Research Article UV Spectrophotometric Simultaneous...

Research ArticleUV Spectrophotometric Simultaneous Determination ofParacetamol and Ibuprofen in Combined Tablets by Derivativeand Wavelet Transforms

Vu Dang Hoang1 Dong Thi Ha Ly1 Nguyen Huu Tho2 and Hue Minh Thi Nguyen3

1 Department of Analytical Chemistry and Toxicology Hanoi University of Pharmacy 13-15 Le Thanh Tong Hanoi Vietnam2College of Education-Gia Lai 126 Le Thanh Ton Pleiku Gia Lai Vietnam3Center for Computational Science and Faculty of Chemistry Hanoi National University of Education Hanoi Vietnam

Correspondence should be addressed to Vu Dang Hoang hoangvdhupeduvn

Received 31 August 2013 Accepted 31 October 2013 Published 19 February 2014

Academic Editors C Pistos and Y Roggo

Copyright copy 2014 Vu Dang Hoang et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The application of first-order derivative and wavelet transforms to UV spectra and ratio spectra was proposed for the simultaneousdetermination of ibuprofen and paracetamol in their combined tablets A new hybrid approach on the combined use of first-orderderivative and wavelet transforms to spectra was also discussed In this application DWT (sym6 and haar) CWT (mexh) andFWT were optimized to give the highest spectral recoveries Calibration graphs in the linear concentration ranges of ibuprofen(12ndash32mgL) and paracetamol (20ndash40mgL) were obtained bymeasuring the amplitudes of the transformed signals Our proposedspectrophotometric methods were statistically compared to HPLC in terms of precision and accuracy

1 Introduction

Fever is a common symptom which consistently causesparental and professional phobia leading to the widespreaduse of antipyretic medications [1 2] Paracetamol(Figure 1(a)) and ibuprofen (Figure 1(b)) are commonlyused over-the-counter (OTC) antipyretic drugs especiallyfor pediatric treatment [3ndash5] because they are on their owneffective safe and relatively inexpensive

It is postulated that paracetamol blocks prostaglandinsynthesis in the hypothalamus via inhibition of cyclo-oxygenase-3 (COX-3) a splice variant of COX-1 that ismainly found in the brain and spinal cord In contrastibuprofenrsquos mechanism of action is to block the productionof prostaglandins by peripherally inhibiting COX-2 Thesedrugs could be served as a maker of soft tissue infectionas persistent high fever is observed in patients receivingibuprofen or paracetamol after varicella [6]

Despite the lack of official recommendations from guide-lines in the United States and United Kingdom it is believedthat combination treatment with ibuprofen and paracetamol

is beneficial over either agent alone for sustained fever reduc-tion in children older than 6 months [7] This combinationtherapy was favored for achieving an afebrile state andsustaining it [8ndash11] It was found that ibuprofen is as or moreefficacious than paracetamol for the treatment of pain andfever in adult and pediatric populations and is equally safe[12] On the other hand the combination was only slightlybetter on a few outcomes than ibuprofen alone but therewas a possible risk of excess dosing with the combination[13] According to Purssellrsquos systematic review there is littleevidence of any benefit or harm from the combined treatmentcompared with the use of each drug alone [14] In theabsence of such data combining paracetamol and ibuprofenfor fever in children was still questioned [15ndash17] Clinicallythis combination was also studied for pain relief [18ndash21] orpostdose symptom alleviation [22]

In the field of applied UV-Vis spectroscopy the analysisof pharmaceutical multicomponent mixtures without priorseparation step is always a difficult task due to overlappingspectral peaks Although the derivative approach still con-tinues to be a promising tool to solve this problem [23 24]

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 313609 13 pageshttpdxdoiorg1011552014313609

2 The Scientific World Journal

HOO

NH

(a) N-Acetyl-p-aminophenol

OH

O

(b) (RS)-2-(4-(2-Methylpropyl)phenyl)pro-panoic acid

Figure 1 Chemical structure of paracetamol (a) and ibuprofen (b)

Spectra

Derivative transform

Sym6 haar

Ratio spectra

Wavelet transform

Ratio spectra

(Sym6 haar)

Spectra (Sym6 haar mexh FWT)

FWT-spectra)

Sym6 haar mexh (FWT-

spectra)

Spectra (1D)

mexh (1Dspectra)

(1D)

1D (Sym6

Figure 2 Derivative and wavelet transforms for the simultaneous determination of IB and PA in binary mixtures

0

1

2

3

210 230 250 270 290

Abso

rban

ce

Wavelength (nm)

IB 20PA 325

IB 20 + PA 325 Addition spectrum

Figure 3 Spectra of IB 20mgL PA 325mgL and their corre-sponding mixture and absorbance addition

it may have drawbacks in some cases such as (i) the higherorder differentiation process diminishes peak amplitude aswell as signal-to-noise ratio (ii) the finding of zero-crossingpoints is very difficult and ratio spectra derivative working

wavelength is undetermined These drawbacks can be elimi-nated by applying wavelet transform approach to the originalabsorption spectra The wavelet transform can be regardedas mathematical functions that cut up data into differentfrequency components and then study each component witha resolutionmatched to its scale It is a powerful tool for signalprocessing in many branches of science and engineering Inthe last decade its applications in analytical chemistry forexample data reduction [25] denoising [26] and baselinecorrection [27] have been recordedThe wavelet-based reso-lution of multicomponent pharmaceutical mixtures has alsobeen exploited for which quaternary mixtures containingparacetamol are an example [28]

In the literature the determination of paracetamol andibuprofen could be simultaneously performed by HPLC CEand HPTLC [29ndash32] Their spectrophotometric determina-tion was also studied by applying chemometrics and deriva-tive approach [33ndash35] It is noteworthy that differentiationand smoothing algorithms forUVderivative spectrophotom-etry in these studies were not clearly indicated whereas theyalways play an important role in determining the sensitivityand accuracy of derivative techniques

The aim of this study was to develop derivativeandwavelet-based UV spectrophotometric methods for thesimultaneous determination of paracetamol and ibuprofen intheir combined tablets using HPLC as a reference method

The Scientific World Journal 3

0

1

2

3

4

225 240 255 270Wavelength (nm)

2493nm2420nm

minus5

minus4

minus3

minus2

minus1

1D

spec

tra

IB 12ndash32mgLPA 20ndash40mgL

(a)

1D

ratio

spec

tra

0

2000

4000

270 280Wavelength (nm)

minus6000

minus4000

minus2000

2748nm

PA (20ndash40mgL)IB 20mgL[PA (20ndash40mgL) + IB 20mgL]IB 20mgL

(b)

1D

ratio

spec

tra

0

2

4


minus8

minus10

minus6

minus4

minus2

2344nm

IB (12ndash32mgL)PA 20mgL[IB (12ndash32mgL) + PA 325mgL]PA 20mgL

(c)

Figure 4 First-order derivatives of spectra (a) and first-order derivatives of ratio spectra (b) and (c)

This study in particular emphasized on exploiting theadvantages of wavelet transform over differentiation algo-rithms (ie continuous discrete and fractional wavelettransform) as well as correcting the shortcomings in theabove-mentioned UV derivative spectrophotometry studies

2 Experimental

21 Apparatus and Software Absorption spectra were regis-tered and treated by using a UNICAM UV 300 double beamspectrophotometer (Thermo Spectronic USA) with a fixedslit width (15 nm) connected to an IBM computer loadedwithThermo Spectronic VISION32 software and 1-cm quartzcellsThe zero-order spectra were recorded in the wavelengthrange of 200ndash325 nm against a blank (phosphate bufferpH 72) at Intelliscan mode to enhance the signal-to-noiseratio of absorbance peaks without extended scan durationwith a Δ120582 = 01 nm (ie 30ndash120 nmmin) For derivative

approach the spectra were differentiated and smoothed byusing Savitzky-Golay filter For wavelet approach the datatreatment was done using MATLAB R2013a software (TheMathWorks Natick MA USA) FWT calculations wereperformed in MATLAB with its code of FWT performed byUnser and Blu

High performance liquid chromatogram (HPLC) analysiswas performed on an Agilent 1100 Series Diode-Array-Detector chromatograph (Agilent Technologies USA) atambient temperature An Eclipse XDB-C18 (3 times 150mm35 120583m) column was used All solutions were filtered througha 045 120583m membrane filter before injection into the chro-matograph All solvents were filtered through a 045 120583mMillipore filter and degassed in an ultrasonic bath

22 Reagents and Standard Solutions Paracetamol PA(995) and ibuprofen IB (1000) were kindly providedby the National Institute of Drug Quality Control (Vietnam)


0

4

8

12

200 220 240 260 280 300Wavelength (nm)

minus8

minus4

2765nm

2568nm


Sym6

-spe

ctra

(a)

0

5

10

200 220 240 260 280 300Wavelength (nm)

Haa

r-sp

ectr

a

2487nm2408nm


minus5

minus10

minus15

(b)

