A Family of Modified Even Order Bernoulli-Type Multiquadric Quasi-Interpolants with Any Degree Polynomial Reproduction Property

Abstract

By using the polynomial expansion in the even order Bernoulli polynomials and using the linear combinations of the shifts of the function f(x)(x∈...) to approximate the derivatives of f(x) , we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants which do not require the derivatives of the function approximated at each node and can satisfy any degree polynomial reproduction property. Error estimate indicates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter c and a nonnegative integer m . Numerical comparisons show that this technique provides a higher degree of accuracy. Finally, applying our operators to the fitting of discrete solutions of initial value problems, we find that our method has smaller errors than the Runge-Kutta method of order 4 and Wang et al.'s quasi-interpolation scheme.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Antonio J. M. Ferreira

School of Mathematics, Jilin University, Changchun 130012, China

Received 5 November 2013; Accepted 14 January 2014; 10 March 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let X={_x0 ,...,_xn } be a set of n+1 distinct points of ... and let f be a function defined on a domain [a,b] containing X . The standard formula for interpolating the function f , where [figure omitted; refer to PDF] has the following form: [figure omitted; refer to PDF] s.t. [figure omitted; refer to PDF] for all j=0,...,n , where ...B3;(·) is an interpolation kernel. Many investigators use radial basis functions to solve the interpolation problems (2)-(3). In particular, the multiquadrics presented by Hardy [1], [figure omitted; refer to PDF] are of especial interest because of their special convergence property, see [2, 3]. Throughout this paper, let the notations _[varphi]j (·) and c denote the multiquadrics and their shape-preserving parameter as in (4), respectively. A review by Franke [4] indicated that the multiquadric interpolation is one of the best schemes among some 29 interpolation methods in terms of accuracy, efficiency, and easy implementation. Although the multiquadric interpolation is always solvable when the scattered points {_xj^}j=0n are distinct [5], the resulting matrix in (2)-(3) quickly becomes ill-conditioned as the number of the scattered points increases. In this paper, we will use the quasi-interpolation technique to overcome the ill-conditioning problem.

A weaker form of (3), well-known as quasi-interpolation, holds for all polynomials of degree no more than m , where m is nonnegative integer; that is, [figure omitted; refer to PDF] where ^...m ={p:deg...(p)[= or <, slanted]m} . Beatson and Powell [6] first proposed a univariate quasi-interpolation operator _{[Lagrangian (script capital L)]B} which reproduces constants, where ...B3;(x) in (2) is a linear combination of the multiquadrics, see (46)-(47). Afterwards, Wu and Schaback [7] proposed another quasi-interpolation operator _{[Lagrangian (script capital L)]D} which possesses shape-preserving and linear reproducing properties. They showed that the error of the operator _{[Lagrangian (script capital L)]D} is ...AA;(^h2 |ln...h|) , when the shape parameter c=...AA;(h) , where h=_{max...1[= or <, slanted]j[= or <, slanted]n} ...{|_xj -_xj-1 |} . Using the operator _{[Lagrangian (script capital L)]D} , Ling [8] constructed a multilevel quasi-interpolation operator and proved that its convergence order is ...AA;(^h2.5 |ln...h|) when c=...AA;(h) . Using the shifts of the cubic multiquadrics, Feng and Li [9] constructed a shape-preserving quasi-interpolation operator. They showed that the operator reproduces all polynomials of degree 2 or less and proved that the convergence rate is ...AA;(^h3 ) as c=...AA;(^h1.5 ) . Combining the operator _{[Lagrangian (script capital L)]B} with Hermite interpolation polynomials, Wang et al. [10] proposed a kind of improved quasi-interpolation operators _{[Lagrangian (script capital L)]H2m-1} which reproduce all polynomials of degree [= or <, slanted]2m-1 . They proved that it converges with a rate of ...AA;(^h2m ) at most. However, the operators _{[Lagrangian (script capital L)]H2m-1} require values of the derivatives at endpoints, which are not convenient for practical purposes. Further, many authors offered some examples using multiquadric quasi-interpolation operator to solve differential equations, see [11-15] for details.

