Weakly Singular Research Articles

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

A Nyström method based on product integration for weakly singular Volterra integral equations with variable exponent

Weakly singular Volterra integral equations with variable exponent have integral kernels of the form (t−s)−α(t) with 0≤α(t)<1. A Nyström method based on product integration and interpolation by piecewise polynomials is constructed and analysed for the numerical solution of this class of problems. To deal with the weak singularity of typical solutions of such problems, suitably graded meshes are used. A rigorous error analysis proves convergence of the computed solution; moreover, superconvergence is obtained if the quadrature points are well chosen. These results also imply error bounds for the solution of a related iterated collocation method. Numerical experiments demonstrate the sharpness of our theoretical results for the Nyström method.

Discontinuous Galerkin finite element method for dynamic viscoelasticity models of power‐law type

AbstractLinear viscoelasticity can be characterized by a stress relaxation function. We consider a power‐law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, symmetric interior penalty Galerkin method (SIPG) for spatial discretization, and the implicit finite difference schemes in time, Crank–Nicolson method. Further, in order to manage the weak singularity in the Volterra kernel, we use a linear interpolation technique. We present a priori stability and error analyses without relying on Grönwall's inequality, and so provide high quality bounds that do not increase exponentially in time. This indicates that our numerical scheme is well‐suited for long‐time simulations. Despite the limited regularity in time, we establish suboptimal fractional order accuracy in time as well as optimal convergence of SIPG. We carry out numerical experiments with varying regularity of exact solutions to validate our error estimates. Finally, we present numerical simulations based on real material data.

About a new nonlinear integro-differential equation of Volterra-Strum-Liouville type with a weakly singular kernel

This present paper proposes an analytical and numerical study of a new integro-differential Volterra-Strum-Liouville equation of the second type, having a weak singularity. We provide conditions that guarantee the existence and uniqueness of the solution of the nonlinear problem. Then, we develop a numerical technique using the production integration method. The numerical application shows the efficiency of the proposed procedure.

A multi-domain hybrid spectral collocation method for nonlinear Volterra integral equations with weakly singular kernel

In this paper, we propose a multi-domain hybrid spectral collocation method for the nonlinear Volterra integral equations (VIEs), whose solutions exhibit weak singularities at the endpoint t=0. In order to efficiently approximate the weakly singular solutions, we divide the interval [0,T] into N subintervals and use the Gauss points of the generalized log orthogonal functions as the collocation points to approximate the weakly singular integral term in the first subinterval. In the remaining subintervals, we use Legendre and Jacobi Gauss points as the collocation points to approximate the corresponding integral terms. We also provide a rigorous hp-version convergence analysis for the hybrid spectral collocation method under L2-norm. A series of examples demonstrate our method is particularly suitable for solutions that have weak singularities at one endpoint.

PERIODICITY AND SYMMETRY IN A CLASS OF INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

PERIODICITY AND SYMMETRY IN A CLASS OF INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

Numerical simulation for two species time fractional weakly singular model arising in population dynamics

ABSTRACT In this work, we analyze and develop an efficient numerical scheme for the Lotka–Volterra competitive population dynamics model involving fractional derivative of order . The fractional derivative is defined in the Caputo sense. The solution exhibits a weak singularity near Using the L1 technique, the fractional operator is discretized. The differential equations are reduced to a system of nonlinear algebraic equations. To solve the corresponding nonlinear system, we employed the generalized Newton–Raphson method. The presence of singularities creates a layer at the origin, and as a result, the proposed scheme fails to achieve its optimal convergence on a uniform mesh. To accelerate the rate of convergence, we used a graded mesh with a suitably chosen grading parameter. The stability analysis and error estimates are derived on a maximum norm. Finally, numerical experiments are conducted to show the validity and applicability of the proposed scheme.

