Integral Equation Scheme Research Articles

Fourth-Order Trapezoid Algorithm with Four Iterative Schemes for Nonlinear Integral Equations

Fourth-Order Trapezoid Algorithm with Four Iterative Schemes for Nonlinear Integral Equations

A neural networks-based numerical method for the generalized Caputo-type fractional differential equations

The paper presents a numerical technique based on neural networks for generalized Caputo-type fractional differential equations with and without delay. We employ the theory of functional connection-based approximation and the physics-informed neural network with extreme learning machine-based learning to solve the differential equation. The proposed method uses the L1 finite difference scheme and the Volterra integral equation scheme to create the loss function. The novelty of this work is the proposal of the neural network-based scheme coupling the idea of the theory of functional connections and a new loss function for the solution of generalized Caputo-type differential equations. The proposed approach is applied to single differential equations and the system of differential equations with single and multiple delays.

Stability Analysis of Nonclassical Difference Schemes for Nonlinear Volterra Integral Equations of the Second Kind

A family of first- and second-order accurate noniterative numerical methods is constructed for solving systems of nonlinear Volterra integral equations of the second kind. The methods are examined for A-, L-, and P-stability. The conclusions are illustrated by numerical results obtained for test equations with stiff and oscillating components

Stability Analysis of Nonclassical Difference Schemes for Nonlinear Volterra Integral Equations of the Second Kind

Stability Analysis of Nonclassical Difference Schemes for Nonlinear Volterra Integral Equations of the Second Kind

Fast $$\theta $$-Maruyama scheme for stochastic Volterra integral equations of convolution type: mean-square stability and strong convergence analysis

Fast $$\theta $$-Maruyama scheme for stochastic Volterra integral equations of convolution type: mean-square stability and strong convergence analysis

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>

Stochastic Volterra integral equations with doubly singular kernels and their numerical solutions

This paper first constructs a Milstein-type scheme for stochastic Volterra integral equations with doubly singular kernels. Then, we also study the Hölder continuity of the solution to these equations and investigate the convergence rate of the Milstein scheme. More precisely, the mean-square convergence rate is min{1−α1−β1,1−2α2−2β2}, where α1,α2,β1,β2 are the singularity exponents of the equations. The difficulty in obtaining our convergence results is the lack of Itô formula for the equations. To get around this problem, we adopt the Taylor formula and then perform a complex analysis of the equations satisfied by the solution. Moreover, we apply the multilevel Monte Carlo technique based on the Euler scheme and the fast Euler scheme for the equations to reduce the computational complexity. More concretely, to achieve given desired accuracy O(ϵ), the multilevel Monte Carlo technique based on the Euler scheme and the fast Euler scheme decrease the computational cost of the standard Euler scheme from Oϵ−2−2α˜ to Oϵ−2α˜ and O(ϵ−1α˜|logϵ|3) when T≈1. Finally, some numerical experiments are given to demonstrate our theoretical results.

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.

Galerkin Approximation for Stochastic Volterra Integral Equations with Doubly Singular Kernels

This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both s=t and s=0. We propose a Galerkin approximate scheme to solve the equation numerically, and we obtain the strong convergence rate for the Galerkin method in the mean square sense. The rate is min{2−2(α1+β1),1−2(α2+β2)} (where α1,α2,β1,β2 are positive numbers satisfying 0<α1+β1<1, 0<α2+β2<12), which improves the results of some numerical schemes for the stochastic Volterra integral equations with regular or weakly singular kernels. Moreover, numerical examples are given to support the theoretical result and explain the priority of the Galerkin method.

Positivity and boundedness preserving nonstandard finite difference schemes for solving Volterra’s population growth model

In this work, we extend the Mickens’ methodology to construct nonstandard finite difference (NSFD) schemes preserving positivity and boundedness of the nonlinear Volterra integro-differential population growth model. A rigorously mathematical study for the positivity, boundedness, convergence and error bounds of the proposed NSFD schemes is provided. It is proved that the NSFD schemes not only converge, but also preserve the positivity and boundedness of the integro-differential model for all finite step sizes. Furthermore, the constructed NSFD schemes can be extended to formulate nonstandard discretization schemes for general Volterra integral equations and fractional-order Volterra integro-differential population growth models. Finally, the theoretical results and advantages of the NSFD schemes over standard ones are supported and illustrated by a set of numerical experiments. The numerical experiments show that some typical standard numerical schemes such as the Euler scheme, the second order and classical fourth order Runge–Kutta schemes fail to correctly preserve the positivity and boundedness for some given step sizes. As a result, they can generate numerical approximations that are completely different from the solutions of the integro-differential model. However, these properties are preserved by the NSFD schemes when using the same step sizes.

