A MODIFIED MECHANICAL QUADRATURE FORMULA AND ITS EXTENSIONS

The typical mechanical quadrature formula is modified as ∫ b a f (x)dx≈ n ∑ i=0 m ∑ j=0 A(i, j) f ( (xi), where f (x) ∈C(m)[a,b] and A(i, j) are the quadrature weights. Based on the Taylor-series expansion technique, the methods for determining the quadrature weights A(i, j) with the known quadrature points xi are given. The corresponding convergence and error estimate are made, then a sequence of Romberg-like quadrature formulae are analyzed. The modified mechanical quadrature formulae are further extended to solve the Riemann-Liouville fractional integral. Numerical results are carried out to show the effectiveness of the proposed methods by comparing some known methods. The proposed methods can be used to solve various linear and nonlinear integral equations with continuous and weakly singular kernels arising in practical physics, mechanics and engineering.


Introduction
Since many definite integrals cannot be solved directly by using the Newton-Leibniz formula, numerical integration formulae are very important to evaluate approximately their values.For f (x) ∈ C[a, b], the typical quadrature formula is given as with the weights A i and the quadrature points x i ∈ [a, b] (i = 0, 1, • • • , n).A method of giving A i and x i is that of evaluating the definite integral I[ f ] such as the Newton-Cotes quadrature formulae, the Gaussian quadrature formulae and so on [1].In the formula (1.1), one only considers the contributions of the values f (x i ) to the integrals.However, in a practical application, it is not enough only to consider the values of f (x) on the points x i , since some experimental data may give the values of f (x i ) and the derivatives f ( j) (x i ) ( j = 1, 2, • • • , m).For example, in order to determine the displacement S of the moving object, one should measure the instantaneous velocity V (t i ) and the acceleration a(t i ) on the discretization points t i ∈ [a, b].According to (1.1), it is easy to evaluate the displacement S as Clearly, in (1.2), one has neglected the values of a(t i ).It motivates us strongly to consider the contributions of a(t i ) to the displacement S. Then we may get more accurate results of the displacement S and decrease the quantity of the measuring points t i .
Moreover, it is noted that a complicated function can be approximated by using the Hermite interpolation and the cubic spline interpolation, where the first-order derivative and the secondorder derivative of the functions are used respectively [2].In order to generally consider the jth-order derivatives of the function f (x) for j = 1, 2, • • • , m, here we propose that the typical mechanical integration formula (1.1) is modified as where f (x) ∈ C (m) [a, b] and f ( j) (x i ) stand for the values of the jth-order derivative of f (x) with respect to x on the points x i .A (i, j) and x i are the quadrature weights and the quadrature points respectively.When m = 0, the formula (1.2) is reduced to that in (1.1).On the other hand, it is noted that the fractional integrals and derivatives have attracted much attention due to their applications in engineering and physics [3,4].It is very important to give numerical methods for evaluating the fractional integrals [5][6][7][8].So the formula in (1.2) will be extended to compute the fractional integrals.
To achieve the objectives of formulating the formula in (1.2) together with its extensions and applications in evaluating the fractional integrals, the paper is organized as follows.In Section 2, a sequence of methods for determining A (i, j) and x i will be given by using the Taylor-series expansion formula.The corresponding convergence and error estimate will also be addressed.
Section 3 will give a sequence of Romberg-like quadrature formulae.The modified mechanical quadrature formula is extended to solve the Riemann-Liouville fractional integral in Section 4.
Numerical results are carried out to show the effectiveness of the proposed methods in Section 5. Section 6 shows the main conclusions.

Determination of the quadrature weights
From the viewpoint of practical applications, the methods of determining the quadrature weights A (i, j) and the quadrature points x i in the formula (1.2) should be given.Usually, similar to Newton-Cotes quadrature formulae, it is convenient to choose the equidistant quadrature points as When f (x i ) and the first-order derivatives f (x i ) are known, the piecewise Hermite interpolation polynomial can be used to determine the coefficients A (i, j) .Furthermore, if we know the values of f (x i ), f (x i ) and f (x i ), the spline interpolation function of degree three is suitable to give the coefficients A (i, j) .In what follows, generally if the values of f are known, we give a simple method of determining the coefficients A (i, j) based on the Taylor-series expansion technique.
Now by using the equidistant quadrature points, we obtain where the variable change of x = x i + hξ is used.It is assumed that f (x i + ξ h) can be expanded as the following Taylor series where 0 and the remainder Then based on the second integral mean value theorem, there exists ξ i ∈ (0, 1) such that where M and m are the maximum and the minimum of f (m+1) (x) on [a, b] respectively, we have ) Comparing (1.3) and (2.4), it is found that From (2.7), it is seen that here we have neglected the values of have no contributions to the numerical integration formula in (2.4).
On the other hand, application of the variable change Under the assumption of with the remainder for η 2 ∈ (a, b).Based on the formula (2.9), it follows Generally, the sum of (2.4) and (2.9) with the parameter ω leads to ) together with the remainder When ω = 0, the formula in (2.10) is reduced to that in (2.4) and when ω = 1, the formula in (2.9) can be derived from that in (2.10).Furthermore, we have the following theorem.