Design and analysis of efficient computational techniques for solving a temporal‐fractional partial differential equation with the weakly singular solution

This work deals with the construction of robust numerical schemes for solving a time‐fractional convection‐diffusion (TFCD) equation with variable coefficients subject to weakly singular solution. The solution of the underlying problem exhibits a weak singularity at time . The methods are designed on both graded and adapted grids, in which the temporal derivative is approximated on the nonuniform grid, in order to tackle the singularity. In the case of adaptive mesh technique, the mesh is constructed adaptively via equidistribution of a positive monitor function. The resulting semi‐discretized equation is further discretized by a high‐order compact finite difference (CFD) scheme to obtain a fully discrete scheme. Stability and convergence of the method on graded mesh are analyzed. Numerical results for one test problem subjected to nonsmooth analytical solution are presented which shows the robustness and efficiency of the proposed numerical schemes. The elapsed computational time (CPU time) for the proposed methods are provided. Numerical results obtained from the methods on graded and adapted grids are compared with those from the scheme on uniform grid. Finally, as an application, the TFCD model and the suggested numerical techniques were employed to price options of a European butterfly spread. The effect of fractional order (FO) derivative on the option price profile is investigated.

A fast time-stepping method based on the [formula omitted]-version spectral collocation method for the nonlinear fractional delay differential equation

In this paper, we propose a fast time-stepping method based on the hp-version spectral collocation method for the nonlinear fractional delay differential equation. To reduce the interaction between the subintervals caused by the delay term, a mesh of bidirectional partition is designed, such that each subinterval after delay falls exactly into a single preceding subinterval. Moreover, with the aid of the sum-of-exponentials approximation for the kernel function (t−s)α−1, a fast time-stepping method is developed to deal with the weakly singular integral, which overcomes the weak singularity of the integral kernel, and greatly reduces the cost for the computation of the history integral. In addition, the hp-convergence of the suggested method is fully analyzed and characterized, which implies that the interplay between h and p can significantly enhance the numerical accuracy. Numerical experiments confirm the theoretical expectations.

A Subdivision Transformation Method for Weakly Singular Boundary Integrals in Thin-structural Problem

Accurate and effective calculation of weakly singular integral in the boundary integral equation of the thin-structural problem is the key to the numerical implementation of the boundary element method. In this paper, a subdivision transformation method evaluated for weakly singular integrals is proposed, and the method is implemented as follows: based on the position of the source points, the shape information of the elements and the nearest distance from the source point to the integral element, a subdivision technology is constructed at first. With this subdivision technology, the integral element can be divided into several integral blocks with good shapes. And then, a simpler coordinate transformation method is constructed to remove the weak singularities of the integral blocks obtained by the subdivision technology. Compared with the conventional polar coordinate transformation method, the present transformation method does not need to calculate their integral interval, which is more simple and effective to implement. Finally, the paper gives three numerical examples to verify the accuracy and validity of the present method.

An [formula omitted]-robust fast algorithm for distributed-order time–space fractional diffusion equation with weakly singular solution

A fast algorithm is proposed for solving two-dimensional distributed-order time–space fractional diffusion equation where the solution has a weak singularity at initial time. The distributed-order fractional problem is firstly transformed into multi-term fractional problem by the Gauss–Legendre quadrature formula. Then the exponential-sum-approximation method on graded mesh is utilized to discretize time Caputo fractional derivatives in time direction, and a standard finite difference method is employed to approximate the spatial Riesz fractional derivatives. The scheme is proved to be α-robust convergent analytically. The discrete linear system possesses symmetric positive definite block-Toeplitz–Toeplitz-block structure and is efficiently solved by conjugate gradient method with the state-of-the-art sine-transformed based preconditioner. Numerical examples confirm the error analysis and the effectiveness of the preconditioner.

Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases

When one solves differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients an to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, u(±1) = 0, popular choices include the “Chebyshev difference basis” ςn(x) ≡ Tn+2(x)−Tn(x) with coefficients here denoted by bn and the “quadratic factor basis” ϱn(x) ≡ (1 − x2)Tn(x) with coefficients cn. If u(x) is weakly singular at the boundary, then the coefficients an decrease proportionally to \({\cal O}\left( {A\left( n \right)/{n^\kappa }} \right)\) for some positive constant κ, where A(n) is a logarithm or a constant. We prove that the Chebyshev difference coefficients bn decrease more slowly by a factor of 1/n while the quadratic factor coefficients cn decrease more slowly still as \({\cal O}\left( {A\left( n \right)/{n^{\kappa - 2}}} \right)\). The error for the unconstrained Chebyshev series, truncated at degree n = N, is \({\cal O}\left( {\left| {A\left( N \right)} \right|/{N^\kappa }} \right)\) in the interior, but is worse by one power of N in narrow boundary layers near each of the endpoints. Despite having nearly identical error norms in interpolation, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x. Meanwhile, for Chebyshev polynomials, the values of their derivatives at the endpoints are \({\cal O}\left( {{N^2}} \right)\), but only \({\cal O}\left( N \right)\) for the difference basis. Furthermore, we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases, solved by the least squares method. We also find an interesting fact that on the face of it, the aliasing error is regarded as a bad thing; actually, the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation. But the premise is under the same basis, and when involving different bases, it may not be established yet.

High-Order Multivariate Spectral Algorithms for High-Dimensional Nonlinear Weakly Singular Integral Equations with Delay

One of the open problems in the numerical analysis of solutions to high-dimensional nonlinear integral equations with memory kernel and proportional delay is how to preserve the high-order accuracy for nonsmooth solutions. It is well-known that the solutions to these equations display a typical weak singularity at the initial time, which causes challenges in developing high-order and efficient numerical algorithms. The key idea of the proposed approach is to adopt a smoothing transformation for the multivariate spectral collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the singularity of the approximate solution can be tailored to that of the exact one, resulting in high-order spectral collocation algorithms. Moreover, we provide a framework for studying the rate of convergence of the proposed algorithm. Finally, we give a numerical test example to show that the approach can preserve the nonsmooth solution to the underlying problems.

An averaged [formula omitted]-type compact difference method for time-fractional mobile/immobile diffusion equations with weakly singular solutions

This paper is concerned with a new numerical method for time-fractional mobile/immobile diffusion equations with weakly singular solutions. We propose an averaged L1-type compact difference method, which improves the temporal convergence order of the traditional L1-type method and is also superior to the WSGD-type and L2−1σ-type methods in terms of regularity requirements. Taking into account the weak singularity of the solution at the initial time, we prove that the proposed method is unconditionally convergent with the convergence order O(τ2|lnτ|+h4), where τ and h are the sizes of the time and spatial steps, respectively. Numerical results confirm the theoretical convergence result.

Boundary element analysis of thin structures using a dual transformation method for weakly singular boundary integrals

This paper focuses on the weakly singular integrals in thin structures of 3D boundary element method. In the pioneering woks for the weakly singular integrals, the singularities are eliminated through introducing transformations whose Jacobian at the singular point is zero. When using these methods for weakly singular boundary integrals on elements with a large-angle and large side length ratio, near singularity will appear. In this paper, a dual transformation method is proposed to accurately calculate weakly singular integrals in which the near singularity is also considered. The numerical implementation is divided into several steps. Firstly, the (α, β) coordinate transformation method is used to eliminate the weak singularity by introducing the α factors in the transformation Jacobian. Then extract the integral form with near singularities about β. Finally, a corresponding distance transformation is constructed to eliminate its near singularity. The results show that the proposed method can accurately integrate for arbitrary element shape and source point position. The dual transformation method may be a good choice to evaluate weakly singular integral for thin structural problem.