0

5

10

15

20

25

30

200 220 240 260 280 300

minus5

minus10

minus15

Mex

h-sp

ectr

a

Wavelength (nm)


2443nm

2726nm

(c)

Figure 5 Wavelet transform of spectra using sym6 (a) haar (b) and mexh (c)

Deionized doubly distilled water was used throughout Allreagents were of an analytical grade Stock solutions of PAand IB (500mgL) were freshly made in phosphate bufferpH 72 A concentration set of standard solutions wereprepared in 25 mL calibrated flasks by using the same stocksolutions

23 Sample Solution Three commercial formulations con-taining paracetamol 325mg + ibuprofen 200mg per tabletwere studied that is Alaxan (United Pharma Vietnam)Dibulaxan (Danapha Pharmaceutical Joint Stock CompanyVietnam) and Febro (OPVPharmaceutical Joint Stock Com-pany Vietnam) For each formulation twenty tablets werefinely pulverized in a mortar A quantity equivalent to onetablet was accurately weighed and dissolved in about 50mLof phosphate buffer pH 72 in a 100 mL volumetric flask bysonication for 20min and subsequently diluted to the mark

with the same solvent Appropriate dilution was then madein a 25 mL volumetric flask to obtain the test solution ca325mgL paracetamol + 20mgL ibuprofen

3 Theoretical Background

The theoretical background of derivative transform andsmoothing of signals using Savitzky-Golay method [36] aswell as fundamentals of CWT DWT [37] and FWT [38] arebriefly described as follows

31 Derivative Transform Approach

311 Savitzky-Golay Method This method determines aderivative spectrumbymoving a spectral window comprising2119899 + 1 measurement points over an absorbance spectrum



minus600

minus200

200

600

Sym6

-rat

io sp

ectr

a

2303nm


(a)

220 240 260 280minus10

10

0

2364nm

Sym6

-rat

io sp

ectr

a

Wavelength (nm)


(b)

500

1500

2500

225 245 265 285

Haa

r-ra

tio sp

ectr

a

minus1500

minus500

Wavelength (nm)

2793nm


(c)

5

15

210 230 250 270

minus5

minus15

minus25

2287nm

Wavelength (nm)

Haa

r-ra

tio sp

ectr

a


(d)

Figure 6 Wavelet transform of ratio spectra using sym6 (a) and (b) and haar (c) and (d)

Then a polynomial of order m is fitted to the measurementpoints inside the spectral window as follows

119875 (120582) = 1198860+ 1198861120582 + 119886

21205822+ sdot sdot sdot + 119886

119898120582119898 (1)

This fit polynomial introduces smoothing which isdependent on the user selectable parameters n and m Fromthe resulting fit parameters 119886

0 119886

119898 the derivatives at the

window center 1205820can be derived easily as follows

119889119875

119889120582

100381610038161003816100381610038161003816100381610038161205820=0

= 1198861+ 21198862120582 + sdot sdot sdot + 119898119886

119898120582119898minus1

= 1198861

1198892119875

1198891205822

1003816100381610038161003816100381610038161003816100381610038161205820=0

= 21198862+ sdot sdot sdot + 119898 (119898 minus 1) 119886

119898120582119898minus2

= 21198862

1198893119875

1198891205823

1003816100381610038161003816100381610038161003816100381610038161205820=0

= 61198863+ sdot sdot sdot + 119898 (119898 minus 1) (119898 minus 2) 119886

119898120582119898minus3

= 61198863

(2)

Once the derivatives are determined at 1205820 the window

is moved one measurement point to the right followed by apolynomial fit inside this newwindow until it reaches the endof the spectrum

32 Wavelet Transform Approach

321 Continuous Wavelet Transform (CWT) Given a time-varying signal f (t) the wavelet transforms consist of comput-ing coefficients which are inner products of the signal anda family of wavelets In a continuous wavelet transform thewavelet corresponding to scale a and time location b can bewritten in terms of the mother wavelet as follows

120595119886119887(119905) =

1

radic|119886|

120595(119905 minus 119887

119886) with 119886 119887 isin 119877 119886 = 0 (3)


0

5

10

15

200 250 300

Sym

6-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

2808nm

2673nm


(a)

0

5

10

15

200 250 300

Haa

r-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

minus15

2595nm

2811nm


(b)

0

20

40

minus20

2342nm

2422nm

200 250 300Wavelength (nm)


Mex

h-FW

T sp

ectr

a

(c)

Figure 7 Wavelet transform (sym6 (a) haar (b) and mexh (c)) of FWT-modified spectra

The continuous wavelet transform (CWT) of f (t) is given by

119882119891(119886 119887) = int

infin

minusinfin

119891 (119905) 120595119886119887(119905) 119889119905 (4)

The inversion back to time domain is given by

119891 (119905) =1

119862120595

int

infin

minusinfin

1

1198862[int

infin

minusinfin

119882119891(119886 119887) 120595

119886119887(119905) 119889119887] 119889119886 (5)

When a continuous wavelet transform is evaluated themother wavelet is scaled and translated to every possiblevalue of a and b Accordingly at each location (translation)of the wavelet information is obtained about the localcontribution of each frequency (scaling) to the entire signal

322 Discrete Wavelet Transform (DWT) When the discretewavelet transform is used to analyze digitized signals thescaling and the translation of the mother wavelet will be

120595119898119899(119905) =

1

1198861198982

0

120595(119905 minus 119899119887

0

119886119898

0

) (6)

The discrete wavelet transform is then written as

119882119891(119898 119899) = int

infin

minusinfin

119891 (119905) 120595119898119899(119905) 119889119905 (7)

Usually a0= 2 and b

0= 1 values are chosen The wavelet

transform calculated is called dyadic when a0= 2

323 Fractional Wavelet Transform (FWT)

B-Spline A B-spline is defined as a generalization of theBezier curve Let a vector known as the knot be defined as119879 = 119905

0 1199051 119905

119898 where119879 is a nondecreasing sequence with

ti isin [0 1] and the control points are defined as1198750 119875119899 Degree


225 250 275

2690nm

2625nm

minus30

minus20

minus10

0

10

20

30

40

Sym6

-1D

spec

tra

Wavelength (nm)


(a)

0

10

20

30

40

50

225 250 275minus30

minus20

minus10

Wavelength (nm)


2725nm

2316nm

Haa

r-1D

spec

tra

(b)

0

20

40

225 250 275

minus40

minus60

minus20

2400nm

2599nm

Wavelength (nm)


Mex

h-1D

spec

tra

(c)

Figure 8 Wavelet transform of first-order derivative spectra using sym6 (a) haar (b) and mexh (c)

is defined as119901 = 119898minus119899minus1The knots 119905119901+1

119905119898minus119901minus1

are calledinternal knots If the basis functional is defined as

1198731198940(119905) =

1 if 119905119894le 119905 lt 119905

119894+1 119905119894lt 119905119894+1

0 otherwise

119873119894119901 (119905) =

119905 minus 119905119894

119905119894+119901minus 119905119894

119873119894119901minus1 (119905) +

119905119894+119901+1

minus 119905

119905119894+119901+1

minus 119905119894+1

119873119894+1119901minus1 (119905)

(8)

then the curve defined by

119862 (119905) =

119899

sum

119894=0

119875119894119873119894119901(119905) (9)

is a B-spline

Fractional B-SplineThe fractional B-spline is defined as

120573120572

+(119909) =

Δ120572+1

+119909120572

+

Γ (120572 + 1)=sum+infin

119896=0(minus1)119896(120572+1

119896) (119909 minus 119896)

120572

+

Γ (120572 + 1) (10)

where Eulerrsquos gamma function is defined as follows

Γ (120572 + 1) = int

+infin

0

119909120572119890minus119909119889119909

(119909 minus 119896)120572

+= max (119909 minus 119896 0)120572

(11)

The forward fractional finite difference operator of order 120572is defined as

Δ120572

+119891 (119909) =

+infin

sum

0

(minus1)119896(120572

119896) 119891 (119909 minus 119896) (12)

where

(120572

119896) =

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (13)

The above-defined B-splines fulfill the convolution propertyas follows

1205731205721

+times 1205731205722

+= 1205731205721+1205722

+ (14)


200 220 240 260 280 300

2

1

0

minus1

minus2

1D

-sym

6sp

ectr

a

Wavelength (nm)


2594nm

2445nm

(a)

0

01

02

225 250 275

minus01

minus02

Wavelength (nm)


1D

-FW

T sp

ectr

a 2592nm

2722nm

(b)

Figure 9 Derivative transform of wavelet transformed spectra by sym6 (a) and FWT (b)

0 1 2 3 4 5(min)

400350300250200150100

500

(mAU

)

0858

Para

ceta

mol

4328

Ibup

rofe

n

Figure 10 Typical liquid chromatogram of a binary mixture of IB20mgL and PA 325mgL