Based on CAIRA-DELL'ACCIO's idea [16], we first define a family of even order Bernoulli-type multiquadric quasi-interpolants _{[Lagrangian (script capital L)]~vm} by combining the multiquadric quasi-interpolation operator _{[Lagrangian (script capital L)]B} in [6] with the polynomial expansion in even order Bernoulli polynomials _vm (x) in [17]. For practical purposes, applying the divided difference formula in [18] to the operators _{[Lagrangian (script capital L)]~vm} , we construct a family of modified even order Bernoulli-type multiquadric quasi-interpolants _{[Lagrangian (script capital L)]vm} which do not require values of the derivatives at nodes. We prove that the operators _{[Lagrangian (script capital L)]vm} reproduce all polynomials of degree [= or <, slanted]2m and have the convergence rate of ...AA;(^h2m+1 ) under a suitable assumption on the shape parameter c . Therefore, our operators _{[Lagrangian (script capital L)]vm} can provide the desired smoothness and precision in the practical applications.

The organization of the remainder of this paper is as follows. In Section 2, we briefly recall the definition of Bernoulli polynomials and even order Bernoulli polynomials giving some useful properties. We also obtain three useful theorems for the error in the even order Bernoulli polynomials expansion. In Section 3, we apply previous results to derive a family of modified even order Bernoulli-type multiquadric quasi-interpolants and get their convergence rate. In Section 4, numerical examples are shown to compare the approximation capacity of our new operators with that of CAIRA-DELL'ACCIO's interpolants and Wang et al.'s quasi-interpolants. In Section 5, we apply our operators to the fitting of discrete solutions of initial value problems for ordinary differential equations. In Section 6, we give the conclusions.

2. Some Remarks on the Polynomial Expansion

2.1. The Generalized Taylor Polynomial

The generalized Taylor polynomial is an expansion in Bernoulli polynomials _Bn (x) , that is, the polynomials of the sequence defined recursively by means of the following relations, see [19]: [figure omitted; refer to PDF] Let function f(x) be in the class ^Cm [a,b](a<b) ; then [figure omitted; refer to PDF] where the polynomial approximation term is considered as follows: [figure omitted; refer to PDF] and the remainder term is defined by [figure omitted; refer to PDF] where [·] denotes the integer part of the argument and h=b-a . The polynomial approximant _Pm [f;a,b](x) has the following result: [figure omitted; refer to PDF] where _Tm [f;a](x) is the r th Taylor polynomial of f with initial point in a . According to (10), we denote by _Pm [f;a,b](x) the generalized Taylor polynomial.

2.2. The Polynomial Expansion in Even Order Bernoulli Polynomials

Let us consider the polynomial sequence defined recursively by the following relations, see [17]: [figure omitted; refer to PDF] By (11), the polynomial sequence {_vn (x)} is related to the following Bernoulli polynomials of even degree, see [17]: [figure omitted; refer to PDF] We denote by _vn (x) the even order Bernoulli polynomials. For any function f in the class ^C2m [a,b] (a<b) , this expansion is realized by the following: [figure omitted; refer to PDF] where the polynomial expansion _Pa,m [f;a,b](x) in even order Bernoulli polynomials is defined by [figure omitted; refer to PDF] and the remainder _Ra,m [f;a,b](x) in its Peano's representation is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] In order to get bounds for remainder (15) even in points outside the interval [a,b] , we consider the operator [figure omitted; refer to PDF] where f∈^Cm [c,d] with c<a and b<d . By applying Peano's kernel theorem [20], we give an integral expression for the remainder (15) as follows.

Theorem 1.

If f∈^C2m [c,d] and x∈[c,d] , then for the remainder [figure omitted; refer to PDF] we have the following integral representations: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and ^(·)+k denotes the positive part of the k th power of the argument; that is, [figure omitted; refer to PDF]

Proof.

