A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations
Introduction
Consider the following two-dimensional nonlinear Volterra integral equation of the second kindwhere the domain is a rectangular, the kernel function K(x, t, y, s, u(y, s)) and the right-hand side function g(x, t) are given, the unknown function u(x, t) must be determined and λ is a non-zero real constant.
Two-dimensional integral equations have significant applications in various fields of applied science and engineering such as, plasma physics [32], the image deblurring problem and its regularization [36], [54], axisymmetric contact problems for bodies with complex rheology [44], diffraction theory [53] and the electrochemical behavior of an inlaid microband electrode for the case of equal diffusion coefficients [45]. These types of integral equations also occur as reformulations from some mixed boundary value problems arising in various branches of applied sciences; for example, solid and fluid mechanics, electrostatics, heat transfer, diffraction and scattering of waves, etc [18], [22], [64].
Several methods have been proposed for the numerical solutions of Volterra integral equations. The modified Newton–Kantorovich scheme combined with the collocation method has been applied for solving Volterra integral equations with piecewise smooth kernels in [50]. The generalized quadrature methods [60], [61] have been used for the numerical solution of singular Volterra integral equation of Abel type. A computational method has been presented for constructing asymptotic approximations to parametric families of solutions of Volterra integral equations of the first kind with piecewise smooth kernel [57] and the existence of continuous solutions has been proved in [59]. The author of [58] has suggested an asymptotic approach for solutions of systems of Volterra integral equations of the first kind with piecewise continuous kernels using logarithmic–power approximations. A suboptimal filtration method for solving convolution–type integral equations of the first kind has been given in [25]. The optimal accuracy (spline–collocation) technique has been investigated for Volterra integral equations with weakly singular kernels [19] and hyper–singular kernels [20], [21].
Obtaining the analytical solution for two-dimensional Volterra integral equations is most difficult, so it is significant to give their numerical solutions. But a few numbers of methods have been proposed for the approximate solution of two-dimensional nonlinear Volterra integral equations. The projection and iterated projection [11], [23], [40] are the commonly used approaches for the numerical solutions of two-dimensional integral equations. The collocation method [24], [34], the iterated collocation method [35], the two-dimensional differential transform (TDDT) method [63], the rationalized Haar functions [15], the geometric series theorem together with Schauder bases [17], the Legendre functions [51], the differential transform method [38], the Euler-type method [38], the block-pulse functions [46], and the Chebyshev polynomials [14] have been applied to obtain the numerical solution of two-dimensional Volterra integral equations.
To handle the Galerkin method for solving two-dimensional integral equations, the solution domains usually require to be divided into non-overlapping triangular fragments [11], [37]. To deliver from these triangulations and mesh refinements, methods based upon the meshless approximations can be useful which estimate a function without any mesh generations on the domain. The significant initial meshfree methods are known in the literature as radial basis functions (RBFs) and the moving least squares (MLS) methods. The MLS scheme as a general case of Shepard’s method has been introduced by Lancaster and Salkauskas [41] applied in the past few decades. The MLS consists of a local weighted least square fitting, valid on a small neighborhood of a point and only based on the information provided by its closet points. This approach is recognized as a meshless method because it is based on a set of scattered points and consequently does not need any domain elements to approximate unknown functions. A valuable advantage of using the MLS approximation is that it sets up and solves many small systems, instead of a single, but large system [33], [65].
The meshless schemes have various applications in different branches of the computational mathematics [27], [28], such as partial differential equations (PDEs) and a large number of papers has presented many numerical methods for solving them. The Galerkin boundary node method (GBNM) [43] has been utilized for two-dimensional exterior Neumann problems. Element-free Galerkin methods [16] have been applied for solving elasticity and heat conduction problems. The meshless local boundary integral equation method [29] has been investigated for the unsteady two-dimensional Schrodinger equation. The boundary node method [66] and the meshless local Petrov–Galerkin (MLPG) approach [13] have been used for the numerical solution of potential problems. The MLS reproducing polynomial meshless method [55] has been introduced for the numerical solution of the parabolic two-dimensional transient heat conduction equation. The local boundary integral equation method [62], [68] has been applied to solve the problem of elasticity with nonhomogeneous material properties.