The centered fractional B-splines of degree 120572 are given by

120573120572

lowast(119909) =

1

Γ (120572 + 1)sum

119896isin119885

(minus1)119896

10038161003816100381610038161003816100381610038161003816

120572 + 1

119896

10038161003816100381610038161003816100381610038161003816

|119909 minus 119896|120572

lowast (15)

where |119909|120572lowasthas the following form

|119909|120572

lowast=

|119909|120572

minus2 sin ((1205872) 120572) 120572 not even

1199092119899 log119909(minus1)1+119899120587 120572 even

(16)

Fractional B-SplineWaveletsThe fractional B-spline waveletsare defined as follows

120595120572

+(119909

2) = sum

119896isin119885

(minus1)119896

2120572sum

1isin119885

(120572 + 1

1)

times 1205732120572+1

lowast(1 + 119896 minus 1) 120573

120572

+(119909 minus 119896)

(17)

The fractional splines wavelets obey

int

+infin

minusinfin

119909119899120595120572

+(119909) 119889119909 = 0 (18)

and the Fourier transform fulfills the following relations

120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

(19)

Here 120572lowast(120603) is symmetric The fractional spline wavelets

behave like fractional derivative operators as indicated by thelast formulas

4 Results and Discussion

For spectrophotometric measurements phosphate buffer pH72 a medium successfully studied for the dissolution test ofPA and IB combined tablets [39] was chosen to solubilizeboth drugs Figure 3 shows the zero-order UV absorptionspectra after being smoothed by Savitzky-Golay algorithm(Order 3 number of coefficients 125) It is clear that (i) theadditivity of absorbances was obeyed for the mixture of IB20mgL + PA 325mgL and (ii) the determination of IBin the mixture was impossible because the spectrum of PA325mgL completely covered the spectrum of IB 20mgLover the range 210ndash290 nm In order to determine simul-taneously IB and PA in binary mixtures their overlappingspectrawere resolved using derivative andwavelet transformsas graphically depicted in Figure 2

In principle derivative and wavelet transforms couldbe applied to spectra or ratio spectra While finding zero-crossing or crossing points is crucial to the transformedspectra the applicability of the transformed ratio spectradepends on finding a point or a region over which thecoincidence of derivative orwavelet signals is observed for theratio spectra of a compound and its corresponding mixture


Table 1 Statistical analysis of calibration graphs of the proposed HPLC and spectrophotometric methods (119899 = 6) IB (12ndash32mgL) and PA(20ndash40mgL)

Method Compound Wavelength (nm) 119886 119887 119878119886

119878119887

119878119910sdot119909

1198772

HPLC IB 2210 57684 minus86417 02861 65906 47873 09999PA 2210 48886 minus25095 02636 82531 42943 09998

Derivative transform1D spectra IB 2420 minus00521 minus00721 00005 00114 00083 09996

PA 2493 minus00731 00689 00019 00608 00331 099701D ratio spectra IB 2304 minus02385 minus05009 00056 01310 00951 09977

PA 2748 72666 89283 13424 41421 22552 09986Wavelet transform

Sym6-spectra IB 2568 00445 minus05066 00005 00116 00084 09997PA 2765 minus00248 00735 00005 00161 00087 09982

Haar-spectra IB 2408 00820 minus06405 00013 00301 00218 09989PA 2487 00964 03651 00018 00584 00318 09984

Mexh-spectra IB 2726 minus01710 minus05047 00051 01196 00869 09963PA 2443 06333 09753 00149 04610 02510 09977

Sym6-ratio spectra IB 2364 minus02416 minus06292 00033 00764 00555 09992PA 2303 minus10291 minus58576 03182 98194 53461 09961

Haar-ratio spectra IB 2287 03189 00935 00040 00922 00670 09993PA 2793 53276 15516 07739 23879 13001 09991

Derivative-Wavelet transforms combined

Sym6-1D spectra IB 2683 minus01691 minus01659 00013 00310 00225 09997PA 2625 minus04605 minus00536 00112 03483 01896 09976

Haar-1D spectra IB 2316 02246 07973 00059 01365 00991 09972PA 2725 minus06666 06464 00188 05828 03173 09967

Mexh-1D spectra IB 2405 minus07507 minus10118 00064 01487 01080 09997PA 2599 minus12642 minus03523 00191 05922 03224 09990

1D-Sym6 spectra IB 2445 00160 minus00071 00002 00060 00044 09989PA 2594 minus00090 minus00904 00004 00124 00068 09921

1D-FWT spectra IB 2722 00025 00007 00001 00003 00002 09998PA 2592 minus00024 00074 00001 00016 00008 09981

Sym6-FWT spectra IB 2808 minus01376 00033 00029 00680 00494 09981PA 2673 02029 01088 00044 01371 00746 09980

Haar-FWT spectra IB 2707 02745 04080 00034 00799 00580 09993PA 2595 minus02289 03976 00084 02595 01413 09946

Mexh-FWT spectra IB 2422 01589 07507 00057 01332 00967 09947PA 2342 minus02808 01765 00112 03469 01888 09936

119884 = 119886119862+ 119887 where 119862 is the concentration in mgL and 119884 in signalrsquos amplitude units (for spectrophotometric methods) or mAU times sec (for HPLC)119886 slope 119887 intercept 119878119886 SD of the slope 119878119887 SD of the intercept 119878119910sdot119909 SD of the residuals 1198772 coefficient of determination

41 Method Development

411 Derivative Transform Figure 4(a) displays the firstderivative spectra of these pure drugs after their originalspectra being differentiated (Order 5 number of coefficients9) and smoothed (Order 3 number of coefficients 501) bySavitzky-Golay algorithm which reveals that there existedzero-crossing points at 2493 and 2420 nm for IB and PArespectively Both wavelengths were subsequently chosenfor the simultaneous determination of PA and IB due totheir derivative amplitudes proportional to the concentrationranges studied of PA (20ndash40mgL) and IB (12ndash32mgL)

Figures 4(b) and 4(c) present the first-order derivatives ofratio spectra after ratio spectra being differentiated (Order5 number of coefficients 9) and smoothed (Order 3number of coefficients 125) by Savitzky-Golay algorithm Tooptimize this technique the influence of divisor standardconcentration was investigated with the concentration rangesfor Lambert-Beerrsquos law compliance A standard spectrumof 20mgL was considered suitable for the determinationof both drugs The determination of each component wasbased on the proportionality of its concentrations to relevantfirst-order derivative amplitudes at a suitable wavelengthThe two points 2748 and 2344 nm at which the highest


Table 2 Assay results for the determination of IB and PA in their combined tablets

of label claim (mean plusmn SD 119899 = 6)

Method Alaxan Dibulaxan FebroIB PA IB PA IB PA

HPLC 994 plusmn 09 998 plusmn 09 998 plusmn 13 994 plusmn 10 1003 plusmn 10 993 plusmn 12

Derivative transform1D spectra 995 plusmn 09 1001 plusmn 14 1003 plusmn 17 996 plusmn 13 994 plusmn 09 993 plusmn 13

1D ratio spectra 998 plusmn 13 991 plusmn 11 1002 plusmn 15 993 plusmn 09 994 plusmn 15 1001 plusmn 14

Wavelet transformSym6-spectra 1003 plusmn 18 996 plusmn 13 1003 plusmn 15 994 plusmn 12 1006 plusmn 08 991 plusmn 10

Haar-spectra 993 plusmn 13 991 plusmn 12 1003 plusmn 09 997 plusmn 17 1005 plusmn 12 994 plusmn 09

Mexh-spectra 1010 plusmn 12 1007 plusmn 08 1005 plusmn 18 988 plusmn 13 994 plusmn 09 996 plusmn 11

Sym6-ratio spectra 1001 plusmn 11 991 plusmn 13 994 plusmn 09 1004 plusmn 09 992 plusmn 11 998 plusmn 09

Haar-ratio spectra 997 plusmn 09 993 plusmn 06 1001 plusmn 10 996 plusmn 11 1006 plusmn 13 995 plusmn 11

Derivative-Wavelet transforms combinedSym6-1D spectra 1008 plusmn 14 1000 plusmn 10 997 plusmn 12 1010 plusmn 11 995 plusmn 14 1005 plusmn 09

Haar-1D spectra 995 plusmn 08 995 plusmn 10 1007 plusmn 11 1009 plusmn 13 1007 plusmn 08 998 plusmn 08

Mexh-1D spectra 997 plusmn 10 995 plusmn 09 998 plusmn 13 1002 plusmn 15 997 plusmn 09 998 plusmn 12

1D-Sym6 spectra 1005 plusmn 11 1005 plusmn 08 998 plusmn 15 993 plusmn 14 1006 plusmn 08 1000 plusmn 11

1D-FWT spectra 999 plusmn 14 1004 plusmn 13 1001 plusmn 13 999 plusmn 12 1004 plusmn 11 1004 plusmn 10

Sym6-FWT spectra 1007 plusmn 11 1003 plusmn 09 1004 plusmn 11 997 plusmn 14 996 plusmn 09 1001 plusmn 09