Direct Operational Vector Scheme for First-Kind Nonlinear Volterra Integral Equations and Its Convergence Analysis

In the current paper, an efficient direct numerical scheme is developed to approximate the solution of nonlinear Volterra integral equations of the first kind. This computational method is based upon operational matrices and vectors which eventually lead to the sparsity of the coefficients matrix of the obtained system. For this purpose, the operational vector for hybrid block pulse functions and Chebyshev polynomials is constructed. Hybrid functions are powerful tools to approximate functions locally, and capable to adjust the orders of block-pulse functions and Chebyshev polynomials to achieve highly accurate numerical solutions. Its simple structure to implement, low computational cost and perfect approximate solutions are the major points of the presented method. The convergence analysis of the numerical approach under the $$L^2$$ -norm is completely studied. Finally, numerical experiments indicate that the proposed approach is effective and powerful to deal with smooth and non-smooth solutions and verify the obtained theoretical results.

ЧИСЛЕННЫЙ АНАЛИЗ ДИНАМИКИ ТРЕХМЕРНЫХ АНИЗОТРОПНЫХ ТЕЛ НА ОСНОВЕ НЕКЛАССИЧЕСКИХ ГРАНИЧНЫХ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ

The application of non-classical approach of the boundary integral equation method in combination with the integral Laplace transform in time to anisotropic elastic wave modeling is considered. In contrast to the classical approach of the boundary integral equation method which is successfully implemented for solving three-dimensional isotropic problems of the dynamic theory of elasticity, viscoelasticity and poroelasticity, the alternative nonclassical formulation of the boundary integral equations method is presented that employs regular Fredholm integral equations of the first kind (integral equations on a plane wave). The construction of such boundary integral equations is based on the structure of the dynamic fundamental solution. The approach employs the explicit boundary integral equations. The inverse Laplace transform is constructed numerically by the Durbin method. A numerical solution of the dynamic problem of anisotropic elasticity theory based on the boundary integral equations method in a nonclassical formulation is presented. The boundary element scheme of the boundary integral equations method is built on the basis of a regular integral equation of the first kind. The problem is solved in anisotropic formulation for the load acting along the normal in the form of the Heaviside function on the cube face weakened by a cubic cavity. The obtained boundary element solutions are compared with finite element solutions. Numerical results prove the efficiency of using boundary integral equations on a single plane wave in solving three-dimensional anisotropic dynamic problems of elasticity theory. The convergence of boundary element solutions is studied on three schemes of surface discretization. The achieved calculation accuracy is not inferior to the accuracy of boundary element schemes for classical boundary integral equations. Boundary element analysis of solutions for a cube with and without a cavity is carried out.

Integral equation solution for three‐dimensional heat transfer in multiple‐fracture enhanced geothermal reservoirs

SummaryFor the prediction of energy production from multiple‐fractured geothermal reservoirs, previous models basically focused on the one‐dimensional conduction in the rock containing evenly distributed fractures of equal scale. Here, a novel model is described to numerically investigate the three‐dimensional heat transfer in geothermal reservoirs with unevenly spaced disc fractures of various sizes including the aperture and radius. In terms of the water flow through each fracture, an approximate analytical solution is obtained on the assumption that the water pressure disturbances, induced by the fracture margin and extraction (injection) operation, at the injection (extraction) well center and at different locations within the injection (extraction) well range were approximately equal. By the integral equation scheme for two‐dimensional planar fractures, the three‐dimensional problem of heat exchange is simulated without the reservoir discretization. The singular integral is analytically calculated in polar coordinates whereas the nonsingular integrand is numerically estimated by the Gaussian quadrature method in Cartesian coordinates. Compared with the one‐dimensional simplification, the three‐dimensional heat conduction remarkably alters the prediction of extraction temperature. In addition, the reservoir temperature field is also significantly influenced by the spacings and dimensions of fractures. The present model may be used for the estimation, design, and optimization of a geothermal reservoir.

Solution of Stokes flow in complex nonsmooth 2D geometries via a linear-scaling high-order adaptive integral equation scheme

We present a fast, high-order accurate and adaptive boundary integral scheme for solving the Stokes equations in complex—possibly nonsmooth—geometries in two dimensions. We apply the panel-based quadratures of Helsing and coworkers to evaluate to high accuracy the weakly-singular, hyper-singular, and super-singular integrals arising in the Nyström discretization, and also the near-singular integrals needed for flow and traction evaluation close to boundaries. The resulting linear system is solved iteratively via calls to a Stokes fast multipole method. We include an automatic algorithm to “panelize” a given geometry, and choose a panel order, which will efficiently approximate the density (and hence solution) to a user-prescribed tolerance. We show that this adaptive panel refinement procedure works well in practice even in the case of complex geometries with large number of corners, or close-to-touching smooth curves. In one example, for instance, a model 2D vascular network with 378 corners required less than 200K discretization points to obtain a 9-digit solution accuracy.