We would like to review some of the most recent works for the numerical solution of integral equations utilizing the meshless schemes. The meshless discrete collocation schemes using RBFs have been investigated for solving two-dimensional integral equations of the second kind with smooth kernels [3] and weakly singular kernels [7], [8]. The RBFs have been applied for the numerical solution of Volterra–Fredholm–Hammerstein integral equations [52] and singular-logarithmic boundary integral equations [5], [9]. The meshless product integration (MPI) method [6] has been proposed to solve one-dimensional linear weakly singular integral equations. The MLS collocation method has been used for solving two-dimensional integral equations [49] and integro-differential equations [30]. An MLS-based Galerkin meshless method has been utilized to solve weakly singular [4] and logarithmic boundary [10], [42] integral equations. Authors of [47], [48] have investigated a domain–type RBF collocation method and a boundary–type RBF collocation method to solve special integro–differential models.
In the current investigation, we present a numerical technique to solve two-dimensional nonlinear Volterra integral equations of the second kind. The method utilizes the MLS scheme constructed on distributed nodal points to approximate the solution in the discrete Galerkin method. Therefore, we require a suitable integration rule to compute integrals in the presented method based on the composite Gauss-Legendre quadrature formula. The scheme reduces the solution of the Volterra integral equation to the solution of a system of algebraic equations. The new technique does not require any domain elements, so it is meshless. The scheme does not increase the difficulties for higher dimensional problems due to the simple adaption of the MLS and more flexible for most classes of Volterra integral equations. We also obtain the error bound and the convergence rate for the proposed method. The efficiency and accuracy of the new approach are examined in various Volterra integral equations.
The outline of the paper is as follows. In Section 2, we review some basic formulations and properties of the MLS method. In Section 3, we present a computational method for solving the two-dimensional Volterra integral Eq. (1) by the meshless local discrete Galerkin scheme. In Section 4, we provide the error analysis of the method. Numerical examples are given in Section 5. Finally, we conclude the article in Section 6.
Section snippets
The MLS approximation
In this section, we apply the MLS technique to approximate two variable functions on the rectangular . Consider the data values of the function u(x, t) at nodal points . The MLS method estimates a function u(x, t) for every point based on the weighted least square sense. We denote the MLS approximation by su, X(x, t) for every and is obtained using the solution of the following problemwhere the
Solving Volterra integral equations
Here, we apply the MLS approximation for solving two-dimensional Volterra integral equations of the second kind, namelywhere the domain the kernel function and the right-hand side function are given, the unknown function u(x, t) must be determined and λ is a non-zero real constant.
Assume that the kernel function K satisfies the following Lipschitz condition [67]:for all
Error estimates
This section includes the error estimate and the convergence rate of the presented method.
We define as a collocation projection operator bywhere and the coefficients determined by solving the following linear systemTo obtain a better understanding of we give an explicit formula for . We introduce a new basis for VN by using the Gram-Schmidt process to create an orthonormal
Numerical examples
In order to demonstrate the effectiveness of the proposed method, four two-dimensional nonlinear Volterra integral equations are solved. For the tests, we used the linear () and quadratic () basis functions and the Gaussian and spline weight functions. We employ 10-points composite dual Gauss-Legendre quadrature rule with for approximating integrals in the scheme. In all computations, we put and for the linear and quadratic cases, respectively [49], [65]. In Gaussian
Conclusion
The main purpose of this article has been to obtain an approximate scheme for solving two-dimensional nonlinear Volterra integral equations of the second kind. The proposed method utilizes the discrete Galerkin method together with shape functions of the MLS approach as a basis. The MLS methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least squares polynomial fitting. In order to compute the integrals cosidered in the proposed
Acknowledgments
The authors are very grateful to the reviewers for their valuable comments and suggestions which have improved the paper.
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