Haar-FWT spectra 998 plusmn 13 999 plusmn 10 997 plusmn 15 1001 plusmn 10 1004 plusmn 12 1006 plusmn 10

Mexh-FWT spectra 1003 plusmn 09 996 plusmn 14 995 plusmn 16 1007 plusmn 08 1006 plusmn 08 997 plusmn 13

amplitude and coincidence of derivative signals were seenwith an error less than 3 were selected as the workingwavelengths for analyzing PA and IB respectively The factthat our data are different from previously published workson spectrophotometric simultaneous determination of IBand PA in their mixture [34 35] could be attributed to thedifference in differentiating and smoothing manner solventand equipment used Nevertheless our experimental setupseems to be better than these studies when referring to (i)the use of water an ecofriendly solvent rather thanmethanoland (ii) higher amplitudes of derivative signals obtained forthe same concentration range

412 Wavelet Transform In practice wavelet transform ofspectra and ratio spectra for the determination of IB and PAwas carried out by transferring spectra data vectors into thewavelet domain and then applying wavelet transform (CWTDWT and FWT) to the signal data in the wavelet domain

For the optimization of the wavelet analysis variouswavelet transform methods at different dilation parameters(a) were tested to identify wavelet transform families in orderto provide the best spectral recovery values In the abovetest Sym6 Haar and Mexh were found to be appropriate forthe transformation of spectral signals of the two compoundsand their mixtures On the other hand several dilationparameters (a) with frequency (f ) for these CWT and DWTapproaches were tested to find the optimal signal processingparameters For this a = 256 with f = 0182 (sym6) f =0249 (haar) and f = 0063 (mexh) were determined Theapplication of these families to resolve spectra and ratiospectra is displayed in Figures 5 and 6

413 Derivative-Wavelet Transforms Combined FWT is anew promising method in signal and image analysis whichoffers the functions of data compression and denoising toeffectively extract the important form of complex originalspectra It is noticeable that the selected columns among thewhole FWT coefficients contain low frequency informationin high scales that is the absorption spectrum is smooth andpossesses high amplitude In this study FWT signal analysisapproach was applied to the zero-order absorption spectra inthe wavelength range of 2000ndash3023 nm (ie 1024 points)Several parameters 120572 and depths of the decomposition (J)were tested for optimizing the fractional signal processing120572 = minus03 and J = 1 were found to be the optimal ones Thetype of B-splines was considered to be causal orthonormalAfter that the FWT spectra were subjected to further wavelettransform (sym6 haar and mexh) to find zero-crossingpoints for the simultaneous determination of IB and PA intheir mixtures (Figures 7(a) 7(b) and 7(c))

In another development the combination of derivativeandwavelet transformswas performed in an effort to increasethe number of zero-crossing points as well as to obtain ahigher sensitivity and selectivity as compared to the originalderivative or wavelet spectra This approach was successfullydone with wavelet transform (sym6 haar and mexh) ofthe first-order derivative spectra (Figures 8(a) 8(b) and8(c)) and the first-order derivative transform of the wavelettransformed spectra (Figures 9(a) and 9(b))

414 High Performance Liquid Chromatography AnalysisThe reversed-phase HPLC for the analysis of binary mixturescontaining IB and PA was developed as a reference method


Table 3 Results of ANOVA and Bartlett tests at the significance level of 5 by applying spectrophotometric and chromatographic methodsto three commercial pharmaceutical formulations

ANOVA testSource of variation Compound Between groups Within groups Total

Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789

Degree of freedom 15 80 95

Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124

Calculated 119865 value

IBI 1207II 0464III 1692

PAI 1377II 1597III 1067

Tabulated 119865 value 1793Bartlett test

Degree of freedom 15


IBI 6923II 6201III 6506

PAI 7352II 5810III 2870

Tabulated 1205942 value 24996I Alaxan II Dibulaxan III Febro

The optimization of HPLC analysis was as follows IB andPA were chromatographically analyzed by isocratic elutionwith a flow rate of 08mLminThemobile phase compositionwas acetonitrile-phosphoric acid 01 (55 45 vv) Injectionvolume was 20120583L and detection wavelength was 2210 nm forboth compounds Under our chromatographic conditionsthe retention time was found to be 086 and 433min for PAand IB respectively (Figure 10)The chromatographic param-eters such as resolution (Rs = 154) peak asymmetry (AF =08) and plate number (ca 100015 cm) were satisfactory forboth compounds obviously confirming the suitability of ourHPLC method

42 Method Validation The validity of the proposed HPLCmethod was assessed by accuracy precision and linearityFor studying the accuracy known quantities of IB and PA(ie 10ndash20 nominal content of their combined tablets) wereadded to each predetermined pharmaceutical formulation

The amount of analyte recovered was expressed as averagepercent recovery with the upper and lower limits of standarddeviation The average percent recoveries obtained were995plusmn09 and 1003plusmn10 for PA and IB respectively indicat-ing the methodrsquos good accuracy and no marked interferenceby common excipients in the tablets studied The within-run precision (repeatability) of HPLC was evaluated byanalyzing six replicates of the same formulation a dayThe lowRSD values (lt20) indicate the methodrsquos good precisionBy analogy the proposed spectrophotometric methods alsoshowed good precision (RSD lt 20)The calibration graphsfor HPLC and UV spectrophotometric determination withthe linear concentration ranges of IB (12ndash32mgL) and PA(20ndash40mgL) are summarized in Table 1

The proposed techniques were successfully applied to thesimultaneous determination of IB and PA in their combinedtablets The spectrophotometric results were statisticallycompared with those obtained by HPLC (Table 2) It isseen that at 95 confidence level there was no significant


difference between the accuracy (evaluated by ANOVA testcalculated F value lt tabulated F value) and precision (evalu-ated by Bartlett test calculated 1205942 value lt tabulated 1205942 value)among all the proposed methods (Table 3)

5 Conclusion

UV spectrophotometricmethods based on first-order deriva-tive transform CWT (mexh) DWT (sym6 and haar) andFWTof spectra and ratio spectrawere developed for the spec-tra resolution of IB and PA in their binary mixtures withoutprior separation step In particular a new hybrid approachon the combined use of derivative and wavelet transformwas also suggested The application of wavelet transformto UV spectra showed some advantages over derivativespectrophotometry such as higher peak intensity obtainedadditional smooth function and scaling factor process elim-inated All the proposed spectrophotometric methods weresimple and statistically compared to liquid chromatographicdata in terms of precision and accuracy It offers possibleinterchangeability between UV spectrophotometric methodsand HPLC for the simultaneous determination of IB and PAin their combined tablets

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research is funded by the Vietnam National Foundationfor Science and Technology Development (NAFOSTED)under Grant no 10407-201258

References

[1] M Crocetti N Moghbeli and J Serwint ldquoFever phobia revis-ited have parental misconceptions about fever changed in 20yearsrdquo Pediatrics vol 107 no 6 pp 1241ndash1246 2001

[2] E Purssell ldquoParental fever phobia and its evolutionary corre-latesrdquo Journal of ClinicalNursing vol 18 no 2 pp 210ndash218 2009

[3] J Tucci E Bandiera R Darwiche Z Medos R Nashed andD Trinh ldquoParacetamol and ibuprofen for paediatric pain andfeverrdquo Journal of Pharmacy Practice and Research vol 39 no 3pp 223ndash225 2009

[4] K McCloskey N Cranswick and T Connell ldquoUse of paraceta-mol and ibuprofen in children in a tertiary care hospital are webecoming complacentrdquo Archives of Disease in Childhood vol97 no 2 article 181 2012

[5] C Smith and R D Goldman ldquoAlternating acetaminophen andibuprofen for pain in childrenrdquo Canadian Family Physician vol58 pp 645ndash647 2012

[6] N D Moore ldquoParacetamol with ibuprofen ibuprofen is amarker of soft tissue infectionrdquo British Medical Journal vol 337Article ID A2072 2008

[7] R R Malya ldquoDoes combination treatment with ibuprofen andacetaminophen improve fever controlrdquo Annals of EmergencyMedicine vol 61 no 5 pp 569ndash570 2013

[8] A D Hay C Costelloe N M Redmond et al ldquoParacetamolplus ibuprofen for the treatment of fever in children (PITCH)randomised controlled trialrdquo British Medical Journal vol 337Article ID A1302 2008

[9] M M Nabulsi H Tamim Z Mahfoud et al ldquoAlternatingibuprofen and acetaminophen in the treatment of febrile chil-dren a pilot studyrdquo BMCMedicine vol 4 article 4 2006

[10] I M Paul S A Sturgis C Yang L Engle H Watts and CM Berlin ldquoEfficacy of standard doses of ibuprofen alone alter-nating and combined with acetaminophen for the treatment offebrile childrenrdquo Clinical Therapeutics vol 32 no 14 pp 2433ndash2440 2010

