DERIVATION OF NEW EFFICIENT QUADRATURE RULES USING

J OURNAL OF I NEQUALITIES AND S PECIAL F UNCTIONS ISSN: ..., URL: HTTP :// WWW. ILIRIAS . COM VOLUME I SSUE 1(2016), PAGES 1-22.

DERIVATION OF NEW EFFICIENT QUADRATURE RULES USING OSTROWSKI TYPE INTEGRAL INEQUALITIES FOR n−TIMES DIFFERENTIABLE MAPPINGS A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

A BSTRACT. In this paper we derive quadrature rules using Ostrowski type inequalities for n−times differentiable mappings by using a more generalized kernel. Some well known inequalities are derived as special cases of the inequalities obtained with the help of the generalized kernel. As a consequence a number of efficient quadrature rules are derived. Specific examples are used to show their effectiveness. A parameter called h is used to control the efficiency of the quadrature rules and the best values of h are identified at which specific quadrature rules give optimum result. In addition, it is also shown that the accuracy of each quadrature rule increases with the increasing value of n.

1. I NTRODUCTION In 1938, Ostrowski [1] obtained a bound for the absolute value of the difference of a function to its average over a finite interval. Theorem 1.1. Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b) , whose derivative f 0 : (a, b) → R is bounded on (a, b) , i.e. kf 0 k∞ = supt∈[a,b] |f 0 (t)| < ∞, then " 2  2 # a+b M b−a + x− , (1.1) |S (f ; a, b)| ≤ 2 2 b−a for all x ∈ [a, b], where the functional S (f ; a, b) represent the deviation of f (x) from its integral mean over [a, b], S (f ; a, b) = f (x) − M (f ; a, b) ,

(1.2)

and 1 M (f ; a, b) = b−a

Zb f (x) dx

(1.3)

a

is the integral mean. If we assume that f 0 ∈ L∞ [a, b] and kf 0 k∞ = ess sup |f 0 (t)| then M in (1.1) may be t∈[a,b]

replaced by kf 0 k∞ . 2000 Mathematics Subject Classification. Primary 26D15, 26D20, 26D99. ˇ ˇ Key words and phrases. Ostrowski inequality, Cebyˆ sev-Gr¨uss inequality and Cebyˆ sev functional. c

2010 Ilirias Publications, Prishtin¨e, Kosov¨e. Submitted March 28, 2016. Published 2016. 1

2

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

The inequality of Ostrowski type is considered the most useful inequality in mathematical analysis. Some of the classical inequalities for means can be derived from [9] for particular choices of the function f. The inequality (1.1), have received considerable attention in the literature, to mention a few, see [6]-[8] and [14]-[16]. Utilizing an integration by parts argument, Dragomir et. al [2]-[3], obtained inequalities of Ostrowski type for mappings whose second derivatives belong to different Lebesgue ˇ spaces. In [4], Pachpatte established Cebyˆ sev type inequalities by using Pecari´c’s extension of the Montgomery identity [5]. Cerone [6] established ostrowski type inequality for one time differentiable mappings. Qayyum et. al in [15] and [16] improved the Ostrowski integral type inequalities. In this current work we will derive Ostrowski’s inequality for n−times differentiable mappings by using a more generalized kernel. With the help of thsi kerenl , we found new error bounds for various quadrature rules. The parameter h has main role to to find the efficiency of various quadrature rules obtained. Efficiency of quadrature rules also depend upon the increasing of n. At the end, we present some graphs also to see the performance of quadrature rules in glance.

2. D ERIVATION OF O STROWSKI INEQUALITY FOR AN n−TIMES DIFFERENTIABLE MAPPINGS

In this section, we will derive the main ostrowski inequality which will be used to derive quadrature rules which is the main objective of this work. Define the peano kernel P (x, .) : [a, b] → R,by

Pn (x, t) =

n

 

C n!

(t − A) , a ≤ t ≤ x,



D n!

(t − B) , x < t ≤ b,

(2.1) n

where

A = B

=

C

=

D

=

b−a , 2 b−a b−h , 2  α 1 , α+β x−a   β 1 , α+β b−x a+h

  b−a for all x ∈ a + h b−a , h ∈ [0, 1] and α, β ∈ R are non negative and not both 2 ,b − h 2 zero. Now we will state and prove the following identity which will be used to prove our main theorem.

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

3

Lemma 2.1. Let f : [a, b] → R be a mapping such that f (n−1) (x) is absolutely continuous on [a, b], then we have the identity: Zb

Pn (x, t)f (n) (t)dt

(2.2)

a n−1 X

n+k+1 h n o (−1) k+1 (k) k+1 (k) C (x − A) f (x) − (a − A) f (a) (k + 1)! k=0 n oi k+1 (k) k+1 (k) +D (b − B) f (b) − (x − B) f (x)  x  Z Zb n + (−1) C f (t) dt + D f (t) dt ,

=

a

for all x ∈ a + h b−a 2 ,b − h n ≥ 1, k ≥ 1. 

x

 b−a 2

and h ∈ [0, 1], where n and k are natural numbers,

Proof: The proof of (2.2) is established using mathematical induction. Let n = 1, then Zb L.H.S of (2.2) = P1 (x, t)f 0 (t)dt a

