Two-dimensional Integral Equations Research Articles

On solvability for two-dimensional non-linear functional integral equations via measure of non-compactness

On solvability for two-dimensional non-linear functional integral equations via measure of non-compactness

Discretization methods and their extrapolations for two-dimensional nonlinear Volterra-Urysohn integral equations

In this paper, a class of nonlinear two-dimensional (2D) integral equations of Volterra type, i.e. Volterra-Urysohn integral equations, is studied. Following the ideas of [24], and assuming that the kernels of the integral equation are Lipschitz functions with respect to the dependent variable, the existence and uniqueness of a solution to the integral equation is shown by a technique based on the Picard iterative method. Then, the Euler and trapezoidal discretization methods are used to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. It is proved that the solution of the Euler method has first order convergence to the exact solution of the integral equation while the solution of the trapezoidal method has quadratic convergence. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Some numerical examples are given which confirm our theoretical results.

Spectral Petrov-Galerkin Method for Solving the Two-Dimensional Integral Equations of the Second Kind

Spectral Petrov-Galerkin Method for Solving the Two-Dimensional Integral Equations of the Second Kind

The effect of different edge conditions on the motion of a submerged elastic disc

The present study examines the effects of various edge conditions of a submerged flexible disc on wave propagation for infinite-depth water within the framework of linear water waves theory. Three types of edge conditions (namely Free edge, simply supported edge, and clamped edge) are taken into consideration for the analysis. The governing boundary value problem (BVP) has been solved by reducing it to a two-dimensional hypersingular integral equation. The notion of modal analysis has been adopted to examine the structural response of the disc on the propagation of waves. Later, the two-dimensional hypersingular integral has been transformed into a second kind one-dimensional Fredholm integral equation by applying Fourier series expansion. Finally, the Nystrom technique based on Gauss–Legendre quadrature nodes is used to obtain an approximate solution of the one-dimensional integral equation. The computed solution is used to evaluate the numerical estimates of the physical quantities such as added mass, damping coefficient, hydrodynamic force, and surface elevation for different edge conditions mentioned earlier.

Optical Soliton solutions for stochastic Davey–Stewartson equation under the effect of noise

In this manuscript, we investigates the stochastic Davey–Stewartson equation under the influence of noise in Ito^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\hat{o}}$$\\end{document} sense. This equations is a two-dimensional integrable equations, are higher-dimensional variations of the nonlinear Schrödinger equation. Plasma physics, nonlinear optics, hydrodynamics, and other fields have made use of the solutions to the stochastic Davey–Stewartson equations. The Sardar subequation method is used that will gives us the the stochastic optical soliton solutions in the form of dark, bright, combine and periodic waves. These exact optical soliton solutions are helpful in understanding a variety of fascinating physical phenomena because of the importance of the Davey- Stewartson equations in the theory of turbulence for plasma waves or in optical fibers. Additionally, we use Mathematica tools to plot our solutions and exhibit a series of three-dimensional, two-dimensional and their corresponding contour graphs to show how the noise affects the exact solutions of the stochastic Davey–Stewartson equation. We show how the stochastic Davey–Stewartson solutions are stabilised at around zero by the multiplicative Brownian motion.

Boubaker operational matrix method for solving fractional weakly singular two-dimensional partial Volterra integral equation

Boubaker operational matrix method for solving fractional weakly singular two-dimensional partial Volterra integral equation

An Improvement Meshless Method for the Numerical Solution of Two-Dimensional Stochastic Fredholm Integral Equations

An Improvement Meshless Method for the Numerical Solution of Two-Dimensional Stochastic Fredholm Integral Equations

Modal analysis of a submerged elastic disk: A hypersingular integral equation approach