[11] L C Kramer P A Richards AMThompson D P Harper andM P Fairchok ldquoAlternating antipyretics antipyretic efficacy ofacetaminophen versus acetaminophen alternated with ibupro-fen in childrenrdquo Clinical Pediatrics vol 47 no 9 pp 907ndash9112008

[12] C A Pierce and B Voss ldquoEfficacy and safety of ibuprofenand acetaminophen in children and adults a meta-analysis andqualitative reviewrdquoThe Annals of Pharmacotherapy vol 44 no3 pp 489ndash506 2010

[13] G M Allan N Ivers and Y Shevchuk ldquoTreatment of pediatricfever are acetaminophen and ibuprofen equivalentrdquo CanadianFamily Physician vol 56 no 8 p 773 2010

[14] E Purssell ldquoSystematic review of studies comparing combinedtreatment with paracetamol and ibuprofen with either drugalonerdquoArchives of Disease in Childhood vol 96 no 12 pp 1175ndash1179 2011

[15] E Purssell ldquoCombining paracetamol and ibuprofen for feverin childrenrdquo British Medical Journal vol 337 Article ID A15902008

[16] M Arpa ldquoDoes acetaminophen in comparison to ibuprofeneffectively reduce fevers in children younger than 18 years ofagerdquo Pediatric Nursing vol 36 no 4 pp 219ndash220 2010

[17] J Crook ldquoFever management evaluating the use of ibuprofenand paracetamolrdquo Paediatric Nursing vol 22 no 3 pp 22ndash262010

[18] K Brune and B Hinz ldquoParacetamol ibuprofenor a com-bination of both drugs against knee pain an excellent newrandomised clinical trial answers old questions and suggestsnew therapeutic recommendationsrdquo Annals of the RheumaticDiseases vol 70 no 9 pp 1521ndash1522 2011

[19] L Wells M Drum J Nusstein A Reader and M BeckldquoWill adding acetaminophen (paracetamol) to ibuprofen bemore effective in relieving postoperative pain on symptomaticnecrotic teethrdquo Evidence-Based Dentistry vol 13 no 4 article105 2012

[20] L K Wells M Drum J Nusstein A Reader and M BeckldquoEfficacy of ibuprofen and ibuprofenacetaminophen on post-operative pain in symptomatic patients with a pulpal diagnosisof necrosisrdquo Journal of Endodontics vol 37 no 12 pp 1608ndash16122011

[21] A F Merry R D Gibbs J Edwards et al ldquoCombinedacetaminophen and ibuprofen for pain relief after oral surgeryin adults a randomized controlled trialrdquo British Journal ofAnaesthesia vol 104 no 1 pp 80ndash88 2010

[22] J D Wark W Bensen C Recknor et al ldquoTreatment withacetaminophenparacetamol or ibuprofen alleviates post-dosesymptoms related to intravenous infusion with zoledronic acid5mgrdquo Osteoporosis International vol 23 no 2 pp 503ndash5122012


[23] V D Hoang N P Nhung and H Y Aboul-Enein ldquoRecentdevelopments and applications of derivative spectrophotometryin pharmaceutical analysisrdquo Current Pharmaceutical Analysisvol 9 pp 261ndash277 2013

[24] C B Ojeda and F S Rojas ldquoRecent applications in derivativeultravioletvisible absorption spectrophotometry 2009ndash2011 areviewrdquoMicrochemical Journal vol 106 pp 1ndash16 2013

[25] J C Cobas P G Tahoces M Martin-Pastor M Penedo and FJ Sardina ldquoWavelet-based ultra-high compression of multidi-mensional NMR data setsrdquo Journal of Magnetic Resonance vol168 no 2 pp 288ndash295 2004

[26] D Chen X Shao B Hu and Q Su ldquoA Background and noiseeliminationmethod for quantitative calibration of near infraredspectrardquoAnalytica Chimica Acta vol 511 no 1 pp 37ndash45 2004

[27] C Ma and X Shao ldquoContinuous wavelet transform applied toremoving the fluctuating background in near-infrared spectrardquoJournal of Chemical Information and Computer Sciences vol 44no 3 pp 907ndash911 2004

[28] E Dinc D Baleanu G Ioele M De Luca and G RagnoldquoMultivariate analysis of paracetamol propiphenazone caffeineand thiamine in quaternary mixtures by PCR PLS and ANNcalibrations applied on wavelet transform datardquo Journal ofPharmaceutical and Biomedical Analysis vol 48 no 5 pp 1471ndash1475 2008

[29] R Bhushan D Gupta and A Mukherjee ldquoLiquid chromato-graphic analysis of certain commercial formulations for non-opioid analgesicsrdquo Biomedical Chromatography vol 21 no 12pp 1284ndash1290 2007

[30] P R Battu andM S Reddy ldquoRPmdashHPLCmethod for simultane-ous estimation of Paracetamol and Ibuprofen in tabletsrdquo AsianJournal of Research in Chemistry vol 2 pp 70ndash72 2009

[31] N Dubey D Jain and S Jadhawani ldquoStability-indicatingHPTLC method for simultaneous estimation of famotidineparacetamol and ibuprofen in combined tablet dosage formsrdquoJournal of Planar Chromatography vol 25 no 2 pp 162ndash1672012

[32] X Yang J Feng Z Chen R Liao and X Li ldquoDeterminationof six analgesics by CE with an improved electromagneticinduction detectorrdquo Chromatographia vol 75 no 1-2 pp 71ndash76 2012

[33] D Basu K K Mahalanabis and B Roy ldquoApplication of leastsquares method in matrix form simultaneous determination ofibuprofen and paracetamol in tabletsrdquo Journal of Pharmaceuti-cal and Biomedical Analysis vol 16 no 5 pp 809ndash812 1998

[34] Y M Issa S I M Zayed and I H I Habib ldquoSimultaneousdetermination of ibuprofen and paracetamol using derivativesof the ratio spectra methodrdquo Arabian Journal of Chemistry vol4 no 3 pp 259ndash263 2011

[35] W S Hassan ldquoDetermination of ibuprofen and paracetamol inbinary mixture using chemometric-assisted spectrophotomet-ric methodsrdquo American Journal of Applied Sciences vol 5 no 8pp 1005ndash1012 2008

[36] A Savitzky and M J E Golay ldquoSmoothing and differentiationof data by simplified least squares proceduresrdquo AnalyticalChemistry vol 36 no 8 pp 1627ndash1639 1964

[37] C Jaideva A K Goswami and W C John Fundamentals ofWavelet Theory Algorithms and Applications Wiley series inMicrowave and Optical Engineering John Wiley amp Sons 2ndedition 2011

[38] M Unser and T Blu ldquoFractional splines and waveletsrdquo SIAMReview vol 42 no 1 pp 43ndash67 2000

[39] Z Yasmeen T Mahatha H Farheen and H K QureshildquoDissolution method development and validation for combi-nation of ibuprofen and paracetamol tabletsrdquo Asian Journal ofPharmaceutical and Clinical Research vol 6 pp 164ndash168 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


HOO

NH

(a) N-Acetyl-p-aminophenol

OH

O

(b) (RS)-2-(4-(2-Methylpropyl)phenyl)pro-panoic acid

Figure 1 Chemical structure of paracetamol (a) and ibuprofen (b)

Spectra

Derivative transform

Sym6 haar

Ratio spectra

Wavelet transform

Ratio spectra

(Sym6 haar)

Spectra (Sym6 haar mexh FWT)

FWT-spectra)

Sym6 haar mexh (FWT-

spectra)

Spectra (1D)

mexh (1Dspectra)

(1D)

1D (Sym6

Figure 2 Derivative and wavelet transforms for the simultaneous determination of IB and PA in binary mixtures

0

1

2

3

210 230 250 270 290

Abso

rban

ce

Wavelength (nm)

IB 20PA 325

IB 20 + PA 325 Addition spectrum

Figure 3 Spectra of IB 20mgL PA 325mgL and their corre-sponding mixture and absorbance addition

it may have drawbacks in some cases such as (i) the higherorder differentiation process diminishes peak amplitude aswell as signal-to-noise ratio (ii) the finding of zero-crossingpoints is very difficult and ratio spectra derivative working

wavelength is undetermined These drawbacks can be elimi-nated by applying wavelet transform approach to the originalabsorption spectra The wavelet transform can be regardedas mathematical functions that cut up data into differentfrequency components and then study each component witha resolutionmatched to its scale It is a powerful tool for signalprocessing in many branches of science and engineering Inthe last decade its applications in analytical chemistry forexample data reduction [25] denoising [26] and baselinecorrection [27] have been recordedThe wavelet-based reso-lution of multicomponent pharmaceutical mixtures has alsobeen exploited for which quaternary mixtures containingparacetamol are an example [28]