After integrating by parts, we get Zb

P1 (x, t)f 0 (t)dt

a

= C [(x − A) f (x) − (a − A) f (a)] + D [(b − B) f (b) − (x − B) f (x)]  x  Z Zb − C f (t)dt + D f (t)dt . (2.3) a

x

Equation (2.3), is identical to the R.H.S of (2.2) for n = 1. To complete the proof by induction, we need to prove (2.2) for n + 1 and assume that it is true for n. That is, we have to prove the equality Zb

Pn+1 (x, t)f (n+1) (t)dt

a

=

n n+k+2 h n o X (−1) k+1 (k) k+1 (k) C (x − A) f (x) − (a − A) f (a) (k + 1)! k=0 n oi k+1 (k) k+1 (k) + D (b − B) f (b) − (x − B) f (x)  x  Z Zb n+1  + (−1) C f (t)dt + D f (t)dt . a

x

(2.4)

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A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

Zb

Pn+1 (x, t)f (n+1) (t)dt

a

 x  Z Zb 1 C (t − A)n+1 f (n+1) (t)dt + D (t − B)n+1 f (n+1) (t)dt (n + 1)!

=

a

x

h i C n+1 (n) n+1 (n) (x − A) f (x) − (a − A) f (a) (n + 1)! h i D n+1 (n) n+1 (n) + (b − B) f (b) − (x − B) f (x) (n + 1)!  x  Z Zb (n + 1)  n n C (t − A) f (n) (t)dt + D (t − B) f (n) (t)dt − (n + 1)!

=

a

=

x

h i C n+1 (n) n+1 (n) (x − A) f (x) − (a − A) f (a) (n + 1)! h i D n+1 (n) n+1 (n) + (b − B) f (b) − (x − B) f (x) (n + 1)! (n−1 o X (−1)n+k+1 h n k+1 (k) k+1 (k) − C (x − A) f (x) − (a − A) f (a) (k + 1)! k=0 n oio k+1 (k) k+1 (k) +D (b − B) f (b) − (x − B) f (x)  x  Z Zb n + (−1) C f (t) dt + D f (t) dt a

=

x

h i C n+1 (n) n+1 (n) (x − A) f (x) − (a − A) f (a) (n + 1)! h i D n+1 (n) n+1 (n) (b − B) f (b) − (x − B) f (x) + (n + 1)! n−1 o X (−1)n+k+2 h n k+1 (k) k+1 (k) + C (x − A) f (x) − (a − A) f (a) (k + 1)! k=0 n oi k+1 (k) k+1 (k) +D (b − B) f (b) − (x − B) f (x)  x  Z Zb n+1  + (−1) C f (t) dt + D f (t) dt a

x

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

=

n n+k+2 h X (−1) k=0

(k + 1)! k+1

+D{(b − B)  + (−1)

n+1

C{(x − A)

k+1

k+1

f (k) (x) − (a − A)

f (k) (b) − (x − B)

Zx

C

k+1

Zb f (t)dt + D

a

5

f (k) (a)}

i f (k) (x)} 

f (t)dt x

= R.H.S of (2.4). This completes the proof of lemma 1. Theorem 2.2. Let f : [a, b] → R be a mapping such that f (n−1) (x) (n ≥ 1) is absolutely continuous on [a, b], then the inequality |τ (x; α, β)|   h i   C (x − A)n+1 − (a − A)n+1  f (n)   k k∞   i h ,   n+1 n+1  (n+1)!   +D (B − x) − (B − b)      f (n) ∈ L∞ [a, b] ,         h i 1     C q (x − A)nq+1 − (a − A)nq+1  q f (n) k kp h i ≤ 1 , nq+1 nq+1  q  q  n!(nq+1)  +D (B − x) − (B − b)      , f (n) ∈ Lp [a, b] , p > 1, p1 + 1q = 1,        (n)  n n   C (x − A) + D (B − x) kf k1   n n  2n! ,  + |C (x − A) − D (B − x) |    f (n) ∈ L1 [a, b] .

(2.5)

where τ (x; α, β) =

n−1 X

(2.6)

n+k+1

h n o (−1) k+1 (k) k+1 (k) C (x − A) f (x) − (a − A) f (a) (k + 1)! k=0 n oi k+1 (k) k+1 (k) f (b) − (x − B) f (x) +D (b − B) n

+ (−1) [αM (f, a, x) + βM (f, x, b)] ,   b−a holds for all x ∈ a + h b−a and h ∈ [0, 1] . 2 ,b − h 2 Proof: From (2.2) and (2.6), we get b Z Zb Zb (n) |τ (x; α, β)| = Pn (x, t)f (t)dt ≤ |Pn (x, t)| dt f (n) (t) dt a

≤

a



(n)

f (t)

a

Zb |Pn (x, t)| dt.

∞ a

(2.7)

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A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

It can easily be shown that Zb |Pn (x, t)| dt a

h   1 n+1 n+1 C (x − A) − (a − A) (n + 1)!  i n+1 n+1 +D (B − x) − (B − b) .