A method based on the hypersingular integral equation approach and the modal analysis is presented to consider the effects of the motion of a submerged elastic disk on the incoming waves. Initially, the governing boundary value problem is reduced to a two-dimensional integral equation with a hypersingular kernel. This integral equation is further reduced to a one-dimensional Fredholm integral equation of the second kind with the help of Fourier series expansions and a newly defined function. As a part of modal analysis, eigenfunction expansion based on natural modes of structural motion is considered to describe the motion of a thin circular elastic disk. Physical quantities, such as hydrodynamic force, added mass, damping coefficient, and surface elevation, are numerically evaluated. The computed numerical results are verified by comparing them with those for the rigid disk horizontally submerged in deep water. Apart from this, as a part of the analytical verification of our present analysis, the reciprocity relation has been included. The effects of different parameters (disk's rigidity, radius, submergence depth, and mode of vibrations) on the aforementioned physical quantities have been studied. The maximum hydrodynamic force occurs around Ka = 0.5, while the maximum added mass and damping coefficient occur around the wavenumber Ka = 0.3 and Ka = 0.5, respectively. The peaks of the hydrodynamic force and free surface elevation become sharper with the increasing values of the disk's size. The numerical results emphasize that the wave focusing can be controlled by changing the submergence depth, size, and rigidity of the disk.

A generalized fuzzy barycentric Lagrange interpolation method for solving two-dimensional fuzzy fractional Volterra integral equations

A generalized fuzzy barycentric Lagrange interpolation method for solving two-dimensional fuzzy fractional Volterra integral equations

DYNAMIC CONTACT PROBLEMS FOR COMPOSITE MEDIA WITH ANISOTROPIC STRUCTURE

The paper for the first time develops a method for solving dynamic contact problems on the effect of a rigid die in the form of a strip of finite width on a layered anisotropic composite. By applying Mandelstam's principle of marginal absorption, the initial boundary value problem is reduced to a boundary value problem with absorption having a single solution. The symbol of the integral equation has no singularities on the real axis. The contact problem is reduced to solving a two-dimensional integral equation with a difference kernel. The application of the Fourier transform along the coordinate along the strip reduces the integral equation to a one-dimensional one containing a free real parameter of the Fourier transform. Integral equations with an exact solution are introduced, with symbols majoring above and below the symbol of the integral equation. Using the factorization method, the initial integral equation is reduced to two integral equations of the second type, the operator of which turns out to be compressive with a sufficiently large bandwidth. By applying the Newton–Kantorovich method to this integral equation, an exact solution of the integral equation is constructed in an operator form. The presence of the majorant symbol of the integral equation allows us to obtain an upper and lower estimate of the constructed exact solution of the integral equation containing the parameter of the Maldenstam limit absorption principle. After that, in the constructed solution, this parameter rushes to zero from above. The solution of the integral equation is obtained in an analytical form and allows us to identify all its singular features. The result is important when searching for harbingers of an increase in seismicity in mountainous areas. The method is applicable in all cases when it is possible to construct a Green function for a non-mixed boundary value problem in a layered anisotropic composite.

A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains

This paper is devoted to the numerical treatment of two-dimensional Fredholm integral equations, defined on general curvilinear domains of the plane. A Nyström method, based on a suitable Gauss-like cubature formula, recently proposed in the literature is proposed. The convergence, stability and good conditioning of the method are proved in suitable subspaces of continuous functions of Sobolev type. The cubature formula, on which the Nyström method is constructed, has an error that behaves like the best polynomial approximation of the integrand function. Consequently, it is also shown how the Nyström method inherits this property and, hence, the proposed numerical strategy is fast when the involved known functions are smooth. Some numerical examples illustrate the efficiency of the method, also in comparison with other methods known in the literature.

Two-dimensional Jacobi pseudospectral quadrature solutions of two-dimensional fractional Volterra integral equations

Two-dimensional Jacobi pseudospectral quadrature solutions of two-dimensional fractional Volterra integral equations

Numerical solution of the two-dimensional first kind Fredholm integral equations using a regularized collocation method

Numerical solution of the two-dimensional first kind Fredholm integral equations using a regularized collocation method

Hybrid Functions Approach via Nonlinear Integral Equations with Symmetric and Nonsymmetrical Kernel in Two Dimensions

The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmetrical kernel is solved in the Banach space L2[0,1]×L2[0,1]. Here, the NIE’s existence and singular solution are described in this passage. Additionally, we use a numerical strategy that uses hybrid and block-pulse functions to obtain the approximate solution of the NIE in a two-dimensional problem. For this aim, the two-dimensional NIE will be reduced to a system of nonlinear algebraic equations (SNAEs). Then, the SNAEs can be solved numerically. This study focuses on showing the convergence analysis for the numerical approach and generating an estimate of the error. Examples are presented to prove the efficiency of the approach.

