduce a collocation-based method for solving a system of two-dimensional Volterra integral equations. The proposed approach is based on the Jacobi wavelets collocation procedure. In addition, the Gegenbauer wavelets method has been implemented to show the comparison of error norms obtained by the proposed method. This proposed approach is applied to reduce the system of Volterra integral equations into a system of algebraic equations. Furthermore, some theorems are elucidated to establish the convergence analysis of the proposed method. Some numerical problems are presented in order to show the accuracy and effectiveness of the proposed scheme. Furthermore, comparison tables and some figures are depicted in support of numerical analysis of the proposed scheme.

Operational matrix approach based on two-dimensional Boubaker polynomials for solving nonlinear two-dimensional integral equations

Two-dimensional First Boubaker polynomials (2D-FBPs) have been formulated and developed as the set of basis for the expansion of bivariate functions. These polynomials are an extension of one-dimensional (1D) FBPs that have previously been used to solve 1D nonlinear integral equations. The dual operational matrix, four operational matrices for integration and operational matrix of product for 2D-FBPs are presented in this study newly. The explicit formula for the operational matrix of product is based on the operational matrix of product by 2D Taylor polynomials (TPs), which has been determined too. Moreover, 2D-FBPs and their operational matrices are used to approximate solution of four types of 2D nonlinear integral equations, that are 2D nonlinear Fredholm, Volterra, Volterra–Fredholm and the system of Volterra integral equations. Using Newton–Cotes collocation nodes, our approach leads to the solution of the set of nonlinear algebraic equations and can be solved uniquely by an iteration method. Since the proposed approach does not require any integration, so all the computations can be easily performed. An explicit and simpler upper bound is obtained for the error vector function of the operational matrix of integration P2Db. In addition, an explicit and simpler upper bound is presented for each bivariate function that expand by 2D-FBPs, as well as the convergence analysis has been proved. Six test problems are given to verify the validity and applicability of the presented methods. The numerical results are compared with exciting methods in literature.

Functional central limit theorems for epidemic models with varying infectivity

In this paper, we prove a functional central limit theorem (FCLT) for a stochastic epidemic model with varying infectivity and general infectious periods recently introduced in R. Forien et al. [Epidemic models with varying infectivity, SIAM J. Appl. Math. 81 (2021), pp. 1893–1930]. The infectivity process (total force of infection at each time) is composed of the independent infectivity random functions of each infectious individual, which starts at the time of infection. These infectivity random functions induce the infectious periods (as well as exposed, recovered or immune periods in full generality), whose probability distributions can be very general. The epidemic model includes the generalized non–Markovian SIR, SEIR, SIS, SIRS models with infection-age dependent infectivity. In the FCLTs for the generalized SIR and SEIR models, the limits of the diffusion-scaled fluctuations of the infectivity and susceptible processes are a unique solution to a two-dimensional Gaussian-driven stochastic Volterra integral equations, and then given these solutions, the limits for the infected (exposed/infectious) and recovered processes are Gaussian processes expressed in terms of the solutions to those stochastic Volterra integral equations. We also present the FCLTs for the generalized SIS and SIRS models.

Hybridization of Block-Pulse and Taylor Polynomials for Approximating 2D Fractional Volterra Integral Equations

This paper proposes an accurate numerical approach for computing the solution of two-dimensional fractional Volterra integral equations. The operational matrices of fractional integration based on the Hybridization of block-pulse and Taylor polynomials are implemented to transform these equations into a system of linear algebraic equations. The error analysis of the proposed method is examined in detail. Numerical results highlight the robustness and accuracy of the proposed strategy.

A Specific Numerical Method for Two-Dimensional Linear Fredholm Integral Equations of the Second Kind by Boubaker Polynomial Bases

A Specific Numerical Method for Two-Dimensional Linear Fredholm Integral Equations of the Second Kind by Boubaker Polynomial Bases

Operational matrix method for solving fractional weakly singular 2D partial Volterra integral equations

The main objective of the present study is to develop the operational matrix for fractional integration. In order to find the numerical solution of non-linear fractional weakly singular two-dimensional partial Volterra integral equations, the operational matrix of the fractional integration of triangular functions was used. By presenting the findings in the form of figures and tables, a deeper investigation and explanation of the proposed approach was provided. In addition, the numerical results show the high reliability and accuracy of the proposed method.

Two-Dimensional Wavelet with Matrix Dilation M = 2I and its Application in Solving Integral Equations

In this study, using a one-dimensionl MRA we constructed a two-dimensional wavelet as well as four masks which are not related to the MRA. Finally, we provide some examples to prove the applicability of our construction in case of finding numerical solution of two-dimensional first kind Fredholm integral equations.