=

From (2.7) and (2.1), we get the first inequality of (2.5). Now, using H¨older’s integral inequality, from (2.7), we get |τ (x; α, β)|  b  q1 Z



q ≤  |Pn (x, t)| dt f (n) . p

a

=

(n)

f p

h 1 q

  nq+1 nq+1 C q (x − A) − (a − A)

n! (nq + 1)  i q1 nq+1 nq+1 +Dq (B − x) . − (B − b) This completes the proof of the second inequality of (2.5). Finally, by using the definition of k.k1 − norm, we get |τ (x; α, β)| ≤



sup |Pn (x, t)| f (n)

t∈[a,b]

=

1

(n)

f n n 1 [(C (x − A) + D (x − B) ) 2n! n n + |C (x − A) − D (x − B) |] ,

which is the last inequality of (2.5). This completes the proof of Theorem 2. Remark. If we substitute n = 1, in (2.5), we get the result of Qayyum et al [15]. Remark. By substituting h = 0 and n = 1, in (2.5), we get Cerone’s result given in [6]. Remark. For n = 2, (2.5) confirms the result of Qayyum et al given in [16]. Remark. Similarly, h = 0 and n = 2, confirms the result given by Qayyum et al in [12]. 3. N EW I NEQUALITIES VIA Cˇ EBY Sˆ EV FUNCTIONAL In this section, we will establish some new results by using Gr¨uss inequality and the ˇ Cebyˆ sev functional. ˇ In 1882, Cebyˆ sev [4] gave the following inequality. |T (f, g)| ≤

1 2 (b − a) kf 0 k∞ kg 0 k∞ , 12

(3.1)

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

7

where f, g : [a, b] → R are absolutely continuous functions, have bounded first derivative, and Zb Zb Zb 1 1 f (x) dx. g (x) dx T (f, g) : = f (x) g (x) dx − 2 b−a (b − a) a

a

a

=

M (f g; a, b) − M (f ; a, b) M (g; a, b) .

(3.2)

In 1935, Gr¨uss [13] proved the following inequality: Zb Zb Zb 1 1 1 b − a f (x) g (x) dx − b − a f (x) dx b − a g (x) dx a

a

(3.3)

a

1 (Φ − ϕ) (Γ − γ) , 4 such that f and g are two integrable functions on [a, b] and ≤

ϕ ≤ f (x) ≤ Φ and γ ≤ g (x) ≤ Γ, for all x ∈ [a, b] . The constant

1 4

(3.4)

is best possible.

Theorem 3.1. Let f : [a, b] → R be a mapping such that f (n−1) (x) is absolutely continuous on [a, b], and α, β are non-negative real numbers. Then h n o n+1 n+1 − (a − A) (3.5) τ (x; α, β) − C (x − A) n oi κ n+1 n+1 +D (b − B) − (x − B) (n + 1)!  21 

1

(n) 2 2 ≤ (b − a) N (x)

f − κ b−a 2 (Γ − γ) (Φ − ϕ) , ≤ (b − a) 2 where τ (x; α, β) is as given by (2.6) and κ :=

f (n−1) (b) − f (n−1) (a) , b−a

N 2 (x) =

(3.7) 1 2

h   2n+1 2n+1 C 2 (x − A) − (a − A)

(b − a) (2n + 1) (n!)  i 2n+1 2n+1 +D2 (b − B) − (x − B)  h   1 n+1 n+1 − C (x − A) − (a − A) (b − a) (n + 1)! n oi2 n+1 n+1 +D (b − B) − (x − B) , Φ

=

sup Pn (x, t) t∈[a,b]

= ϕ =

(3.6)

h i 1 n−1 n−1 α (x − a) + β (x − b) , (α + β) n! inf Pn (x, t) = 0,

t∈[a,b]

8

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

if n is even, and Φ

=

sup Pn (x, t) = t∈[a,b]

1 n−1 α (x − a) , (α + β) n!

and n−1

ϕ =

inf Pn (x, t) =

t∈[a,b]

−β (b − x) (α + β) n!

if n is odd. Proof: ˇ Gr¨uss type results involving the Cebyˆ sev functional is T (f, g) = M (f g; a, b) − M (f ; a, b) M (g; a, b) , where M is the integral mean defined in (1.3). Associating f (t) with Pn (x, t) and g (t) with f (n) (t), we obtain   T Pn (x, .) , f (n) (.); a, b     = M Pn (x, .) f (n) (.); a, b − M (Pn (x, .) ; a, b) M f (n) (.); a, b . Using identity (2.2),we obtain   (b − a) T Pn (x, .) , f (n) (.); a, b = τ (x; α, β) − (b − a) M (Pn (x, .) ; a, b) κ, (3.8) From (2.2) and (3.2),

=

(b − a) M (Pn (x, .) ; a, b) Zb Pn (x, t)dt

(3.9)

a

=

h   1 n+1 n+1 C (x − A) − (a − A) (n + 1)!  i n+1 n+1 + D (b − B) − (x − B) .

Now to find (3.5), we will Combine (3.9) with (3.7), the left hand side of (3.5) will be obtained. Let f, g : [a, b] → R and f g : [a, b] → R be integrable on [a, b], then [6] |T (f, g)|

1

1

≤ T 2 (f, f ) T 2 (g, g) (Γ − γ) 1 ≤ T 2 (f, f ) 2 1 ≤ (Φ − ϕ) (Γ − γ) 4

(f, g ∈ L2 [a, b]) (γ ≤ g (x) ≤ Γ, t ∈ [a, b]) (ϕ ≤ f (x) ≤ Φ, t ∈ [a, b]) .

