Abstract
In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses a high accuracy but also has the posterior error estimate which can be used to design self-adaptive algorithm. Numerical experiments demonstrate the efficiency and applicability of the proposed method.
1. Introduction
In recent years, Volterra integral equation with delay has received a considerable amount of attention. This paper considers the following weakly singular Volterra integral equations with proportional delays:withandwhere and are known continuous functions defined on the domains and , respectively. is a known function and is an unknown function. In practice, the delay arguments are consistent with the real phenomena which make the models more realistic for simulation. Delay integral equation and partial differential equation have been widely used in many population growth and relevant phenomena in mathematical biology [1–3].
Numerical algorithms for implementing delay models should be designed specially according to the nature of the equations. There are many numerical techniques for the delay differential equations [4–8] and integral-differential equations [9–11]. Simultaneously, vast researchers also focused their interests on the numerical techniques of delay integral equations with continuous kernels, such as least squares approximation method [12], spectral method [13], Bernoulli wavelet method [14], and collocation method [15]. Xie et al. in [16] handled the Volterra integral equation with the delay function vanishing at the initial point in the given interval; they found that the iterated collocation solution possessed local superconvergence at the mesh points. In [17], the authors adopted multistep method based on Hermite collocation method and it turned out that the numerical method had uniform order with collocation points and previous time steps. In [18], an hp-spectral collocation method was used for nonlinear Volterra integral equations with vanishing variable delays.
There are a few researches on the delay integral equation with weakly singular kernels, such as [19]. In this paper, we concentrate on (1) whose upper limit of integral is a delay function and the integral kernels are weakly singular functions, which increase the computational complexity and the theoretical difficulty. To the best of our knowledge, there are no studies in (1) by the mechanical quadrature method in recent years. A further advantage of this method for (1) is that the error has an asymptotic expansion. Thus, we can improve the accuracy order of approximation. Simultaneously, the theoretical analysis is complete and the calculation is simplified.
In this paper, we firstly deduce the improved Gronwall inequality; the existence and uniqueness of the solution for (1) are testified via the improved Gronwall inequality. Then the equation is approximated by employing the floor technique to the delay argument and by adopting the quadrature formula [20] to the weakly integrals. Next, the approximate equation for (1) is constructed by combining the mechanical quadrature method and the interpolation technique; then, we use the iterative method for solving the approximate equation. The existence and uniqueness of the solution for the approximate equation are testified via the discrete Gronwall inequality. Finally, we prove that the convergent order is . In order to achieve a higher accuracy order , the Richardson- extrapolation based on the asymptotic expansion of error is adopted; moreover, a posterior error estimate is realized conveniently.
The layouts of this paper are as follows. In Section 2, we prove the existence and uniqueness of the solution for (1). In Section 3, we introduce the quadrature method, iterative method, and interpolation technique. In Section 4, the existence and uniqueness of the solution for the approximate equation are discussed. In Section 5, the convergence and the error estimation are obtained to ensure the reliability of the method. In Section 6, the asymptotic expansion of the error is achieved, a higher accuracy order is realized by extrapolation, and a posterior error estimate is derived. In Section 7, some numerical examples are demonstrated to illustrate the theoretical results. Some concluding remarks are provided in Section 8.
2. The Existence and Uniqueness of the Solution for the Original Equation
In this section, we will verify the existence and uniqueness of the solution for (1). We first prove the improved Gronwall inequality.
Lemma 1. Suppose that , , and are nonnegative integrable functions, , , and ; based on the inequality we have
Proof. Due to the fact that , andLet ; we can deduceIntegrate on both sides and then namely, The proof of the Lemma 1 is completed.
Theorem 2. Assume that are known continuous functions defined on the domains and , respectively; then the solution of (1) is existent uniquely.
Proof. We construct the sequence , satisfying where and are defined in (3), with . are continuous functions; then there exists a constant such that . We havewith ; . Now, we can deduceBy means of the mathematical induction, when , we obtainNext, we prove that is the basic sequence in ; in fact, For sufficiently small , there exists a positive integer such that when and any , we haveAccording to Cauchy’s test for convergence, the sequence () is convergent uniformly to which is the solution of (1).
Suppose that both and are the solutions of (1); let ; then we getNext, we verify that is integrable; According to Lemma 1, we can derive that ; the solution is unique.
3. The Quadrature Method and the Iterative Algorithm
Let the delay function , , satisfy the following conditions [21]:
(1) , and with (vanishing delay);
(2) on for some and ; are constants;
(3) .
For , the special case is ; we get ; (1) is a weakly integral equation with vanishing delay. In order to solve (1), the modified trapezoid quadrature formula is considered to deal with the integrals and .
It is challenging because the integral upper limit is a delay function and the integral kernels are weakly singular at the endpoint.
Lemma 3 (see [22]). Let , , and ; then the modified trapezoidal rule iswhere is the zeta function. Further, has the following asymptotic expansion:where , are the Bernoulli numbers.
Lemma 4. Let , and with ; then there is
Proof. The Taylor expansion of function at the point isSimilarly, the Taylor expansion of function at the point isCombining (22) with (23), we derive (21).
Now, the concrete approximate scheme of (1) will be constructed on the basis of the above lemmas. We divide into subintervals with the uniform step size ; . Let be replaced by in (1); that is to say,where denotes the maximum integer less than . It is obvious that , , are not the node values, and we can turn to the adjacent points and by the linear interpolation approximation with . Then, we have that are established for with , and
Now, we structure the quadrature algorithm; by Lemma 3 and the trapezoidal quadrature rule, we obtainwhere , , and are error functions, which are, respectively, equal towhere The discrete forms of (1) are obtained:Let be the approximate solution of and ignore the error function; (32) becomesThe iterative algorithm is built to solve (33).