On one hand, there are evaluations of derivatives of f up to the order 2m-1 on points a and b of [c,d] in the approximation term (14); on the other hand, the exactness of (14) on the set ^...2m denotes the exactness of the operator _Pa,m [f;a,b] on the subset ^...2m-1 . Applying Peano's kernel theorem, we then obtain [figure omitted; refer to PDF] where (20) is given by applying the linear functional f[arrow right]_Ra,m [f;a,b](x) to a function (x-t^)+2m-1 in x . Let x∈[c,a] ; then [figure omitted; refer to PDF] Let t∈[c,x] ; then [figure omitted; refer to PDF] where (x-t^)2m-1 is considered a polynomial in x of degree 2m-1 . By the expression of (·^)+k , (20) is equal to zero in the interval b<t<d . Thus, we prove the first case of (19). The remaining cases of (19) can be got in an analogous manner.

By Theorem 1, we can get the following result.

Theorem 2.

If f∈^C2m [c,d] and x∈[c,d] , then for the remainder (18) we get [figure omitted; refer to PDF] where _||·||∞ denotes the sup-norm on [c,d] and [figure omitted; refer to PDF]

Proof.

Let c<x<a ; then we find from (19) that [figure omitted; refer to PDF] Let x<t<a ; then [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] In [21], we have the following known identity: [figure omitted; refer to PDF] By the identities [17], using relations (30), we get [figure omitted; refer to PDF] In [17], we have [figure omitted; refer to PDF] Therefore, by applying (31), we obtain the following form from (29): [figure omitted; refer to PDF] Further, by applying the third case of (32) and the identities (33), we have [figure omitted; refer to PDF] Note that the integrands are of type g(t)^f(2m) (t) with a g(t) that does not change sign in [x,a] . By applying the first mean value theorem for integrals to (34), we find for some ξ , _ξj , _ηj , _θj ∈[c,d] , j=1,...,m that [figure omitted; refer to PDF] After some calculations in (35), we obtain [figure omitted; refer to PDF] Let a<t<b ; then we have [figure omitted; refer to PDF] By the first mean value theorem for integrals, we can get after some calculations [figure omitted; refer to PDF] where _λj ∈[c,d] , j=1,...,m . By applying relations (36) and (38) to (27), we have [figure omitted; refer to PDF] By identities (32), we obtain [figure omitted; refer to PDF] By using (40) in (39), we have after some simplifications that [figure omitted; refer to PDF] Because [figure omitted; refer to PDF] we obtain the first case of expression (25). Similarly, we can prove the remaining cases.

Since the polynomials _Pa,m [f;a,b](x) of degree are not greater than 2m , we can obtain the desired bounds in an analogous manner.

Theorem 3.

If f∈^C2m+1 [c,d] and x∈[c,d] , then for the remainder (18) we get [figure omitted; refer to PDF] where _||·||∞ denotes the sup-norm on [c,d] and [figure omitted; refer to PDF]

3. The Modified Even Order Bernoulli-Type Quasi-Interpolants

The multiquadric quasi-interpolant _{[Lagrangian (script capital L)]B} [6] is defined by the following: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for i=1,...,n-1 , where {_Bi (t):t∈...} is the hat function that has the nodes {_xi-1 ,_xi ,_xi+1 } , that is identically zero outside the interval _xi-1 [= or <, slanted]t[= or <, slanted]_xi+1 and that satisfies the normalization condition _Bi (_xi )=1 . The operator _{[Lagrangian (script capital L)]B} reproduces constants. Based on the operator _{[Lagrangian (script capital L)]B} , we first define a family of even order Bernoulli-type multiquadric quasi-interpolants _{[Lagrangian (script capital L)]~vm} as follows: [figure omitted; refer to PDF] where _{Pxi ,m} [f;_xi ,_xi+1 ](x) is the natural extension of the polynomial expansion defined in (14) and _xn+1 =_xn-1 . The operators _{[Lagrangian (script capital L)]~vm} possess the polynomial reproduction property as follows.

Theorem 4.

The operators _{[Lagrangian (script capital L)]~vm} reproduce all univariate polynomials of degree no more than 2m .

Proof.

The argument _{[Lagrangian (script capital L)]~vm} [p;a,b](x)=p follows from the well-known property [figure omitted; refer to PDF] since _{Pxi ,m} [p;_xi ,_xi+1 ](x)=p for i=0,...,n , where p∈^...2m .