(3.10)

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

9

Also, note that 0

  1 ≤ T 2 f (n) (.), f (n) (.)  i 21 h   = M f (n) (.)2 ; a, b − M 2 f (n) (.); a, b 



Rb

2  12

(n) (t) dt   f  1 Zb a (n) 2   (t) dt −  =  f  b−a   b − a a

(3.11)

   

 21

1

(n) 2 2

f − κ b−a 2 (Γ − γ) , 2

 = ≤

where γ ≤ f 00 (t) ≤ Γ, t ∈ [a, b] . Now, for the bounds on (3.8), we have to determine 1 T 2 (Pn (x, .) , Pn (x, .)) , φ and Φ such that ϕ ≤ Pn (x, .) ≤ Φ. From the definition of Pn (x, t), we have
T (Pn (x, .) , Pn (x, .)) = M Pn2 (x, .) ; a, b − M 2 (Pn (x, .) ; a, b) .

(3.12)

From (3.9) we obtain M (Pn (x, .) ; a, b) h   1 n+1 n+1 C (x − A) − (a − A) = (b − a) (n + 1)!  i n+1 n+1 +D (b − B) − (x − B) and
(b − a) M Pn2 (x, .) ; a, b   Zx Zb 1  2 2n 2n = (t − A) dt + D2 (t − B) dt 2 C (n!) a x h   1 2n+1 2n+1 2 C (x − A) − (a − A) = 2 (2n + 1) (n!) i  2n+1 2n+1 +D2 (b − B) − (x − B) . Thus, substituting the above results into (3.12) gives 1

0 ≤ N (x) := T 2 (Pn (x, .) , Pn (x, .)) which is given explicitly by (3.7). Combining (3.8), (3.11) and (3.12) gives from the first inequality in (3.10), the first inequality in (3.5). Utilizing the inequality in (3.11) produces the second result in (3.5). Further, it may be noticed from the definition of Pn (x, t) in (2.1)

10

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

that for α, β ≥ 0 Φ

=

sup Pn (x, t) t∈[a,b]

h i 1 n−1 n−1 α (x − a) + β (b − x) (α + β) n! inf Pn (x, t) = 0,

= ϕ =

t∈[a,b]

if n is even and Φ

=

sup Pn (x, t) = t∈[a,b]

1 n−1 α (x − a) (α + β) n! n−1

ϕ =

inf Pn (x, t) =

t∈[a,b]

−β (b − x) (α + β) n!

,

if n is odd. 4. N EW O STROWSKI T YPE I NEQUALITIES FOR α = x − a AND β = b − x Theorem 4.1. Let f : [a, b] → R be a mapping such that f (n−1) (x)  is absolutely contin b−a uous on [a, b], and γ ≤ f (n) (t) ≤ Γ, ∀ t ∈ [a, b] , then for all x ∈ a + h b−a , 2 ,b − h 2 we have M |L| ≤ (S − γ) (4.1) 2 (b − a) (n + 1)! and |L| ≤

M (Γ − S) , 2 (b − a) (n + 1)!

(4.2)

|L| ≤

O (S − γ) , 2 (b − a) (n + 1)!

(4.3)

|L| ≤

O (Γ − S) , 2 (b − a) (n + 1)!

(4.4)

if n is even and if n is odd,

where L :

=

n−1 X

(−1)

n+k+1

2

k=0

(b − a) (k + 1)!

h  (k) k+1 k+1 (x − A) − (x − B) f (x)

i k+1 (k) (b) − (a − A) f (a) h 1 n+1 n+1 − (x − A) − (x − B) 2 (b − a) (n + 1)! #  n+1 h i b−a n + h (1 + (−1) ) × f (n−1) (b) − f (n−1) (a) 2 k+1

+ (b − B)

(k)

Zb

n

+

f

(−1)

f (t) dt,

2

(b − a)

a

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

M

:

11

  n b−a x− a+h [n + 1 − 2 (x − a) + h (b − a)] 2   n b−a + x− b−h [n + 1 − 2 (b − x) + h (b − a)] 2    n  n  b−a b−a , + (n + 1) x − a + h. − x− b−h 2 2 =

and

O

:

  n b−a {[n + 1 − 2 (x − a) + h (b − a)]} x− a+h 2   n+1  n+1 b−a b−a +2 x − b − h −4 h 2 2  n  b−a . + (n + 1) x − a + h 2 =

Proof: Substituting α = x − a and β = b − x, in (2.1) and using

1 b−a

Zb

f (n) (t)dt =

f (n−1) (b) − f (n−1) (a) , b−a

(4.5)

a

we obtain

1 b−a

Zb Pn (x, t)dt

(4.6)

a

=

"  n+1   n+1 1 b−a b−a x− a+h − x− b−h (b − a) (n + 1)! 2 2  n+1 # b−a +2 h , 2

,

=

h   1 n+1 n+1 C (x − A) − (a − A) (n + 1)!  i n+1 n+1 + D (b − B) − (x − B) .

12

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

Now

1 b−a

Zb Pn (x, t)f

(n)

(t)dt −

a

=

n−1 X

(−1)

n+k+1

h

2

k=0

(b − a) (k + 1)!

+ (b − B) +

(−1)

f

(k)

k+1

f

(a)

k+1



f

(k)

(x)

i

f (t) dt

2 a

1

−

− (x − B) (k)

(4.7)

a

a

k+1

(b) − (a − A)

f (n) (t)dt

Zb

n

(b − a)

k+1

Pn (x, t)dt

2

(b − a)

(x − A)

Zb

Zb

1

h

n+1

n+1

(x − A) − (x − B) 2 (b − a) (n + 1)! # n+1  b−a n (1 + (−1) ) + h 2   × f (n−1) (b) − f (n−1) (a) ,

Let

1 Rn (x) = b−a

Zb

Zb

1

Zb

f (n) (t)dt.