Iterative Algorithm
Step 1. Take sufficiently small and set ; ;
Step 2. Let ; then compute as follows:
Step 3. If , set and and return to Step 2; else, let and return to Step 2.
4. The Existence and Uniqueness of the Solution for the Approximate Equation
Now, we prove the existence and uniqueness of the solution for the approximate equation. We first introduce the following lemma.
Lemma 5 (see [23]). Suppose that the sequence , , satisfieswhere and are nonnegative constants. Let with ; then we can derive
Theorem 6. Assume that is sufficiently small; then the solution of (34) is existent uniquely, and the algorithm converges at a geometrical rate.
Proof. From the nature of the delay function , we discuss the existence and uniqueness of the solution for the approximate equation under two situations.
First, we prove that the solution of (34) is existent under two situations.
(1) One situation is ; that is, when , we can easily obtain Let for a sufficiently small ; then holds.
(2) The other situation is ; that is, when , thenLet for a sufficiently small ; then holds.
With the discussion of the above two situations, one can conclude that the iterative algorithm is convergent geometrically, and the limit is the solution of (34); therefore, the solution of (34) is existent.
Next, we prove that the solution of (34) is unique. If and are solutions of (34), the difference can be represented as , , andand , because are continuous on bounded domains, with ; thenwith . Let .
(1) The first situation is , that is, when . Let for a sufficiently small ; then (40) can be written as follows:namely,where(2) The second situation is , that is, when . Let for a sufficiently small ; then (40) can be written asnamely,AndBased on Lemma 5, with and the solution of the discrete equation (34) is unique. The proof of Theorem 6 is completed.
5. The Error Estimation
In this section, we give the error estimate between the approximation solution and the exact solution of (1).
Theorem 7. Let be the exact solution of (1); the kernel functions , , , and the functions and are continuous in the domains and , respectively. Then there is a positive constant independent of such that and have the following estimation:
Proof. From (26) and (27), we haveBy the trapezoidal formula and Lemma 4, we havewhereThen, we have where . Subtracting (34) from (51), we getLetand the analysis is the same as (41) and (44); we haveFrom Lemma 5, there is a positive constant independent of such thatThe proof of Theorem 7 is completed.
6. Error Asymptotic Expansion and Extrapolation Algorithm
In this section, we present the main theoretical result of the error asymptotic expansions and the relevant extrapolation algorithm.
Theorem 8. Based on the conditions of Theorem 7, there exist continue functions satisfying the asymptotic expansion
Proof. Suppose that satisfies the auxiliary delay equations: and , , satisfy the approximation equations:Similar to the proof of Theorem 7, we obtainLetWe haveFrom Lemma 1, there exists a constant such thatThe asymptotic expansion is (56).
Based on Theorem 8, we adopt the Richardson extrapolation to improve the accuracy.
Extrapolation Algorithm
Step 1. Assume that , and halve the step length to obtainBy combining (56) with (63), we get
Step 2. We implement Richardson extrapolation:Combining (64) with (65), we getMoreover, a posterior asymptotic error estimateThe error is bounded by , which is essential to construct adaptable algorithms.
7. Numerical Experiments
In this section, three examples will be presented to show the efficiency of the quadrature method. We design a set of grids on the interval ; the absolute error is denoted bywith ; and are the exact solution and the approximate solution at , respectively. Set , and the convergence order is defined by
Example 1. Consider the following equation:where , and the analytical solution is .
The numerical results at the point with the partitions are addressed in Table 1. By the mechanical quadrature method and iterative method, the obtained absolute errors with more refined partitions show a more accurate approximate solution, and the convergent rate is adjacent to -order. Based on the Richardson extrapolation, the errors are closer to the exact solution, and the convergence order is improved to 2-order. The posteriori error is also achieved. The two kinds of convergence orders are consistent with the theoretical analysis.
Example 2. Consider the following delay Volterra integral equation with weakly singular kernel aswith , , and the initial value , and is determined by the analytical solution .
We denote the approximate solution by with . The error results at some interior points in the interval with different partition are listed in Table 2. The comparison of the exact solution and the approximate solution with partition is shown in Figure 1. It is obvious that this paper provides a high accuracy algorithm for the weakly singular Volterra integral equation with proportional delays.
Example 3. Consider the following delay Volterra integral equation with weakly singular kernel aswith , and the initial value , and is determined by the analytical solution .
The absolute error , and the posteriori errors at some interior points are listed in Table 3 when , which can observe that the error results decay quickly with the increasing of . The absolute errors between the exact solution and the approximate solution with partition are shown in Figure 2, which indicate that our algorithm is effective.
8. Conclusion
In this paper, we use the mechanical quadrature method and Romberg extrapolation for weakly Volterra integral equation with proportional delays. Most papers analyze the delay Volterra integral equation with continuous kernels; the study for the delay Volterra integral equation with weakly singular kernels still faces a real challenge for the relevant researchers in both numerical computation and theoretical analysis. The improved Gronwall inequality is adopted to prove the existence and uniqueness of the solution of the original equation. At the same time, the discrete Gronwall equality and iterative method are adopted to prove the existence and uniqueness of the solution of the approximate equation. Moreover, according to the error asymptotic expansion of the mechanical quadrature method and the extrapolation, a high order of accuracy can be achieved and a posterior error estimation can be obtained. Both the theoretical analysis and the numerical examples show that the presented method is efficient.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This work was supported by the financial support from the National Natural Science Foundation of China (Grant no. 11371079).