Although the quasi-interpolants _{[Lagrangian (script capital L)]~vm} reproduce all polynomials of degree [= or <, slanted]2m , they require the derivative of f at every node, which are very difficult to measure in practice. Therefore, we use divided difference operator ^DA2j-1 f in following Definition 5 to approximate ^f(2j-1) in the operators _{[Lagrangian (script capital L)]~vm} and then get a family of modified even order Bernoulli-type multiquadric quasi-interpolants _{[Lagrangian (script capital L)]vm} .

Definition 5 (see [18]).

Let ...={f|"f:...[arrow right]...} and let A be a discrete subset of ... , j∈... . Suppose that ^D2j-1 is the order 2j-1 derivative. An operator ^DA2j-1 :...[arrow right]... is said to be a _...2m -exact A -discretization of ^D2j-1 if and only if

(i) there exists a real vector λ=(_λa)a∈A s.t. for any f∈... , [figure omitted; refer to PDF]

(ii) for any p∈_...2m , [figure omitted; refer to PDF]

In such situation, we also say that ^DA2j-1 f is a _...2m -exact A -discretization of ^D2j-1 f . Let the points be distinct in the set A ; then ^DA2j-1 is determined uniquely.

Let |·| denote the number of elements in set. Let the points in set A be distinct and |A|=2m+1 ; then by Definition 5 and [18], a _...2m -exact A -discretization of the order 2j-1 derivative ^f(2j-1) is [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

By virtue of the location of each pair _xi , _xi+1 (i=0,...,n) , we choose suitable sets _Axi , _Axi+1 and then replace ^f(2j-1) (_xi ) and ^f(2j-1) (_xi+1 ) with ^{D_Axi 2j-1} f(_xi ) and ^{D_Axi+1 2j-1} f(_xi+1 ) , respectively. Thus, the modification quasi-interpolants _{[Lagrangian (script capital L)]vm} can be expressed as follows: [figure omitted; refer to PDF] Note that the expressions of ^{D_Axi 2j-1} f(_xi ) (i=0,...,n, j=1,...,m) in the modification operators _{[Lagrangian (script capital L)]vm} are provided by the following theorem.

Theorem 6.

For any A⊂... and x∈... , let A be a _...2m -unisolvent set and _Ax =A-x denote the set of points e∈... of the form e=a-x , where a∈A . Let A={_xi ,...,_xi+2m } (i=0,...,m-1) , {_xi-m ,...,_xi ,...,_xi+m } (i=m,...,n-m) , and {_xi-2m ,...,_xi } (i=n-m+1,...,n) . Then, for each j=1,...,m , we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

For each x∈... , we set _Ax ={_xi -x,...,_xi+2m -x} (i=0,...,m-1) , {_xi-m -x,...,_xi -x,...,_xi+m -x} (i=m,...,n-m) , and {_xi-2m -x,...,_xi -x} (i=n-m+1,...,n) . According to (50) and (53), we get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Therefore, we have [figure omitted; refer to PDF] Let us set x=_xi ; then we get the proof of the Theorem 6.

Remark 7.

For m=1 , we give the expression of the modification operator _{[Lagrangian (script capital L)]v1} as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

3.1. The Polynomial Reproduction Properties of the Operators _{[Lagrangian (script capital L)]vm}

Theorem 8.

The operators _{[Lagrangian (script capital L)]vm} reproduce all univariate polynomials of degree no more than 2m .

Proof.

By using the proof of Theorem 4 and formulas (51)-(54), we get the proof of Theorem 8 immediately.

3.2. The Convergence Rate of the Operators _{[Lagrangian (script capital L)]vm}

In order to obtain the convergence rate of the modified multiquadric quasi-interpolants _{[Lagrangian (script capital L)]vm} , we make use of the following notations: [figure omitted; refer to PDF] where #(·) denotes the cardinality function. So, 2h=_{max...1[= or <, slanted]i[= or <, slanted]n} ...|_xi -_xi-1 | and M denotes the maximum number of points from X contained in an interval _Ih (x) . At first, for the quasi-interpolants _{[Lagrangian (script capital L)]~vm} , we then give the error estimates as follows.

Theorem 9.