Romberg-like quadrature formulae
In the above section, several methods for determining the coefficients A (i, j) are given by using the Taylor-series expansion technique.In order to derive high-order approximation methods from low-order ones, it is seen that the extrapolation method of Richardson is effective.Here we further apply the ideas of Romberg quadrature formulae to the proposed methods and the derived ones are called as Romberg-like quadrature formulae.
Based on the formula (2.4), we define From the remainder R 1 , after some computations, one can get and In what follows, we analyze the error estimate of the Romberg-type quadrature formula RT 1 .
One has the following theorem Romberg-type quadrature formula RT 1 has the following error estimate proof We can calculate that Moreover from the expression of R 1 and h = (b − a)/n, it is seen that where the integral mean value theorem and the intermediate value theorem have been used with . Similarly, we have where y i = a + ih, y i < ỹi < y i + h and η 4 ∈ [a, b].Now the error estimate can be rewritten as where η is between η 3 and η 4 .The proof is completed.
Moreover, from (2.9) and (2.10), we define Then the Romberg-like quadrature formulae RT 2 and RT 3 can be given as For the errors of RT 2 and RT 3 , we have the following theorem.
-times continuously differentiable.The Romberg-type quadrature formulae RT 2 and RT 3 have the following error estimate Proof The error estimate of RT 2 can be obtained similar to that of RT 1 and the proof procedure has been omitted here.In what follows, we focus on the error estimate of RT 3 .It is calculated that Furthermore, under the consideration of R 3 and h = (b − a)/n, it follows where η 5,6 ∈ [a, b].On the other hand, we have This completes the proof.Comparison between Theorems 1 and 2 shows that when m ≥ 1, the Romberg-like quadrature formulae are suitable to accelerate the modified mechanical quadrature formula.

Quadrature formulae for the R-L fractional integral
Recently, fractional integrals and their applications have attracted much attention [3][4][5][6].Numerical methods of evaluating fractional order integrals and solving fractional order differential equations are popular [7,8].In the section, the modified mechanical integration formulae will be extended to compute the Riemann-Liouville fractional integral expressed as Now the equidistant quadrature points are chosen as and one has 1 Similar to the former section, application of the Taylor-series expansion (2.3) yields 1 where B β i ( j + 1, α) is the incomplete Beta function defined as Under the assumption of f (x) ∈ C (m+1) [a,t], the remainder can be computed as ) On the other hand, applying the variable change x = x i+1 − hξ , one has 1 Based on the Taylor-series expansion in (2.3), it follows 1 The remainder is In addition, the convex combination of (4.2) and (4.4) with the weight ω is 1 and the remainder In the end, we have the following theorem.

Numerical examples
In order to illustrate the effectiveness of the proposed methods, several numerical examples are carried out in the section.For convenience, the parameters m and n are written as a pair of (m, n) in the following numerical computations.
, and we calculate with the exact result I = π 4 .Based on the present method, we choose (m, n) = (2, 2), (2, 4), (2, 8), (4, 2), (4, 4) and (4, 8) to compute.The weight ω is chosen as −1.0, −0.5, 0, 0.5 and 1.0, respectively.The absolute errors between the approximate solution I (m,n,ω) and the exact solution I are given in Table 1.It is seen from Table 1 that with the increasing of m or n for a fixed ω, the absolute errors are decreasing.In particular, it is found that the parameter of ω has large influences on the accuracy of the approximate solutions.The variations of the absolute errors versus ω are depicted in Figures 1 and 2 for some pairs of m and n.It is seen from Figures 1 and 2 that for each pair of m and n, there is a point of ω where the absolute error is tending to zero.That is to say, we can obtain a good approximation by choosing small values of m and n, then searching a suitable value of ω. respectively.
Table 1 The absolute errors of the approximate and exact solutions for Example 1.Moreover, it is significant to compare the present method with the composite trapezoid (CT) formula and the composite Simpson (CS) formula.In Table 2, we choose m = 2 and ω = 0.5 to carry out Example 1 again by using the present method.The absolute errors in Table 2 show that the proposed method is effective.On the other hand, it is noted that the analysis and applications of Abel integral equation have been studied widely [9].Abel integral equation is applied to model a fractional-order system and it is written as [3] 1 with 0 < α < 1.Its solution is expressed as the following well-known formula (5.2) Moreover, some numerical methods have been proposed to give the approximate solution of Abel integral equation (5.1) such as the quadrature methods [9], the Chebyshev polynomials method [10] and the Taylor expansion method [11].Obviously, based on the present method, the approximate solution can be obtained effectively by evaluating the integral in (5.1) and the following example is given.
The absolute errors are given in Table 3 for t = 0.2, 0.4, 0.6, 0.8, 1 and ω = 0.5 under (m, n) = (2, 4) and (m, n) = (2, 8), respectively.For convenience, the corresponding absolute errors by using the known method in [11] are computed and given in Table 3.It is found from Table 3 that the present method is effective and suitable to solve Abel integral equations.
Table 3 The absolute errors of approximate and exact solutions for Example 2.

The absolute errors
The present method: ω = 0.

Conclusions
The typical mechanical quadrature formula has been modified as a novel numerical integration formula by considering the derivatives of integrand.The Taylor-series expansion technique has been applied to obtain the coefficients of the modified mechanical quadrature formula.In order to accelerate the given quadrature formula, the Romberg-like quadrature formulae have been analyzed.The corresponding convergence and error estimate have been given.The proposed method is further extended to numerically solving Riemann-Liouville fractional integral.
Numerical results show the effectiveness of the proposed formulae.In the future, the given methods will be used to numerically solve various linear and nonlinear integral equations arising from ordinary differential equations, physics, mechanical and engineering.

with η5, 6 ∈
[a, b].So one can rewrite the error estimate of RT 3 as

Table 2
The absolute errors of approximate and exact solutions for Example 1.