(4.8)

Rn (x)   Zb  Zb  1 1 = f (n) (t) − C Pn (x, t) − P (n) (x, s)ds dt b−a b−a

(4.9)

Pn (x, t)f a

(n)

(t)dt −

(b − a)

Pn (x, t)dt

2 a

a

If C ∈ R is an arbitrary constant, then we have

a

a

Furthermore, it is a straightforward exercise to show that

|Rn (x)| ≤

(4.10)

b Z Zb 1 1 (n) max Pn (x, t) − P (x, s)ds f (n) (t) − C dt. b − a t∈[a,b] b−a a

a

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

13

Consider n to be an even integer, then

Zb (n) Pn (x, t) − 1 P (x, s)ds b−a

(4.11)

a

=

=

1 (b − a) (n + 1)!    n   b−a b−a × max | x − a + h n+1−x+a+h 2 2 "  n+1  n+1 # b−a b−a + x− b−h −2 h |, 2 2   n   b−a b−a | x− b−h n+1+x−b+h 2 2 (  n+1  n+1 ) # b−a b−a − x − a + h. −2 h | 2 2 1 2 (b − a) (n + 1)!   n b−a × x− a+h [n + 1 − 2 (x − a) + h (b − a)] 2   n b−a + x− b−h [n + 1 − 2 (b − x) + h (b − a)] 2   n   n   b−a b−a . + (n + 1) x − a + h − x− b−h 2 2

14

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

Now consider n to be an odd integer, then Zb 1 (n) Pn (x, t) − P (x, s)ds b − a

(4.12)

a

=

=

1 (b − a) (n + 1)!    n   b−a b−a × max | x − a + h. n + 1 − x + a + h. 2 2 "  n+1 n+1 #  b−a b−a −2 h |, + x− b−h 2 2   n+1   n+1 b−a b−a | x− b−h − x − a + h. 2 2 n+1   n+1  n+1 #   b−a b−a b−a − x − a + h. −2 h | | x− b−h 2 2 2 1 2 (b − a) (n + 1)!   n b−a × x− a+h {[n + 1 − 2 (x − a) + h (b − a)]} 2   n+1  n+1 b−a b−a +2 x − b − h −4 h 2 2   n  b−a . + (n + 1) x − a + h. 2

We also have Zb (n) f (t) − γ dt = (S − γ) (b − a) .

(4.13)

a

Zb (n) f (t) − Γ dt = (Γ − S) (b − a) .

(4.14)

a

Therefore, we can obtain (4.1) and (4.4) by using (4.5) to (4.14) taking C = γ and C = Γ in (4.10).

5. D ERIVATION OF N UMERICAL Q UADRATURE RULES We propose some new quadrature rules involving higher order derivatives of the function f . In fact, the following new quadrature rules can be obtained while investigating error bounds using theorem 5:

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

15

We will discuss the efficiency of these rules in light of n and the parameter h. Zb Qn,1 (f ) :=

f (t) dt a

"  k+1  k+1 ! (k) 1 h h k+1 k+1 ≈ (b − a) (−1) −1 − f (a) (k + 1)! 2 2 k=0 # k+1    (k) b−a k (k) (−1) f (b) + f (a) + h 2 "  n+1  n+1 ! h h 1 n+1 n+1 − (b − a) (−1) −1 + (n + 1)! 2 2 #  n+1 h i b−a n + h (1 + (−1) ) × f (n−1) (b) − f (n−1) (a) , 2 n−1 X

Zb Qn,2 (f )

:

=

f (t) dt a

≈

" k+1    (k)  a + b  (b − a) 1 k+1 k+1 (h − 1) (−1) −1 f (k + 1)! 2k+1 2 k=0 #  k+1   (k) b−a k (k) + h (−1) f (b) + f (a) 2 " n+1   (b − a) 1 n+1 n+1 (h − 1) (−1) − 1 + (n + 1)! 2n+1 #  n+1 h i b−a n + h (1 + (−1) ) × f (n−1) (b) − f (n−1) (a) , 2 n−1 X

Zb Qn,3 (f )

:

=

f (t) dt a

"  k+1  k+1 !   k+1 (k) 1 (b − a) 1 −3 3a + b k k+1 ≈ (−1) − h + (−1) + h f (k + 1)! 2k+1 2 2 4 k=0 #  k+1   (k) b−a k (k) + h (−1) f (b) + f (a) 2 "  n+1  n+1 ! n+1 1 1 (b − a) −3 n n+1 + (−1) −h + (−1) +h (n + 1)! 2n+1 2 2 #  n+1 h i b−a n + h (1 + (−1) ) × f (n−1) (b) − f (n−1) (a) . 2 n−1 X

16

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

To investigate the efficiency of these quadrature rules we integrate various types of functions including Polynomial, Exponential and Logarithmic. The results of the numerical integrations are given in tables 1, 2 and 3, with absolute error and the value of n used to obtain the mentioned accuracy.