Let c satisfy [figure omitted; refer to PDF] where D is a positive constant and l is a positive integer. Let f(x)∈^C2m [a,b] ; then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and ^{C[variant prime]} is a positive constant independent of f , x , and X .

Proof.

Let each pair _xi , _xi+1 ∈[a,b] , be fixed and let _xi <_xi+1 . For each x∈[a,b] we make use of the following settings: [figure omitted; refer to PDF] By applying (14) to (48), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Assume that [figure omitted; refer to PDF] where the set ^...j=-NN_Qh (x+2hj) denotes the covering of [a,b] with half open intervals. Thus, for every i∈{0,...,n} , there exists a unique j∈{0,...,N} s.t. _xi ∈_Tj . Then, we get the following inequalities: [figure omitted; refer to PDF] where j=2,...,N and _τi ∈[_xi-1 ,_xi+1 ] . Therefore, we have from (70) [figure omitted; refer to PDF] We also obtain from the definition of M [figure omitted; refer to PDF] On the other hand, when _x0 ∈_Tj , j=2,...,N , we get, after some calculations, by applying the first mean value theorem for integrals to (46), [figure omitted; refer to PDF] where _τ0 ∈[_x0 ,_x1 ] . When _xn ∈_Tj ,j=2,...,N , we obtain in an analogous manner [figure omitted; refer to PDF] When _xi (i=1,...,n-1)∈_yj (j=2,...,N) , we also obtain [figure omitted; refer to PDF] where _τi ∈[_xi-1 ,_xi+1 ] . Then, for (68), we have [figure omitted; refer to PDF] where the last inequality follows from [figure omitted; refer to PDF] Let 2m<2l-1 ; then 2^h2m +^{D2h2l+2m-2∑j=1N} ...^j2m-2 =...AA;(^h2m ) .

Let 2m[= or >, slanted]2l-1 ; then 2^h2m +^{D2h2l+2m-2∑j=1N} ...^j2m-2 =...AA;(^h2l-1 ) .

Applying Theorem 3, we can obtain the desired error estimates of the operator _{[Lagrangian (script capital L)]~vm} in an analogous manner.

Theorem 10.

Let c satisfy [figure omitted; refer to PDF] where D is a positive constant and l is a positive integer. Let f(x)∈^C2m+1 [a,b] ; then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and ^{C[variant prime][variant prime]} is a positive constant independent of f , x , and X .

Because of disadvantage with the derivatives in the operators _{[Lagrangian (script capital L)]~vm} , we give the following desired error estimates of the modification quasi-interpolants _{[Lagrangian (script capital L)]vm} .

Theorem 11.

Let c satisfy [figure omitted; refer to PDF] where D is a positive constant and l is a positive integer. Let f(x)∈^C2m+1 [a,b] ; then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and C is a positive constant independent of f , x , and X .

Proof.

Consider [figure omitted; refer to PDF] The second term of the right-hand sides in (84) has been obtained from Theorem 10, so we only need to prove the first term.

We denote by _hmax... and _hmin... the maximum and the minimum distance between adjacent nodes, respectively. Let h=_hmax... /_hmin... [= or >, slanted]1 and _C0 =_{||^f(2m+1) (x)||∞} .

Let _C1 be a constant; then, according to [18], we get [figure omitted; refer to PDF] Therefore, we obtain [figure omitted; refer to PDF] Let _C2 ,_C3 ,...,_C2m ,C¯,C_,C be constants; then [figure omitted; refer to PDF] Finally, applying Theorem 10 and (87) to (84), we get [figure omitted; refer to PDF] Let 2m+1<2l-1 , then 2^h2m+1 +^D2h2l-1 =...AA;(^h2m+1 ) .

Let 2m+1[= or >, slanted]2l-1 , then 2^h2m+1 +^D2h2l-1 =...AA;(^h2l-1 ) .

4. Numerical Examples

We consider the following functions on the interval [0,1] , which are firstly used in [16]: [figure omitted; refer to PDF] We apply the interpolation operators _SBm , the quasi-interpolation operators _{[Lagrangian (script capital L)]H2m-1} , and the quasi-interpolants _{[Lagrangian (script capital L)]vm} on the above functions with c=^(2h)l , where _SBm and _{[Lagrangian (script capital L)]H2m-1} are defined by [10, 16], respectively.