In the literature the determination of paracetamol andibuprofen could be simultaneously performed by HPLC CEand HPTLC [29ndash32] Their spectrophotometric determina-tion was also studied by applying chemometrics and deriva-tive approach [33ndash35] It is noteworthy that differentiationand smoothing algorithms forUVderivative spectrophotom-etry in these studies were not clearly indicated whereas theyalways play an important role in determining the sensitivityand accuracy of derivative techniques

The aim of this study was to develop derivativeandwavelet-based UV spectrophotometric methods for thesimultaneous determination of paracetamol and ibuprofen intheir combined tablets using HPLC as a reference method


0

1

2

3

4


2493nm2420nm

minus5

minus4

minus3

minus2

minus1

1D

spec

tra


(a)

1D

ratio

spec

tra

0

2000

4000


minus6000

minus4000

minus2000

2748nm


(b)

1D

ratio

spec

tra

0

2

4


minus8

minus10

minus6

minus4

minus2

2344nm


(c)



2 Experimental






0

4

8

12

200 220 240 260 280 300Wavelength (nm)

minus8

minus4

2765nm

2568nm


Sym6

-spe

ctra

(a)

0

5

10

200 220 240 260 280 300Wavelength (nm)

Haa

r-sp

ectr

a

2487nm2408nm


minus5

minus10

minus15

(b)

0

5

10

15

20

25

30

200 220 240 260 280 300

minus5

minus10

minus15

Mex

h-sp

ectr

a

Wavelength (nm)


2443nm

2726nm

(c)











minus600

minus200

200

600

Sym6

-rat

io sp

ectr

a

2303nm


(a)

220 240 260 280minus10

10

0

2364nm

Sym6

-rat

io sp

ectr

a

Wavelength (nm)


(b)

500

1500

2500

225 245 265 285

Haa

r-ra

tio sp

ectr

a

minus1500

minus500

Wavelength (nm)

2793nm


(c)

5

15

210 230 250 270

minus5

minus15

minus25

2287nm

Wavelength (nm)

Haa

r-ra

tio sp

ectr

a


(d)



119875 (120582) = 1198860+ 1198861120582 + 119886

21205822+ sdot sdot sdot + 119886

119898120582119898 (1)


0 119886



119889119875

119889120582

100381610038161003816100381610038161003816100381610038161205820=0

= 1198861+ 21198862120582 + sdot sdot sdot + 119898119886

119898120582119898minus1

= 1198861

1198892119875

1198891205822

1003816100381610038161003816100381610038161003816100381610038161205820=0


119898120582119898minus2

= 21198862

1198893119875

1198891205823

1003816100381610038161003816100381610038161003816100381610038161205820=0


119898120582119898minus3

= 61198863

(2)





120595119886119887(119905) =

1

radic|119886|

120595(119905 minus 119887

119886) with 119886 119887 isin 119877 119886 = 0 (3)


0

5

10

15

200 250 300

Sym

6-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

2808nm

2673nm


(a)

0

5

10

15

200 250 300

Haa

r-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

minus15

2595nm

2811nm


(b)

0

20

40

minus20

2342nm

2422nm



Mex

h-FW

T sp

ectr

a

(c)



119882119891(119886 119887) = int

infin

minusinfin

119891 (119905) 120595119886119887(119905) 119889119905 (4)


119891 (119905) =1

119862120595

int

infin

minusinfin

1

1198862[int

infin

minusinfin

119882119891(119886 119887) 120595

119886119887(119905) 119889119887] 119889119886 (5)



120595119898119899(119905) =

1

1198861198982

0

120595(119905 minus 119899119887

0

119886119898

0

) (6)


119882119891(119898 119899) = int

infin

minusinfin

119891 (119905) 120595119898119899(119905) 119889119905 (7)

Usually a0= 2 and b





0 1199051 119905




225 250 275

2690nm

2625nm

minus30

minus20

minus10

0

10

20

30

40

Sym6

-1D

spec

tra

Wavelength (nm)


(a)

0

10

20

30

40

50

225 250 275minus30

minus20

minus10

Wavelength (nm)


2725nm

2316nm

Haa

r-1D

spec

tra

(b)

0

20

40

225 250 275

minus40

minus60

minus20

2400nm

2599nm

Wavelength (nm)


Mex

h-1D

spec

tra

(c)



119905119898minus119901minus1


1198731198940(119905) =

1 if 119905119894le 119905 lt 119905

119894+1 119905119894lt 119905119894+1

0 otherwise

119873119894119901 (119905) =

119905 minus 119905119894

119905119894+119901minus 119905119894

119873119894119901minus1 (119905) +

119905119894+119901+1

minus 119905

119905119894+119901+1

minus 119905119894+1

119873119894+1119901minus1 (119905)

(8)


119862 (119905) =

119899

sum

119894=0

119875119894119873119894119901(119905) (9)

is a B-spline


120573120572

+(119909) =

Δ120572+1

+119909120572

+

Γ (120572 + 1)=sum+infin

119896=0(minus1)119896(120572+1

119896) (119909 minus 119896)

120572

+

Γ (120572 + 1) (10)


Γ (120572 + 1) = int

+infin

0

119909120572119890minus119909119889119909

(119909 minus 119896)120572

+= max (119909 minus 119896 0)120572

(11)


Δ120572

+119891 (119909) =

+infin

sum

0

(minus1)119896(120572

119896) 119891 (119909 minus 119896) (12)

where

(120572

119896) =

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (13)


1205731205721

+times 1205731205722

+= 1205731205721+1205722

+ (14)


200 220 240 260 280 300

2

1

0

minus1

minus2

1D

-sym

6sp

ectr

a

Wavelength (nm)


2594nm

2445nm

(a)

0

01

02

225 250 275

minus01

minus02

Wavelength (nm)


1D

-FW

T sp

ectr

a 2592nm

2722nm

(b)


0 1 2 3 4 5(min)

400350300250200150100

500

(mAU

)

0858

Para

ceta

mol

4328

Ibup

rofe

n



120573120572

lowast(119909) =

1

Γ (120572 + 1)sum

119896isin119885

(minus1)119896

10038161003816100381610038161003816100381610038161003816

120572 + 1

119896

10038161003816100381610038161003816100381610038161003816

|119909 minus 119896|120572

lowast (15)


|119909|120572

lowast=

|119909|120572


1199092119899 log119909(minus1)1+119899120587 120572 even

(16)


120595120572

+(119909

2) = sum

119896isin119885

(minus1)119896

2120572sum

1isin119885

(120572 + 1

1)

times 1205732120572+1

lowast(1 + 119896 minus 1) 120573

120572

+(119909 minus 119896)

(17)


int

+infin

minusinfin

119909119899120595120572

+(119909) 119889119909 = 0 (18)


120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

(19)









119878119887

119878119910sdot119909

1198772




















































Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789


Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124


IBI 1207II 0464III 1692

PAI 1377II 1597III 1067




IBI 6923II 6201III 6506

PAI 7352II 5810III 2870








5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0

1

2

3

4


2493nm2420nm

minus5

minus4

minus3

minus2

minus1

1D

spec

tra


(a)

1D

ratio

spec

tra

0

2000

4000


minus6000

minus4000

minus2000

2748nm


(b)

1D

ratio

spec

tra

0

2

4


minus8

minus10

minus6

minus4

minus2

2344nm


(c)



2 Experimental






0

4

8

12

200 220 240 260 280 300Wavelength (nm)

minus8

minus4

2765nm

2568nm


Sym6

-spe

ctra

(a)

0

5

10

200 220 240 260 280 300Wavelength (nm)

Haa

r-sp

ectr

a

2487nm2408nm


minus5

minus10

minus15

(b)

0

5

10

15

20

25

30

200 220 240 260 280 300

minus5

minus10

minus15

Mex

h-sp

ectr

a

Wavelength (nm)


2443nm

2726nm

(c)











minus600

minus200

200

600

Sym6

-rat

io sp

ectr

a

2303nm


(a)

220 240 260 280minus10

10

0

2364nm

Sym6

-rat

io sp

ectr

a

Wavelength (nm)


(b)

500

1500

2500

225 245 265 285

Haa

r-ra

tio sp

ectr

a

minus1500

minus500

Wavelength (nm)

2793nm


(c)

5

15

210 230 250 270

minus5

minus15

minus25

2287nm

Wavelength (nm)

Haa

r-ra

tio sp

ectr

a


(d)



119875 (120582) = 1198860+ 1198861120582 + 119886

21205822+ sdot sdot sdot + 119886

119898120582119898 (1)


0 119886



119889119875

119889120582

100381610038161003816100381610038161003816100381610038161205820=0

= 1198861+ 21198862120582 + sdot sdot sdot + 119898119886

119898120582119898minus1

= 1198861

1198892119875

1198891205822

1003816100381610038161003816100381610038161003816100381610038161205820=0


119898120582119898minus2

= 21198862

1198893119875

1198891205823

1003816100381610038161003816100381610038161003816100381610038161205820=0


119898120582119898minus3

= 61198863

(2)





120595119886119887(119905) =

1

radic|119886|

120595(119905 minus 119887

119886) with 119886 119887 isin 119877 119886 = 0 (3)