5.1. Case. 1: When h = 1, we get Zb f (t) dt

Qn,1 (f ) := a

" # k+1   (k) 1 b−a k (k) ≈ (−1) f (b) + f (a) (k + 1)! 2 k=0  n+1 h i b−a 1 n (1 + (−1) ) f (n−1) (b) − f (n−1) (a) + (n + 1)! 2 n−1 X

Zb Qn,2 (f ) :=

f (t) dt a

≈

" # k+1   (k) 1 b−a k (k) (−1) f (b) + f (a) (k + 1)! 2 k=0  n+1 h i 1 b−a n + (1 + (−1) ) f (n−1) (b) − f (n−1) (a) , (n + 1)! 2

n−1 X

Qn,3 (f ) " # k+1   (k) b−a 1 k (k) (−1) f (b) + f (a) : = (k + 1)! 2 k=0  n+1 h i b−a 1 n + (1 + (−1) ) f (n−1) (b) − f (n−1) (a) . (n + 1)! 2 n−1 X

5.2. Case 2: When h = 21 , we get Zb Qn,1 (f ) :=

f (t) dt a

≈

n−1 X k=0

k+1

1 (b − a) (k + 1)! 4k+1

n+1

+

1 (b − a) (n + 1)! 4n+1





k+1

(3)

n+1

(3)

f

(k)

k

(a) + (−1) f n

+ (−1)

(k)

(b)



h i f (n−1) (b) − f (n−1) (a)

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

Zb Qn,2 (f ) :=

f (t) dt a

" k+1   (k)  a + b  1 (b − a) k ≈ 1 + (−1) f (k + 1)! 4k+1 2 k=0 #  k+1   (k) b−a k (k) + (−1) f (b) + f (a) 4 n+1 h i 2 (b − a) 1 n (n−1) (n−1) (1 + (−1) ) f + (b) − f (a) , (n + 1)! 4n+1 n−1 X

Zb Qn,3 (f ) :=

f (t) dt a

"   k+1 (k) (b − a) 3a + b 1 f ≈ (k + 1)! 2k+1 4 k=0 #  k+1   (k) b−a k (k) + (−1) f (b) + f (a) 4 " #  n+1 n+1 b−a 1 (b − a) n + (1 + (−1) ) + (n + 1)! 2n+1 4 h i × f (n−1) (b) − f (n−1) (a) . n−1 X

5.3. Case. 3: When h = 43 , we get

Zb Qn,1 (f ) :=

f (t) dt a

≈

n−1 X

k+1 h  (k) (b − a) k+1 k+1 (5) − (3) f (a) 8k+1 (k + 1)! k=0  i (k) k+1 k (k) + (3) (−1) f (b) + f (a) n+1

+

h ih i (b − a) n+1 n n+1 (n−1) (n−1) (5) + 3 (−1) f (b) − f (a) . 8n+1 (n + 1)!

17

18

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

Zb Qn,2 (f ) :=

f (t) dt a

≈

n−1 X

k+1   (k)  a + b  (b − a) k 1 + (−1) f 8k+1 (k + 1)! 2 k=0  i (k) k+1 k (k) + (3) (−1) f (b) + f (a)

+

n+1  h i (b − a) n n+1 (n−1) (n−1) (1 + (−1) ) 1 + (3) f (b) − f (a) , 8n+1 (n + 1)!

Qn,3 (f ) :

n−1 X

k+1   (k)  3a + b  (b − a) 1 k+1 −1 + (3) = f (k + 1)! 8k+1 4 k=0  i (k) (k) k+1 k + (3) (−1) f (b) + f (a) n+1 h i (b − a) 1 n+1 n+1 n −1 + 2 (3) + (3) (−1) (n + 1)! 8n+1 h i × f (n−1) (b) − f (n−1) (a) .

+

To investigate the efficiency of these quadrature rules we integrate various types of functions including Polynomial, Exponential, Logarithmic. The results of the numerical integrations are given in table.1 with absolute error and the value of n used to obtain the mentioned accuracy 5.4. Performance of the proposed quadrature rules. In this section we discuss the performance of the proposed quadrature rules using numerous examples and for all values of the parameter h. The functions used for the analysis are listed below. f1 (x)

=

x2 + x + 2,

f2 (x) = x sin x,

f3 (x)

=

ex sin x ,

f4 (x) = x2 + sin x,

f5 (x)

=

ex ,

f7 (x)

=

x + cos x,

2

(5.1)

f6 (x) = ex cos (ex − 2x) ,

 f8 (x) = log x2 + 2 sin log x2 + 2 .

From the above tables, we observe that all three quadrature rules show exact value of the integral of f1 for n = 2 for all three cases i.e h = 1, h = 21 and h = 34 . The polynomial of degree k, n = k + 1 will give exact value of the integral f1 for all three cases. Acceptable error estimates can be obtained for smaller values of n to save computational time. In case of h = 21 , for integral of f6 , Qn,2 (f ) (for h = 12 ) gives an error of the order of 10−6 for n = 6 while Qn,1 (f ) and Qn,3 (f ) give a similar error for n = 10 and n = 11 (for h = 1 and h = 34 respectively). Similarly for all other functions Qn,2 (f ) (for h = 12 ) report errors of the order of 10−5 or 10−6 for relatively smaller values of n as compared to the other two quadrature rules. Specifically, Qn,2 (f ) (for h = 12 ) give an excellent estimate for the integrals of f6 and f8 at n = 6 and n = 4 respectively. In general Qn,2 (f ) (for h = 21 ) give better results as compared to the rest of the quadrature rules for much smaller values of n. Therefore we can surmise that Qn,2 (f ) (for h = 21 ) is computationally more efficient both in terms of error