We use uniform grids of 21 points for the operators _SBm , _{[Lagrangian (script capital L)]H2m-1} , and _{[Lagrangian (script capital L)]vm} in Tables 1, 2, 3, and 4. In order to estimate the errors as accurate as possible, we compute the approximation functions at the points i/101 , i=1,...,100 . Tables 1-4 show the mean and max errors which are computed for different values of the parameters μ , l , and m . The numerical results show that our quasi-interpolants _{[Lagrangian (script capital L)]vm} have better approximation behavior.

Table 1: Numerical results of the operators _SBm and _{[Lagrangian (script capital L)]vm} for the saddle function.

S B m f 1	[Lagrangian (script capital L)] v m f 1
( μ , m )	[straight epsilon] mean	[straight epsilon] max ...	( l , m )	[straight epsilon] mean	[straight epsilon] max ...
( 2,1 )	1.002 × 1 0 - 3	4.786 × 1 0 - 3	( 2,1 )	2.237 × 1 0 - 4	1.126 × 1 0 - 3
( 2,2 )	5.836 × 1 0 - 4	3.654 × 1 0 - 3	( 2,2 )	3.356 × 1 0 - 6	2.745 × 1 0 - 5
( 2,3 )	4.148 × 1 0 - 4	2.776 × 1 0 - 3	( 2,3 )	4.020 × 1 0 - 7	2.300 × 1 0 - 6
( 3,1 )	4.100 × 1 0 - 4	2.578 × 1 0 - 3	( 3,1 )	2.063 × 1 0 - 4	1.097 × 1 0 - 3
( 3,2 )	1.626 × 1 0 - 4	1.200 × 1 0 - 3	( 3,2 )	1.698 × 1 0 - 6	2.623 × 1 0 - 5
( 3,3 )	6.893 × 1 0 - 5	2.332 × 1 0 - 4	( 3,3 )	1.325 × 1 0 - 8	2.021 × 1 0 - 7
( 4,1 )	3.008 × 1 0 - 4	3.643 × 1 0 - 3	( 4,1 )	1.824 × 1 0 - 4	1.053 × 1 0 - 3
( 4,2 )	1.822 × 1 0 - 4	2.004 × 1 0 - 4	( 4,2 )	9.620 × 1 0 - 7	2.363 × 1 0 - 5
( 4,3 )	1.844 × 1 0 - 5	5.491 × 1 0 - 4	( 4,3 )	9.891 × 1 0 - 8	2.031 × 1 0 - 7

Table 2: Numerical results of the operators _{[Lagrangian (script capital L)]H2m-1} and _{[Lagrangian (script capital L)]vm} for the saddle function.

[Lagrangian (script capital L)] H 2 m - 1 f 1	[Lagrangian (script capital L)] v m f 1
( l , m )	[straight epsilon] mean	[straight epsilon] max ...	( l , m )	[straight epsilon] mean	[straight epsilon] max ...
( 2,1 )	2.654 × 1 0 - 4	1.200 × 1 0 - 3	( 2,1 )	2.237 × 1 0 - 4	1.126 × 1 0 - 3
( 2,2 )	6.678 × 1 0 - 6	4.005 × 1 0 - 5	( 2,2 )	3.356 × 1 0 - 6	2.745 × 1 0 - 5
( 2,3 )	3.400 × 1 0 - 6	1.280 × 1 0 - 5	( 2,3 )	4.020 × 1 0 - 7	2.300 × 1 0 - 6
( 3,1 )	2.541 × 1 0 - 4	1.180 × 1 0 - 3	( 3,1 )	2.063 × 1 0 - 4	1.097 × 1 0 - 3
( 3,2 )	6.435 × 1 0 - 6	4.160 × 1 0 - 5	( 3,2 )	1.698 × 1 0 - 6	2.623 × 1 0 - 5
( 3,3 )	1.236 × 1 0 - 7	1.230 × 1 0 - 6	( 3,3 )	1.325 × 1 0 - 8	2.021 × 1 0 - 7
( 4,1 )	2.540 × 1 0 - 4	1.180 × 1 0 - 3	( 4,1 )	1.824 × 1 0 - 4	1.053 × 1 0 - 3
( 4,2 )	6.435 × 1 0 - 6	4.160 × 1 0 - 5	( 4,2 )	9.620 × 1 0 - 7	2.363 × 1 0 - 5
( 4,3 )	1.185 × 1 0 - 7	1.230 × 1 0 - 6	( 4,3 )	9.891 × 1 0 - 8	2.031 × 1 0 - 7