Numerical solution of two-dimensional weakly singular Volterra integral equations of the first kind via bivariate rational approximants

We present a numerical method for solving the two-dimensional weakly singular Volterra integral equations of the first kind. The integral equation is first converted into algebraic form using the two-dimensional Laplace transform. We then derived the series expansion for large values, which is inverted term by term to provide the convergent series expansion of the solution for small values. The asymptotic expansion of the solution is extracted from the series expansions of the two-dimensional Laplace transform around the singularity points. The bivariate homogeneous two-point Padé approximants are used to improve convergence. Numerical results are provided to illustrate the accuracy of the method.

Spectral technique with convergence analysis for solving one and two-dimensional mixed Volterra-Fredholm integral equation.

A numerical approach based on shifted Jacobi-Gauss collocation method for solving mixed Volterra-Fredholm integral equations is introduced. The novel technique with shifted Jacobi-Gauss nodes is applied to reduce the mixed Volterra-Fredholm integral equations to a system of algebraic equations that has an easy solved. The present algorithm is extended to solve the one and two-dimensional mixed Volterra-Fredholm integral equations. Convergence analysis for the present method is discussed and confirmed the exponential convergence of the spectral algorithm. Various numerical examples are approached to demonstrate the powerful and accuracy of the technique.

The Use of Homotopy Regularization Method for Linere and Nonlinner Fredholm Integral Equations of the First Kind

Recently, Wazwaz has studied the regularization method to the one-dimensional linear Fredholm integral equations of the first kind [Wazwaz, 2011]. In this work, we develop this method for the linear and nonlinear two-dimensional Fred-holm integral equations of the first kind. Indeed, the regularization method is used for linear integral equations directly. But nonlinear integral equations of the first kind are transformed to linearintegral equations of the first kind by a change of variable; then, The Regularization-Homotopy Method is applied. The combination of the regularization method and the homotopy perturbation method, or shortly, the regularization-homotopy method, is used to find a solution to the equation. Some examples will be used to highlight the reliability of the generalized of Regularization-Homotopy Method.

Higher-order uniform accurate numerical scheme for two-dimensional nonlinear fractional Hadamard integral equations

<abstract><p>In this paper, we consider a higher-order numerical scheme for two-dimensional nonlinear fractional Hadamard integral equations with uniform accuracy. First, the high-order numerical scheme is constructed by using piecewise biquadratic logarithmic interpolations to approximate an integral function based on the idea of the modified block-by-block method. Secondly, for $ 0 < \gamma, \lambda < 1 $, the convergence of the high order numerical scheme has the optimal convergence order of $ O(\Delta_{s}^{4-\gamma}+\Delta_{t}^{4-\lambda }) $. Finally, two numerical examples are used for experimental testing to support the theoretical findings.</p></abstract>

A higher-order numerical scheme for system of two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy

<abstract><p>We give a modified block-by-block method for the nonlinear fractional order Volterra integral equation system by using quadratic Lagrangian interpolation based on the classical block-by-block method. The core of the method is that we divide its domain into a series of subdomains, that is, block it, and use piecewise quadratic Lagrangian interpolation on each subdomain to approximate $ \mathit{\boldsymbol{\kappa}}(x, y, s, r, u(s, r)) $. Our proposed method has uniform accuracy and its convergence order is $ O(h_x^{4-\alpha}+h_y^{4-\beta}) $. We give a strict proof for the error analysis of the method, and give several numerical examples to verify the correctness of the theoretical analysis.</p></abstract>

On the solubility of a class of two-dimensional integral equations on a quarter plane with monotone nonlinearity

In the paper we study a class of two-dimensional integral equations on a quarter-plane with monotone nonlinearity and substochastic kernel. With specific representations of the kernel and nonlinearity, an equation of this kind arises in various fields of natural science. In particular, such equations occur in the dynamical theory of $p$-adic open-closed strings for the scalar field of tachyons, in the mathematical theory of the geographical spread of a pandemic, in the kinetic theory of gases, and in the theory of radiative transfer in inhomogeneous media. \newline We prove constructive theorems on the existence of a nontrivial nonnegative and bounded solution. For one important particular case, the existence of a one-parameter family of nonnegative and bounded solutions is also established. Moreover, the asymptotic behavior at infinity of each solution from the given family os studied. At the end of the paper, specific particular examples (of an applied nature) of the kernel and nonlinearity that satisfy all the conditions of the proven statements are given.