0

5

10

15

200 250 300

Sym

6-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

2808nm

2673nm


(a)

0

5

10

15

200 250 300

Haa

r-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

minus15

2595nm

2811nm


(b)

0

20

40

minus20

2342nm

2422nm



Mex

h-FW

T sp

ectr

a

(c)



119882119891(119886 119887) = int

infin

minusinfin

119891 (119905) 120595119886119887(119905) 119889119905 (4)


119891 (119905) =1

119862120595

int

infin

minusinfin

1

1198862[int

infin

minusinfin

119882119891(119886 119887) 120595

119886119887(119905) 119889119887] 119889119886 (5)



120595119898119899(119905) =

1

1198861198982

0

120595(119905 minus 119899119887

0

119886119898

0

) (6)


119882119891(119898 119899) = int

infin

minusinfin

119891 (119905) 120595119898119899(119905) 119889119905 (7)

Usually a0= 2 and b





0 1199051 119905




225 250 275

2690nm

2625nm

minus30

minus20

minus10

0

10

20

30

40

Sym6

-1D

spec

tra

Wavelength (nm)


(a)

0

10

20

30

40

50

225 250 275minus30

minus20

minus10

Wavelength (nm)


2725nm

2316nm

Haa

r-1D

spec

tra

(b)

0

20

40

225 250 275

minus40

minus60

minus20

2400nm

2599nm

Wavelength (nm)


Mex

h-1D

spec

tra

(c)



119905119898minus119901minus1


1198731198940(119905) =

1 if 119905119894le 119905 lt 119905

119894+1 119905119894lt 119905119894+1

0 otherwise

119873119894119901 (119905) =

119905 minus 119905119894

119905119894+119901minus 119905119894

119873119894119901minus1 (119905) +

119905119894+119901+1

minus 119905

119905119894+119901+1

minus 119905119894+1

119873119894+1119901minus1 (119905)

(8)


119862 (119905) =

119899

sum

119894=0

119875119894119873119894119901(119905) (9)

is a B-spline


120573120572

+(119909) =

Δ120572+1

+119909120572

+

Γ (120572 + 1)=sum+infin

119896=0(minus1)119896(120572+1

119896) (119909 minus 119896)

120572

+

Γ (120572 + 1) (10)


Γ (120572 + 1) = int

+infin

0

119909120572119890minus119909119889119909

(119909 minus 119896)120572

+= max (119909 minus 119896 0)120572

(11)


Δ120572

+119891 (119909) =

+infin

sum

0

(minus1)119896(120572

119896) 119891 (119909 minus 119896) (12)

where

(120572

119896) =

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (13)


1205731205721

+times 1205731205722

+= 1205731205721+1205722

+ (14)


200 220 240 260 280 300

2

1

0

minus1

minus2

1D

-sym

6sp

ectr

a

Wavelength (nm)


2594nm

2445nm

(a)

0

01

02

225 250 275

minus01

minus02

Wavelength (nm)


1D

-FW

T sp

ectr

a 2592nm

2722nm

(b)


0 1 2 3 4 5(min)

400350300250200150100

500

(mAU

)

0858

Para

ceta

mol

4328

Ibup

rofe

n



120573120572

lowast(119909) =

1

Γ (120572 + 1)sum

119896isin119885

(minus1)119896

10038161003816100381610038161003816100381610038161003816

120572 + 1

119896

10038161003816100381610038161003816100381610038161003816

|119909 minus 119896|120572

lowast (15)


|119909|120572

lowast=

|119909|120572


1199092119899 log119909(minus1)1+119899120587 120572 even

(16)


120595120572

+(119909

2) = sum

119896isin119885

(minus1)119896

2120572sum

1isin119885

(120572 + 1

1)

times 1205732120572+1

lowast(1 + 119896 minus 1) 120573

120572

+(119909 minus 119896)

(17)


int

+infin

minusinfin

119909119899120595120572

+(119909) 119889119909 = 0 (18)


120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

(19)









119878119887

119878119910sdot119909

1198772




















































Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789


Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124


IBI 1207II 0464III 1692

PAI 1377II 1597III 1067




IBI 6923II 6201III 6506

PAI 7352II 5810III 2870








5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0

4

8

12

200 220 240 260 280 300Wavelength (nm)

minus8

minus4

2765nm

2568nm


Sym6

-spe

ctra

(a)

0

5

10

200 220 240 260 280 300Wavelength (nm)

Haa

r-sp

ectr

a

2487nm2408nm


minus5

minus10

minus15

(b)

0

5

10

15

20

25

30

200 220 240 260 280 300

minus5

minus10

minus15

Mex

h-sp

ectr

a

Wavelength (nm)


2443nm

2726nm

(c)











minus600

minus200

200

600

Sym6

-rat

io sp

ectr

a

2303nm


(a)

220 240 260 280minus10

10

0

2364nm

Sym6

-rat

io sp

ectr

a

Wavelength (nm)


(b)

500

1500

2500

225 245 265 285

Haa

r-ra

tio sp

ectr

a

minus1500

minus500

Wavelength (nm)

2793nm


(c)

5

15

210 230 250 270

minus5

minus15

minus25

2287nm

Wavelength (nm)

Haa

r-ra

tio sp

ectr

a


(d)



119875 (120582) = 1198860+ 1198861120582 + 119886

21205822+ sdot sdot sdot + 119886

119898120582119898 (1)


0 119886



119889119875

119889120582

100381610038161003816100381610038161003816100381610038161205820=0

= 1198861+ 21198862120582 + sdot sdot sdot + 119898119886

119898120582119898minus1

= 1198861

1198892119875

1198891205822

1003816100381610038161003816100381610038161003816100381610038161205820=0


119898120582119898minus2

= 21198862

1198893119875

1198891205823

1003816100381610038161003816100381610038161003816100381610038161205820=0


119898120582119898minus3

= 61198863

(2)





120595119886119887(119905) =

1

radic|119886|

120595(119905 minus 119887

119886) with 119886 119887 isin 119877 119886 = 0 (3)


0

5

10

15

200 250 300

Sym

6-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

2808nm

2673nm


(a)

0

5

10

15

200 250 300

Haa

r-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

minus15

2595nm

2811nm


(b)

0

20

40

minus20

2342nm

2422nm



Mex

h-FW

T sp

ectr

a

(c)



119882119891(119886 119887) = int

infin

minusinfin

119891 (119905) 120595119886119887(119905) 119889119905 (4)


119891 (119905) =1

119862120595

int

infin

minusinfin

1

1198862[int

infin

minusinfin

119882119891(119886 119887) 120595

119886119887(119905) 119889119887] 119889119886 (5)



120595119898119899(119905) =

1

1198861198982

0

120595(119905 minus 119899119887

0

119886119898

0

) (6)


119882119891(119898 119899) = int

infin

minusinfin

119891 (119905) 120595119898119899(119905) 119889119905 (7)

Usually a0= 2 and b





0 1199051 119905




225 250 275

2690nm

2625nm

minus30

minus20

minus10

0

10

20

30

40

Sym6

-1D

spec

tra

Wavelength (nm)


(a)

0

10

20

30

40

50

225 250 275minus30

minus20

minus10

Wavelength (nm)


2725nm

2316nm

Haa

r-1D

spec

tra

(b)

0

20

40

225 250 275

minus40

minus60

minus20

2400nm

2599nm

Wavelength (nm)


Mex

h-1D

spec

tra

(c)



119905119898minus119901minus1


1198731198940(119905) =

1 if 119905119894le 119905 lt 119905

119894+1 119905119894lt 119905119894+1

0 otherwise

119873119894119901 (119905) =

119905 minus 119905119894

119905119894+119901minus 119905119894

119873119894119901minus1 (119905) +

119905119894+119901+1

minus 119905

119905119894+119901+1

minus 119905119894+1

119873119894+1119901minus1 (119905)

(8)


119862 (119905) =

119899

sum

119894=0

119875119894119873119894119901(119905) (9)

is a B-spline


120573120572

+(119909) =

Δ120572+1

+119909120572

+

Γ (120572 + 1)=sum+infin

119896=0(minus1)119896(120572+1

119896) (119909 minus 119896)

120572

+

Γ (120572 + 1) (10)


Γ (120572 + 1) = int

+infin

0

119909120572119890minus119909119889119909

(119909 minus 119896)120572

+= max (119909 minus 119896 0)120572

(11)


Δ120572

+119891 (119909) =

+infin

sum

0

(minus1)119896(120572

119896) 119891 (119909 minus 119896) (12)

where

(120572

119896) =

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (13)


1205731205721

+times 1205731205722

+= 1205731205721+1205722

+ (14)


200 220 240 260 280 300

2

1

0

minus1

minus2

1D

-sym

6sp

ectr

a

Wavelength (nm)


2594nm

2445nm

(a)

0

01

02

225 250 275

minus01

minus02

Wavelength (nm)