Table 3: Numerical results of the operators _SBm and _{[Lagrangian (script capital L)]vm} for the sphere function.

S B m f 2	[Lagrangian (script capital L)] v m f 2
( μ , m )	[straight epsilon] mean	[straight epsilon] max ...	( l , m )	[straight epsilon] mean	[straight epsilon] max ...
( 2,1 )	2.012 × 1 0 - 3	6.732 × 1 0 - 3	( 2,1 )	2.356 × 1 0 - 4	4.664 × 1 0 - 4
( 2,2 )	1.521 × 1 0 - 4	9.474 × 1 0 - 4	( 2,2 )	1.001 × 1 0 - 6	2.045 × 1 0 - 6
( 2,3 )	1.900 × 1 0 - 4	8.204 × 1 0 - 4	( 2,3 )	9.097 × 1 0 - 8	9.763 × 1 0 - 7
( 3,1 )	4.834 × 1 0 - 4	1.255 × 1 0 - 3	( 3,1 )	2.304 × 1 0 - 4	4.024 × 1 0 - 4
( 3,2 )	2.801 × 1 0 - 5	1.189 × 1 0 - 4	( 3,2 )	9.972 × 1 0 - 8	3.665 × 1 0 - 7
( 3,3 )	2.711 × 1 0 - 5	1.080 × 1 0 - 4	( 3,3 )	8.851 × 1 0 - 9	3.541 × 1 0 - 9
( 4,1 )	3.991 × 1 0 - 4	1.332 × 1 0 - 3	( 4,1 )	2.636 × 1 0 - 4	3.964 × 1 0 - 4
( 4,2 )	1.989 × 1 0 - 5	1.278 × 1 0 - 4	( 4,2 )	8.614 × 1 0 - 8	3.524 × 1 0 - 7
( 4,3 )	1.249 × 1 0 - 5	5.604 × 1 0 - 5	( 4,3 )	9.811 × 1 0 - 9	2.004 × 1 0 - 8

Table 4: Numerical results of the operators _{[Lagrangian (script capital L)]H2m-1} and _{[Lagrangian (script capital L)]vm} for the sphere function.

[Lagrangian (script capital L)] H 2 m - 1 f 2	[Lagrangian (script capital L)] v m f 2
( l , m )	[straight epsilon] mean	[straight epsilon] max ...	( l , m )	[straight epsilon] mean	[straight epsilon] max ...
( 2,1 )	3.037 × 1 0 - 4	5.727 × 1 0 - 4	( 2,1 )	2.356 × 1 0 - 4	4.664 × 1 0 - 4
( 2,2 )	1.590 × 1 0 - 6	4.065 × 1 0 - 6	( 2,2 )	1.001 × 1 0 - 6	2.045 × 1 0 - 6
( 2,3 )	9.838 × 1 0 - 7	2.290 × 1 0 - 6	( 2,3 )	9.097 × 1 0 - 8	9.763 × 1 0 - 7
( 3,1 )	2.848 × 1 0 - 4	5.653 × 1 0 - 4	( 3,1 )	2.304 × 1 0 - 4	4.024 × 1 0 - 4
( 3,2 )	2.217 × 1 0 - 7	3.077 × 1 0 - 6	( 3,2 )	9.972 × 1 0 - 8	3.665 × 1 0 - 7
( 3,3 )	1.581 × 1 0 - 8	3.135 × 1 0 - 8	( 3,3 )	8.851 × 1 0 - 9	3.541 × 1 0 - 9
( 4,1 )	2.848 × 1 0 - 4	5.653 × 1 0 - 4	( 4,1 )	2.636 × 1 0 - 4	3.964 × 1 0 - 4
( 4,2 )	6.017 × 1 0 - 7	3.107 × 1 0 - 6	( 4,2 )	8.614 × 1 0 - 8	3.524 × 1 0 - 7
( 4,3 )	1.730 × 1 0 - 8	6.000 × 1 0 - 8	( 4,3 )	9.811 × 1 0 - 9	2.004 × 1 0 - 8