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

19

approximation, and
simplicity, 
 time, see figure (2). As a rough estimate we integrated log x2 + 2 sin log x2 + 2 using the builtin algorithms of Mathematica 10.0 which took 26.30 seconds to give its approximate answer. To obtain similar approximation for the integral of f8 , Qn,2 (f ) (for h = 12 ) took less than a second. In order to complete the analysis of the proposed quadrature rules, we fix n at 7 and plot their error versus h ∈ [0, 1] . We see from figure. 1, which is for Qn,1 (f ) , (for h = 1) that the minimum error occurs (for all the test functions), when h is close to 1. The minimum eror in this case is of order 10−5 . In case of Qn,2 (f ), (for h = 12 ) the minimum error is of the order 10−8 and occurs when h is close to 0.5, see figure (2). Coincidently this rule is of the midpoint type rule. Similarly the minimum error for Qn,3 (f ) (for h = 34 ) occurs when h is between 0.6 and 0.8. The order of the minimum error is 10−7 (see figure 3). Based on this analysis, we can conjecture that Qn,2 (f ) is the most efficient quadrature rule, Qn,3 (f ) comes second in terms of performance. It should be noted that if desired the value of n can be adjusted to improve the error bounds or decrease computational time.

F IGURE 1. Error Analysis of Qn,1 (f ) for all values of h

20

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

F IGURE 2. Error Analysis of Qn,2 (f ) for all values of h

F IGURE 3. Error Analysis of Qn,3 (f ) for all values of h

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

21

TABLE 1. Performance of the proposed quadrature rules when h = 1

Sr. No. 1.

n : Qn,1 (f )

n : Qn,2 (f )

n : Qn,3 (f )

Exact Value

2 : 2.83333

2 : 2.83333

2 : 2.83333

2.83333

0

0

0

f2 (x)dx

6 : 0.30117

6 : 0.30117

6 : 0.30117

Error:

1.1381×10−6

1.1381×10−6

1.1381×10−6

f3 (x)dx

6 : 0.909328

6 : 0.909328

6 : 0.909328

Error:

2.33999×10−6

2.33999×10−6

2.33999×10−6

f4 (x)dx

5 : 0.793022

5 : 0.793022

5 : 0.793022

Error:

8.63182×10−6

8.63182×10−6

8.63182×10−6

11 : 1.46266

11 : 1.46266

11: 1.46266

−6

−6

−6

Method R1 f1 (x)dx 0

Error: 2.

R1 0

3.

R1 0

4.

R1 0

5.

R1

f5 (x)dx

0

6.

R1

Error:

5.8789×10

5.8789×10

f6 (x)dx

11:

11:

Error

7.37624×10−6

7.37624×10−6

7.37624×10−6

6 : 1.34147

6 : 1.34147

6 : 1.34147

Error:

1.4808×10−7

1.4808×10−7

1.4808×10−7

f8 (x)dx

9 : 0.62977

9 : 0.62977

9 : 0.62977

1.18074×10−6

1.18074×10−6

1.18074×10−6

0

7.

R1

f7 (x)dx

0

8.

R1

1.31384

1.31384

5.8789×10 11:

1.31384

0

Error:

0.301169

0.909331

0.793031

1.46265

1.31383

1.34147

0.629769

22

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

TABLE 2. Performance of the proposed quadrature rules when h =

Sr. No. 1.

Method R1 f1 (x)dx

n : Qn,1 (f )

n : Qn,2 (f )

n : Qn,3 (f )

2:

2:

2:

2.83333

2.83333

2.83333

1 2

Exact Value 2.83333

0

Error: 2.

R1

3.

4.

0.301161

4 : 0.301167

Error:

7.29406×10−6

1.85379×10−6

f3 (x)dx

8:

0.909333

Error:

2.72597×10

f4 (x)dx

6:

Error: 5.

6.

R1

8.

0.90933

9.99927×10

f5 (x)dx

14:

7:

10:

Error:

4.99942×10−6

f6 (x)dx

16:

Error

6.96805×10−6

f7 (x)dx

6:

1.46266

1.31382

1.34147

Error:

3.98943×10−6

f8 (x)dx

13:

Error:

1.46266

5.7003×10−6 6 : 1.31383 6.61067×10−6 4:

1.34147

0.909331

−7

2.4354×10−6

0.793021

4:

6:

1.7914×10−7

0

R1

4.30288×10

−6

0.301169

1.31344×10−6

9.70313×10−6

0

7.

4 : 0.909335

0.301167

5: 0.793029

0

R1

−6

6:

0.793031

0

R1

0

7:

0

R1

0

f2 (x)dx

0

R1

0

1.46266

0.793031

1.46265

4.80974×10−6 11:

1.31383

1.31383

2.58963×10−6 5:

1.34147

3.27914×10−7

3.40064×10−6

0.629771

4 : 0.629765

6: 0.629768

2.05259×10−6

3.76193×10−6

2.17411×10−7

0

1.34147

0.629769

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

23

TABLE 3. Performance of the proposed quadrature rules when h =

Sr. No. 1.

Method R1 f1 (x)dx

n : Qn,1 (f )

n : Qn,2 (f )

n : Qn,3 (f )

2:;

2: 2.83333

2:

2.83333

2.83333

3 4

Exact Value 2.83333

0

Error: 2.

R1

f2 (x)dx

0

3.

R1

0 6:

0.30116

Error:

8.51239×10−6

f3 (x)dx

7:

0.909333

0

4.

R1

Error:

2.49554×10

f4 (x)dx

6:

Error:

2.74945×10−6

f5 (x)dx

12:

Error:

0

5.

