Solving the third-kind Volterra integral equation via the boundary value technique: Lagrange polynomial versus fractional interpolation

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Abstract

The solution to the third-kind Volterra integral equation (VIE3) usually has unbounded derivatives near the original point t=0, which brings difficulties to numerical computation. In this paper, we analyze two kinds of modified multistep collocation methods for VIE3: collocation boundary value method with the fractional interpolation (FCBVM) and that with Lagrange interpolation (CBVMG). The former is developed based on the non-polynomial interpolation which is particularly feasible for approximating functions in the form of tη with the real number η0. The latter is devised by using classical polynomial interpolation. The application of the boundary value technique enables both approaches to efficiently solve long-time integration problems. Moreover, we investigate the convergence properties of these two kinds of algorithms by Grönwall’s inequality.

Introduction

In this paper, we investigate the numerical solution to the following VIE3tβu(t)=f(t)+0t(ts)αK(t,s)u(s)ds,tI:=[0,T],where α+β(0,1),f(·)C(I) and u(·) is an unknown function. Moreover, the continuous function K(·,·) is defined on D={(t,s):0stT}, and f(t) may be unbounded around t=0, which definitely affects the regularity of the exact solution u(·).

In the literature, VIE3 originates from the study on the first-kind VIE (see [1]). Suppose the given functions g(·) and H(·,·) are sufficiently smooth. Then the first-kind VIE0t(ts)α1H(t,s)u(s)ds=g(t),tI,with 0<α<1, can be transformed by differentiating both sides with respect to t intoHα(t,t)u(t)=Gα(t)0tHα(t,s)tu(s)ds,whereHα(t,t)=Γ(1α)Γ(α)H(t,t),Gα(t)=0t(ts)αg(s)ds.If the kernel function H(t,t) vanishes on the subset of I, then the existence of the unique solution u(·)C(I) will no longer be guaranteed, and VIE3 has to be studied to analyze the solvability of the above equation.

VIE3 is also closely related with the cordial VIE (see [2], [3])u(t)=g(t)+(Tk,α,βu)(t),where the integral operator Tk,α,β is defined by(Tk,α,βu)(t)=0ttβ(ts)αK(t,s)u(s)ds.By employing the relation between VIE3 and cordial VIE, the existence, uniqueness and regularity of solutions of VIE3 (1) were intensively investigated in Allaei et al. [4]. In the case of 0<α+β<1, the integral operator Tk,α,β is compact, and the linear system generated by the collocation method is uniquely solvable for sufficiently small stepsize. In the case of α+β1, the kernel function takes the form of K(t,s)=sα+β1K1(t,s), where K1(·,·)C(D), and the operator Tk,α,β in the case of K1(0,0)0 is noncompact. Therefore, the solvability of the linear system can not be guaranteed in general.

In [5], Allaei et al. introduced a modified graded mesh for devising the collocation method. The solvability and convergence of collocation solutions to VIE3 (1) were investigated. Similar techniques were discussed in Song et al. [6]. Afterwards, the multistep collocation method for VIE3 (1) was constructed and the solvability and convergence were studied in detail in Shayanfard et al. [7]. By studying the operational matrix of the fractional integration and hat functions, efficient approaches were studied in Nemati and Lima [8]. By fixing the integral interval with the variable transformation, a spectral collocation method based on Jacobi wavelets and Gauss–Jacobi quadrature was proposed in Nemati et al. [9].

In addition, special classes of VIE3 (1) arise extensive attention during past decades, which are usually closely connected with heat conduction problems. For example, consider the following integral equationu(t)=g(t)+0ttβsβ1K1(t,s)u(s)ds,t[0,T],where g(·) is given and K1(·,·) is smooth. For β>1 and K1(t,s)1, the Hermite-type collocation method was devised in Diogo et al. [10], where the approximation was constructed by the cubic spline. By introducing proper function spaces, the existence of the asymptotic error expansion for Euler’s method was proved in Lima and Diogo [11]. Besides, the numerical approximation was improved by Richardson’s extrapolation. In [12], a class of product integration methods was proposed. An innovative equation was obtained by dealing with the nonsmooth solution through transformation. Then the resulting equation was solved by the product trapezoidal and Simpson’s rules.