5. An Application of the New Operators

After solving the following initial value problems: [figure omitted; refer to PDF] by virtue of a discrete method, we often need to master the solution on a set of points that differs from the grid. Here we use our operators _{[Lagrangian (script capital L)]vm} to solve the problems. In fact, combinations of our operators _{[Lagrangian (script capital L)]vm} with discrete solvers of ODEs provide approximations of the solution of the problems (90) on [a,b] . An algorithm for constructing these quasi-interpolants is given as follows. The discrete solver produces an approximation _y~i of the exact solution y(_xi ) at nodes _xi , i=0,...,n in [a,b] . Substituting the exact values mentioned above into the definition of our operators _{[Lagrangian (script capital L)]vm} by their respective approximations, we get the proposed quasi-interpolants. We consider the initial value problems as follows.

Problem A [figure omitted; refer to PDF]

Problem B [figure omitted; refer to PDF] The exact solutions of Problems A and B are y(x)=(x-1)^ex and y(x)=-2^cos...2 x , respectively. By the Runge-Kutta method of order 4, we obtain the _y~i on a uniform grid of 21 nodes in [0,1] . Calculating the approximative functions at points i/101 , i=1,...,100 , we get the mean and max errors in Table 5. Comparing the approximation capacity of our proposed quasi-interpolants with that of Runge-Kutta scheme of order 4 and Wang et al.'s quasi-interpolation scheme [10] in Table 5, we find that our technique has smaller errors in the Problems A and B.

Table 5: Numerical results of the initial value Problems A and B.

	Problem A	Problem B
	[straight epsilon] mean	[straight epsilon] max ...	[straight epsilon] mean	[straight epsilon] max ...
y ~ i	6.34 × 1 0 - 7	1.94 × 1 0 - 6	2.09 × 1 0 - 6	1.23 × 1 0 - 5
[Lagrangian (script capital L)] H 2 m - 1 , ( l , m ) = ( 4,2 )	2.30 × 1 0 - 7	6.18 × 1 0 - 7	7.00 × 1 0 - 7	2.43 × 1 0 - 6
[Lagrangian (script capital L)] H 2 m - 1 , ( l , m ) = ( 4,3 )	8.05 × 1 0 - 8	5.72 × 1 0 - 7	3.84 × 1 0 - 7	4.02 × 1 0 - 6
[Lagrangian (script capital L)] v m , ( l , m ) = ( 4,2 )	6.02 × 1 0 - 8	9.17 × 1 0 - 8	9.32 × 1 0 - 8	3.43 × 1 0 - 7
[Lagrangian (script capital L)] v m , ( l , m ) = ( 4,3 )	1.67 × 1 0 - 9	2.44 × 1 0 - 7	8.72 × 1 0 - 8	6.59 × 1 0 - 7

6. Conclusions

In this paper, we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants _{[Lagrangian (script capital L)]vm} which reproduce polynomials of higher degree. There is no demand for the derivatives of function f approximated at each node in our operators _{[Lagrangian (script capital L)]vm} , so they do not increase the orders of smoothness of the function f . Under a certain assumption, we give an expected result on the convergence rate of our operators _{[Lagrangian (script capital L)]vm} . The numerical examples show that our operators _{[Lagrangian (script capital L)]vm} produce higher degree of accuracy. Furthermore, applying the operators _{[Lagrangian (script capital L)]vm} to the fitting of discrete solutions of initial value problems, we find that our operators _{[Lagrangian (script capital L)]vm} provide more accurate approximation solver.

Acknowledgment

The work was supported by National Natural Science Foundation of China (Grant nos. 61373003 and 11271041).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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