R1 0

6.

R1

f6 (x)dx

0

Error 7.

R1

f7 (x)dx

0

Error: 8.

R1

−6

f8 (x)dx

0

0

6: 0.301169

4: 0.301174

1.68896×10−7

5.73257×10−6

6: 0.90933

6: 0.90933

3.48606×10

−7

5.00862×10

5: 0.793029

3.20833×10−6

2.02841×10−6

9: 1.46266

9: 1.46266

4.78583×10−6

5.84777×10−6

6.98186×10−6

12: 1.31384

10: 1.31383

10: 1.31383

3.66334×10−6

1.45706×10−6

9.09394×10−7

4:; 1.34148

5:; 1.34147

1.07741×10−6

5.87281×10−6

3.54508×10−6

9: 0.629775

7: 0.629767

7: 0.629765

6.22791×10−6

1.77057×10−6

3.90261×10−6

6:;

1.46266

1.34147

0

Error:

0.909331

−7

4: 0.793034

0.793028

0.301169

0.793031

1.46265

1.31383

1.34147

0.629769

24

A. QAYYUM, A. R. KASHIF, M. SHOAIB, I. FAYE

R EFERENCES ¨ [1] A. Ostrowski, Uber die Absolutabweichung einer di erentienbaren Funktionen von ihren Integralimittelwert, Comment. Math. Hel. 10, (1938), 226-227. [2] S. S. Dragomir and S. Wang, A new inequality Ostrowski’s type in Lp - norm, Indian Journal o f Mathematics, 40 (1998), 299-304. [3] S. S. Dragomir and S. Wang, A new inequality Ostrowski’s type in L1 - norm and applications to some special means and some numerical quadrature rules, Tamkang Journal of Mathematics. 28 (1997), 239244. ˇ [4] P. L. Cebyˆ sev, Sur less expressions approximatives des integrales definies par les autres prises entre les memes limites, Proc. Math. Soc. Charkov 2 (1882), 93–98. [5] D. S. Mitrinovi´c , J. E. Pecari´c and A. M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, (1994). [6] P. Cerone, A new Ostrowski Type Inequality Involving Integral Means Over End Intervals, Tamkang Journal Of Mathematics Volume 33, No. 2, (2002). [7] A. Qayyum and S. Hussain, A new generalized Ostrowski Gr¨uss type inequality and applications, Applied Mathematics Letters 25 (2012) 1875–1880. [8] S. S. Dragomir and S. Wang, An inequality of Ostrowski-Gr¨us type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Computer and Mathematics with Application, 33 (11), 15-20, (1997). [9] M. Alomari, M. Darus, Some Ostrowski’s type inequalities for convex functions with applications, RGMIA, 13(1) Art No: 3 (2010). [10] P. Cerone, S.S. Dragomir, J. Roumeliotis, An inequality of Ostrowski type for mappings whose second derivatives are bounded and applications, RGMIA Research Report Collection 1 (2) (1998). [11] A. Sofo, S.S. Dragomir, An inequality of Ostrowski type for twice differentiable mappings in term of the Lp norm and applications, Soochow Journal of Mathematics, 27 (1), (2001), 97–111. [12] A. Qayyum, M. Shoaib, A. E. Matouk, and M. A. Latif, On New Generalized Ostrowski Type Integral Inequalities, Abstract and Applied Analysis, Volume 2014: (2014), Article ID 275806. Rb 1 [13] G. Gr¨uss, Uber das Maximum des absoluten Betrages von b−a f (x) g (x) dx − a

1 (b−a)2

Rb a

Rb f (x) dx g (x) dx, Math. Z. 39 (1935), 215–226. a

[14] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type Inequalities for n-time differentiable mappings and applications, RGMIA Research Report Collection, 1(1), ( 1998). [15] A. Qayyum, I. Faye and M. Shoaib, Improvement of Ostrowski Integral Type inequalities with application, Filomat 30:6 (2016), 1441?1456. [16] A. Qayyum, M. Shoaib and I. Faye, Some new generalized results on ostrowski type integral inequalities with application, Journal of computational analysis and applications, vol. 19, No.4, (2015).

A. Q AYYUM 1- D EPARTMENT OF FUNDAMENTAL AND APPLIED SCIENCES , U NIVERSITI T EKNOLOGI PETRONAS, 32610 BANDAR S ERI I SKANDAR , P ERAK DARUL R IDZUAN , M ALAYSIA

2- D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF H A’ IL , 2440, S AUDI A RABIA E-mail address: [email protected] A. R. K ASHIF D EPARTMENT OF M ATHEMATICS , C APITAL U NIVERSITY OF S CIENCES AND T ECHNOLOGY, I SLAMABAD , PAKISTAN E-mail address: [email protected] M. S HOAIB A BU D HABI M ENS C OLLEGE , H IGHER C OLLEGES OF T ECHNOLOGY, P.O. B OX 25035, A BU D HABI , U NITED A RAB E MIRATES E-mail address: [email protected]

DERIVATION OF NEW EFFICIENT QUADRATURE RULES

25

I. FAYE D EPARTMENT OF FUNDAMENTAL AND APPLIED SCIENCES , U NIVERSITI T EKNOLOGI PETRONAS, 32610 BANDAR S ERI I SKANDAR , P ERAK DARUL R IDZUAN , M ALAYSIA E-mail address: ibrahima [email protected]