The aim of this paper is to study high-order and stable approaches for VIE3 (1). We focus on two classes of multistep schemes which are able to provide higher order approximation to the solution of VIE3 than one-step methods with the same collocation grid. On the other hand, since the regularity of the solution to VIE3 (1) is affected by both tβ and f(t), it is difficulty to determine the singularity of u(t) at t=0 in computational practice. Thus, we always suppose that the singularity of u(t) is unknown and are interested in the computational performance of numerical methods for solving VIE3 (1) with unpredictable singularities. In order to get over the weak singularity of the solution to VIE3 (1), we discuss two approaches based on the graded mesh and fractional interpolation. In contrast to the uniform grid, the clustering of points around t=0 enables the graded mesh to deal with the weak singularity effectively (see [13], [14]). Meanwhile, the fractional interpolation is able to efficiently approximate the unknown singular solution due to the smoothing transformation (see [15]). Another difficulty we encounter in the design of numerical methods is the long-time integration problem, which can be partly overcame by the boundary value technique. The boundary value method was firstly developed to solve initial value problems of differential equations (see [16]). The main idea of such a technique is to transform the initial value problem into the boundary value problem, and to employ modified linear multistep formulae to get a stable numerical solution. Inspired by this methodology, several authors studied the boundary value method for solving VIEs (see [17], [18], [19], [20], [21]). In [22], the second author and Xiang developed the collocation boundary value method (CBVM) for the second-kind VIEs, where the local polynomial interpolation relied on several unknown approximate values. Such a approach was able to compute high-order numerical solutions without adding collocation points and maintained stable within a wide region. In [15], the second author and Liu discussed CBVM for the weakly singular VIE through fractional interpolation.

In the remaining part, we investigate FCBVM and CBVMG for solving VIE3 (1). Both approaches are presented in Section 2 with the help of fractional polynomial interpolation and the graded mesh. Some numerical examples are conducted in Section 3 to test the performance of the proposed algorithms. Afterwards, we analyze the solvability and convergence property of both methods in Section 4. Some remarks are concluded in the final section.

Section snippets

Design of FCBVM

In this subsection, we investigate a class of non-polynomial approximation methods which is particularly efficient in approximating weakly singular functions behaving as tη around t=0 and devise FCBVM.

Firstly, consider the uniform mesh XN with the constant step size h=TN. Then define the graded meshXNγ={ψ(tj):ψ(tj)=T1γtjγ,j=0,1,,N},with tjXN,γ>1, and let the modified fundamental interpolation function beϕn,jk,γ(t)=i=0,ijk+1ψ^1(t)tn+itn+jtn+i,j=0,,k+1,where ψ^(t)=T1μtμ. In this paper,

Numerical evidences

In this section, we examine the numerical properties of FCBVM and CBVMG by considering several VIE3s. In all experiments, we let log2eNe2N denote the convergence order, where the error vector eN consists of the absolute errors generated from numerical methods and eN denotes its maximum.

Example 1

Consider VIE3 with the interval [0,1]t1/3u(t)=f(t)+0t(ts)1/2u(s)ds.Here we carefully choose the force term f(t) so that the referenced solution is u(t)=t5/3.

The computed results are listed in Tables 1

Solvability and convergence

Based on the previous numerical observations, we examine the theoretical properties of numerical solutions computed by FCBVM and CBVMG in this section. We focus on the asymptotic order of the absolute errors with respect to the quantity of collocation nodes N, and all constants arising in the theoretical results are denoted as C for simplicity, which is independent of N. Besides, we suppose that the exact solution for VIE3 behaves as O(tη) near t=0 with an unknown singular parameter η.

Conclusion

We find from the numerical and theoretical results both FCBVM and CBVMG are able to compute weakly singular solutions to VIE3. It is also found that the boundary value technique enables the numerical approaches to solve long-time integration problems (see Example 3). Besides, FCBVM may perform better than CBVMG once the smoothing parameter γ is chosen properly (see Tables 7 and 10). Therefore, we can conclude FCBVM has tremendous potentials in solving weakly singular problems.

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    This work was supported by National Natural Science Foundation of China (No. 11901133) and Science and Technology Foundation of Guizhou Province (No. QKHJC[2020]1Y014).

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