Weakly Singular Symmetric Galerkin Boundary Element Method for Fracture Analysis of Three-Dimensional Structures Considering Rotational Inertia and Gravitational Forces

The Symmetric Galerkin Boundary Element Method is advantageous for the linear elastic fracture and crackgrowth analysis of solid structures, because only boundary and crack-surface elements are needed. However, for engineering structures subjected to body forces such as rotational inertia and gravitational loads, additional domain integral terms in the Galerkin boundary integral equation will necessitate meshing of the interior of the domain. In this study, weakly-singular SGBEM for fracture analysis of three-dimensional structures considering rotational inertia and gravitational forces are developed. By using divergence theorem or alternatively the radial integration method, the domain integral terms caused by body forces are transformed into boundary integrals. And due to the weak singularity of the formulated boundary integral equations, a simple Gauss-Legendre quadrature with a few integral points is suffcient for numerically evaluating the SGBEM equations. Some numerical examples are presented to verify this approach and results are compared with benchmark solutions.

BEM analysis of multilayer thin structures using a composite transformation method for boundary integrals

In this paper, a composite transformation method is developed to evaluate the singular and nearly singular boundary integrals in the boundary element analysis for multilayer thin structures. The composite transformation method is implemented as follows: Firstly, a sinh transformation method is developed to eliminate the influence of nearly singular integrals in the multilayer thin-structural problem; and then a complex transformation method is improved to deal with the weakly singular integrals. The complex transformation formulations of weakly singular integrals are derived from the composition of the (α, β) coordinate transformation and the sinh transformation in β direction. With this method for weakly singular integrals, the weak singularity in the radial direction and the potentially near singularity in the circumferential direction can be removed. Finally, the proposed transformation formulations of singular and nearly singular integrals are used to analyze multi-domain elasticity problem for multilayer thin structures. The numerical results for several examples are presented to demonstrate the computational accuracy of the proposed method.

A boundary weak singularity elimination method for multilayer structures

In the numerical discretization for the models of multilayer thin structures, the elements with narrow strip shape may be appeared. When using the previous methods to integrate these elements, the potentially near singularity will arise in the circumferential direction. In this paper, a weak singularity elimination method for multilayer structures of 3D boundary element method is presented to accurately calculate weakly singular integrals. Its numerical implementation is divided into the following two parts: (1) (α, β) coordinate transformation is applied to deal with the weakly singular integral arising in boundary integral equation and separate out the integral in the circumferential direction (β direction); (2) Based on the integral form of β, a corresponding sinh transformation method is developed to eliminate the near singularity in β direction. Finally, the proposed transformation formulation is extended to multi-domain potential problem and applied to the boundary analysis for multilayer structures. Several numerical examples are given to verify the computational accuracy of the proposed method. And the numerical results are promising and encouraging.

A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the weak singularity, fractional order integration in time is managed approximately by linear interpolation so that we can formulate a fully discrete problem. In this paper, we present a stability bound as well as a priori error estimates. Furthermore, we carry out numerical experiments with varying regularity of exact solutions at the end.

Two Finite Difference Schemes for Multi-Dimensional Fractional Wave Equations with Weakly Singular Solutions

Abstract A time-fractional initial-boundary value problem of wave type is considered, where the spatial domain is ( 0 , 1 ) d (0,1)^{d} for some d ∈ { 1 , 2 , 3 } d\in\{1,2,3\} . Regularity of the solution 𝑢 is discussed in detail. Typical solutions have a weak singularity at the initial time t = 0 t=0 : while 𝑢 and u t u_{t} are continuous at t = 0 t=0 , the second-order derivative u t ⁢ t u_{tt} blows up at t = 0 t=0 . To solve the problem numerically, a finite difference scheme is used on a mesh that is graded in time and uniform in space with the same mesh size ℎ in each coordinate direction. This scheme is generated through order reduction: one rewrites the differential equation as a system of two equations using the new variable v := u t v:=u_{t} ; then one uses a modified L1 scheme of Crank–Nicolson type for the driving equation. A fast variant of this finite difference scheme is also considered, using a sum-of-exponentials (SOE) approximation for the kernel function in the Caputo derivative. The stability and convergence of both difference schemes are analysed in detail. At each time level, the system of linear equations generated by the difference schemes is solved by a fast Poisson solver, thereby taking advantage of the fast difference scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of both numerical methods.