Towards Space‐Time Boundary Element Methods for Retarded Potential Integral Equations

AbstractA space‐time discretization scheme for time domain boundary integral equations of the 3d wave equation, so‐called retarded potential boundary integral equations, is proposed. The principal idea of this approach is to treat time like an additional spatial coordinate. This leads to discretization methods in 3+1 dimensions based on simplex meshes of the lateral boundary of the space‐time cylinder. We discuss an explicit representation of the integral operators of the wave equation appropriate for the space‐time setting. A robust and accurate quadrature scheme for the numerical evaluation of the occurring integrals is presented and verified by means of numerical experiments.

Explicit unconditionally stable methods for the heat equation via potential theory

We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite difference or finite element schemes for the heat equation are stable only if the time step $\Delta t$ is of the order $O(\Delta x^2)$, where $\Delta x$ is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions $d\ge 1$, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an $L^2$-norm of the solution to the integral equation is bounded by $c_dT^{d/2}$ times the norm of the right hand side. For the Robin problem on the half space in any dimension, with constant Robin (heat transfer) coefficient $\kappa$, we exhibit a constant $C$ such that the forward Euler scheme is stable if $\Delta t < C/\kappa^2$, independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in $L^\infty$-norm.

Approximate solutions of Volterra integral equations by an interpolation method based on ramp functions

In the present paper we develop a numerical collocation scheme for integral equations based on the interpolation method which uses the so-called neural network operators activated by ramp functions. The proposed collocation scheme allows to solve both linear and nonlinear Volterra integral equations of the second kind; the convergence of the numerical method has been proved under standard assumptions. Numerical examples have been provided and discussed; the proposed method has been compared with the classical piecewice polynomials collocation method and a method based on sigmoidal functions, such as the Heaviside and the logistic functions.

Dynamic rupture propagation on fault planes with explicit representation of short branches

An active fault zone is home to a plethora of complex structural and geometric features that are expected to affect earthquake rupture nucleation, propagation, and arrest, as well as interseismic deformation. Simulations of these complexities have been largely done using continuum plasticity or scalar damage theories. In this paper, we use a highly efficient novel hybrid finite element-spectral boundary integral equation scheme to investigate the dynamics of fault zones with small scale pre-existing branches as a first step towards explicit representation of anisotropic damage features in fault zones. The hybrid computational scheme enables exact near-field truncation of the elastodynamic field allowing us to use high resolution finite element discretization in a narrow region surrounding the fault zone that encompasses the small scale branches while remaining computationally efficient. Our results suggest that the small scale branches may influence the rupture in ways that may not be realizable in homogenized continuum models. Specifically, we show that these short secondary branches significantly affect the post event stress state on the main fault leading to strong heterogeneities in both normal and shear stresses and also contribute to the enhanced generation of high frequency radiation. The secondary branches also affect off-fault plastic strain distribution and suggest that co-seismic inelasticity is sensitive to pre-existing damage features. We discuss our results in the larger context of the need for modeling earthquake ruptures with high resolution fault zone physics.

A Fast Derivative-Free Iteration Scheme for Nonlinear Systems and Integral Equations

Derivative-free schemes are a class of competitive methods since they are one remedy in cases at which the computation of the Jacobian or higher order derivatives of multi-dimensional functions is difficult. This article studies a variant of Steffensen’s method with memory for tackling a nonlinear system of equations, to not only be independent of the Jacobian calculation but also to improve the computational efficiency. The analytical parts of the work are supported by several tests, including an application in mixed integral equations.

Forward plane‐wave electromagnetic model in three dimensions using hybrid finite volume–integral equation scheme

ABSTRACTWe present a concept of the hybrid finite volume–integral equation technique for solving Maxwell's equation in a quasi‐static form. The divergence correction was incorporated to improve the convergence and stability of the governing linear system equations which pose a challenge on the discretization of the curl–curl Helmholtz equation. A staggered finite volume approach is applied for discretizing the system of equations on a structured mesh and solved in a secondary field technique. The bi‐conjugate gradient stabilizer was utilized with block incomplete lower‐upper factorization preconditioner to solve the system of equation. To obtain the electric and magnetic fields at the receivers, we use the integral Green tensor scheme. We verify the strength of our hybrid technique with benchmark models relative to other numerical algorithms. Importantly, from the tested models, our scheme was in close agreement with the semi‐analytical solution. It also revealed that the use of a quasi‐analytical boundary condition helps to minimize the runtime for the linear system equation. Furthermore, the integral Green tensor approach to compute at the receivers demonstrates better accuracy compared with the conventional interpolation method. This adopted technique can be applied efficiently to the inversion procedure.