Integrability and high-order localized waves of the (4 + 1)-dimensional nonlinear evolution equation

Constructing the exact solutions of high-dimensional nonlinear evolution equations and exploring their dynamics have always been important and open problems in real-world applications. The celebrated Korteweg-de Vries equation [KdV] and Kadomtsev-Petviashvili [KP] equation are typical examples of one-dimensional and two-dimensional integrable equations respectively. A natural idea is to investigate the integrable analogues of these equations in higher dimensional space. In this paper, an integrable extension of the Kadomtsev-Petviashvili equation, the (4 + 1)-dimensional nonlinear evolution equation (NLEE) [Physical Review Letters 96, (2006) 190201], is investigated. By dimensionality reduction we obtain three new nonlinear equations, namely (3 + 1)-dimensional NLEE, (2 + 1)-dimensional NLEE and (1 + 1)-dimensional NLEE. Bäcklund transformations and multi-soliton solutions are important characteristics of integrable equations. Based on the binary Bell polynomials and Hirota bilinear method we derive bilinear Bäcklund transformations and N-soliton solutions of these new equations, which show that these equations are integrable. We also give the multiple rational solutions of these new equations. These new integrable NLEEs enrich the models of integrable systems and help understand the new characteristics of nonlinear dynamics in real-world applications.

Numerical solution of two-dimensional linear and nonlinear Volterra integral equations using Taylor collocation method

The main purpose of this work is to provide a numerical approach for two-dimensional Volterra integral equations (2D-VIEs). An algorithm based on the use of Taylor polynomials is developed to construct a collocation solution v∈Sp−1,p−1(−1)(ΠN,M) for approximating the solution of 2D-VIEs. It is shown that this algorithm is convergent. Some numerical examples are included to demonstrate the accuracy of the presented method.

A Higher-Order Numerical Scheme for Two-Dimensional Nonlinear Fractional Volterra Integral Equations with Uniform Accuracy

In this paper, based on the modified block-by-block method, we propose a higher-order numerical scheme for two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy. This approach involves discretizing the domain into a large number of subdomains and using biquadratic Lagrangian interpolation on each subdomain. The convergence of the high-order numerical scheme is rigorously established. We prove that the numerical solution converges to the exact solution with the optimal convergence order O(hx4−α+hy4−β) for 0<α,β<1. Finally, experiments with four numerical examples are shown, to support the theoretical findings and to illustrate the efficiency of our proposed method.

A Comparison of General Solutions to the Non-Axisymmetric Frictionless Contact Problem with a Circular Area of Contact: When the Symmetry Does Not Matter

The non-axisymmetric problem of frictionless contact between an isotropic elastic half-space and a cylindrical punch with an arbitrarily shaped base is considered. The contact problem is formulated as a two-dimensional Fredholm integral equation of the first type in a fixed circular domain with the right-hand side being representable in the form of a Fourier series. A number of general solutions of the contact problem, which were published in the literature, are discussed. Based on the Galin–Mossakovskii general solution, new formulas are derived for the particular value of the contact pressure at the contact center and the contact stress-intensity factor at the contour of the contact area. Since the named general solution does not employ the operation of differentiation of a double integral with respect to the coordinates that enter it as parameters, the form of the general solution derived by Mossakovskii as a generalization of Galin’s solution for the special case, when the contact pressure beneath the indenter is bounded, is recommended for use as the most simple closed-form general solution of the non-axisymmetric Boussinesq contact problem.

Идентификация квадратичного ядра уравнения Вольтерра для моделирования нелинейных динамических систем

In the last two decades, integral models have been used to describe the dynamics of stationary nonlinear systems in which the terms of the Volterra series are the kernels. The most commonly used are the linear term (the impulse transition function depends on one variable) and the quadratic term (depending on two variables). An active experiment in which a special combination of rectangular pulses is fed to the input of the system is carried out to select two of its components in the output signal of the identified system - the output of the linear "subsystem" and the output of the "quadratic" subsystem. After isolating the output of the "quadratic" subsystem, the identification of the quadratic term of the Volterra series is reduced to solving a two-dimensional integral equation of the first kind. In the literature, inversion formulas are given in which the quadratic kernel function is obtained as a result of arithmetic operations with second-order derivatives of the output signal. Differentiation of functions is an incorrectly posed problem, when small errors in the assignment of a function (measurement noise) cause large errors in derivatives (especially in second-order derivatives). The paper proposes the use of smoothing cubic splines for stable calculation of derivatives. To calculate the mixed second-order derivative, a spline with two variables is built - a smoothing bicubic spline. The main problem that arises in practice when processing the data of a real experiment is the selection of a smoothing parameter, on the value of which a smoothing error of noisy data depends. As a rule, the value of the variance of the measurement noise is unknown in the experiment. Therefore, in this work, it is proposed to use an algorithm based on the L-curve method to select the smoothing parameter in the constructed splines (especially in the bicubic one), which does not require setting the variance of the measurement noise. The proposed identification algorithm has a high computational efficiency. The performed computational experiment showed a small methodical error (about 1%) and good resistance to noise in measurements of the output signals of the identified system. To reduce the random component of the identification error, it is proposed to use post-processing with a local-spatial compound filter.

Alternating bounded solutions of a class of nonlinear two-dimensional convolution-type integral equations

This paper is devoted to studying a class of nonlinear two-dimensional convolution-type integral equations on R 2 \mathbb {R}^2 . This class of equations has applications in the theory of p p -adic open-closed strings and in the mathematical theory of the spread of epidemics in space and time. The existence of an alternating bounded solution is proved. The asymptotic behaviour of the constructed solution is also studied in a particular case. At the end of the paper, specific applied examples of these equations are given to illustrate the results. UDK 517.968.4.