1D

-FW

T sp

ectr

a 2592nm

2722nm

(b)


0 1 2 3 4 5(min)

400350300250200150100

500

(mAU

)

0858

Para

ceta

mol

4328

Ibup

rofe

n



120573120572

lowast(119909) =

1

Γ (120572 + 1)sum

119896isin119885

(minus1)119896

10038161003816100381610038161003816100381610038161003816

120572 + 1

119896

10038161003816100381610038161003816100381610038161003816

|119909 minus 119896|120572

lowast (15)


|119909|120572

lowast=

|119909|120572


1199092119899 log119909(minus1)1+119899120587 120572 even

(16)


120595120572

+(119909

2) = sum

119896isin119885

(minus1)119896

2120572sum

1isin119885

(120572 + 1

1)

times 1205732120572+1

lowast(1 + 119896 minus 1) 120573

120572

+(119909 minus 119896)

(17)


int

+infin

minusinfin

119909119899120595120572

+(119909) 119889119909 = 0 (18)


120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

(19)









119878119887

119878119910sdot119909

1198772




















































Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789


Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124


IBI 1207II 0464III 1692

PAI 1377II 1597III 1067




IBI 6923II 6201III 6506

PAI 7352II 5810III 2870








5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



minus600

minus200

200

600

Sym6

-rat

io sp

ectr

a

2303nm


(a)

220 240 260 280minus10

10

0

2364nm

Sym6

-rat

io sp

ectr

a

Wavelength (nm)


(b)

500

1500

2500

225 245 265 285

Haa

r-ra

tio sp

ectr

a

minus1500

minus500

Wavelength (nm)

2793nm


(c)

5

15

210 230 250 270

minus5

minus15

minus25

2287nm

Wavelength (nm)

Haa

r-ra

tio sp

ectr

a


(d)



119875 (120582) = 1198860+ 1198861120582 + 119886

21205822+ sdot sdot sdot + 119886

119898120582119898 (1)


0 119886



119889119875

119889120582

100381610038161003816100381610038161003816100381610038161205820=0

= 1198861+ 21198862120582 + sdot sdot sdot + 119898119886

119898120582119898minus1

= 1198861

1198892119875

1198891205822

1003816100381610038161003816100381610038161003816100381610038161205820=0


119898120582119898minus2

= 21198862

1198893119875

1198891205823

1003816100381610038161003816100381610038161003816100381610038161205820=0


119898120582119898minus3

= 61198863

(2)





120595119886119887(119905) =

1

radic|119886|

120595(119905 minus 119887

119886) with 119886 119887 isin 119877 119886 = 0 (3)


0

5

10

15

200 250 300

Sym

6-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

2808nm

2673nm


(a)

0

5

10

15

200 250 300

Haa

r-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

minus15

2595nm

2811nm


(b)

0

20

40

minus20

2342nm

2422nm



Mex

h-FW

T sp

ectr

a

(c)



119882119891(119886 119887) = int

infin

minusinfin

119891 (119905) 120595119886119887(119905) 119889119905 (4)


119891 (119905) =1

119862120595

int

infin

minusinfin

1

1198862[int

infin

minusinfin

119882119891(119886 119887) 120595

119886119887(119905) 119889119887] 119889119886 (5)



120595119898119899(119905) =

1

1198861198982

0

120595(119905 minus 119899119887

0

119886119898

0

) (6)


119882119891(119898 119899) = int

infin

minusinfin

119891 (119905) 120595119898119899(119905) 119889119905 (7)

Usually a0= 2 and b





0 1199051 119905




225 250 275

2690nm

2625nm

minus30

minus20

minus10

0

10

20

30

40

Sym6

-1D

spec

tra

Wavelength (nm)


(a)

0

10

20

30

40

50

225 250 275minus30

minus20

minus10

Wavelength (nm)


2725nm

2316nm

Haa

r-1D

spec

tra

(b)

0

20

40

225 250 275

minus40

minus60

minus20

2400nm

2599nm

Wavelength (nm)


Mex

h-1D

spec

tra

(c)



119905119898minus119901minus1


1198731198940(119905) =

1 if 119905119894le 119905 lt 119905

119894+1 119905119894lt 119905119894+1

0 otherwise

119873119894119901 (119905) =

119905 minus 119905119894

119905119894+119901minus 119905119894

119873119894119901minus1 (119905) +

119905119894+119901+1

minus 119905

119905119894+119901+1

minus 119905119894+1

119873119894+1119901minus1 (119905)

(8)


119862 (119905) =

119899

sum

119894=0

119875119894119873119894119901(119905) (9)

is a B-spline


120573120572

+(119909) =

Δ120572+1

+119909120572

+

Γ (120572 + 1)=sum+infin

119896=0(minus1)119896(120572+1

119896) (119909 minus 119896)

120572

+

Γ (120572 + 1) (10)


Γ (120572 + 1) = int

+infin

0

119909120572119890minus119909119889119909

(119909 minus 119896)120572

+= max (119909 minus 119896 0)120572

(11)


Δ120572

+119891 (119909) =

+infin

sum

0

(minus1)119896(120572

119896) 119891 (119909 minus 119896) (12)

where

(120572

119896) =

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (13)


1205731205721

+times 1205731205722

+= 1205731205721+1205722

+ (14)


200 220 240 260 280 300

2

1

0

minus1

minus2

1D

-sym

6sp

ectr

a

Wavelength (nm)


2594nm

2445nm

(a)

0

01

02

225 250 275

minus01

minus02

Wavelength (nm)


1D

-FW

T sp

ectr

a 2592nm

2722nm

(b)


0 1 2 3 4 5(min)

400350300250200150100

500

(mAU

)

0858

Para

ceta

mol

4328

Ibup

rofe

n



120573120572

lowast(119909) =

1

Γ (120572 + 1)sum

119896isin119885

(minus1)119896

10038161003816100381610038161003816100381610038161003816

120572 + 1

119896

10038161003816100381610038161003816100381610038161003816

|119909 minus 119896|120572

lowast (15)


|119909|120572

lowast=

|119909|120572


1199092119899 log119909(minus1)1+119899120587 120572 even

(16)


120595120572

+(119909

2) = sum

119896isin119885

(minus1)119896

2120572sum

1isin119885

(120572 + 1

1)

times 1205732120572+1

lowast(1 + 119896 minus 1) 120573

120572

+(119909 minus 119896)

(17)


int

+infin

minusinfin

119909119899120595120572

+(119909) 119889119909 = 0 (18)


120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

(19)









119878119887

119878119910sdot119909

1198772




















































Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789


Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124


IBI 1207II 0464III 1692

PAI 1377II 1597III 1067




IBI 6923II 6201III 6506

PAI 7352II 5810III 2870








5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0

5

10

15

200 250 300

Sym

6-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

2808nm

2673nm


(a)

0

5

10

15

200 250 300

Haa

r-FW

T sp

ectr

a

Wavelength (nm)

minus5

minus10

minus15

2595nm

2811nm


(b)

0

20

40

minus20

2342nm

2422nm



Mex

h-FW

T sp

ectr

a

(c)



119882119891(119886 119887) = int

infin

minusinfin

119891 (119905) 120595119886119887(119905) 119889119905 (4)


119891 (119905) =1

119862120595

int

infin

minusinfin

1

1198862[int

infin

minusinfin

119882119891(119886 119887) 120595

119886119887(119905) 119889119887] 119889119886 (5)



120595119898119899(119905) =

1

1198861198982

0

120595(119905 minus 119899119887

0

119886119898

0

) (6)


119882119891(119898 119899) = int

infin

minusinfin

119891 (119905) 120595119898119899(119905) 119889119905 (7)

Usually a0= 2 and b





0 1199051 119905




225 250 275

2690nm

2625nm

minus30

minus20

minus10

0

10

20

30

40

Sym6

-1D

spec

tra

Wavelength (nm)


(a)

0

10

20

30

40

50

225 250 275minus30

minus20

minus10

Wavelength (nm)


2725nm

2316nm

Haa

r-1D

spec

tra

(b)

0

20

40

225 250 275

minus40

minus60

minus20

2400nm

2599nm

Wavelength (nm)


Mex

h-1D

spec

tra

(c)



119905119898minus119901minus1


1198731198940(119905) =

1 if 119905119894le 119905 lt 119905

119894+1 119905119894lt 119905119894+1

0 otherwise

119873119894119901 (119905) =

119905 minus 119905119894

119905119894+119901minus 119905119894

119873119894119901minus1 (119905) +

119905119894+119901+1

minus 119905

119905119894+119901+1

minus 119905119894+1

119873119894+1119901minus1 (119905)

(8)


119862 (119905) =

119899

sum

119894=0

119875119894119873119894119901(119905) (9)

is a B-spline


120573120572

+(119909) =

Δ120572+1

+119909120572

+

Γ (120572 + 1)=sum+infin

119896=0(minus1)119896(120572+1

119896) (119909 minus 119896)

120572

+

Γ (120572 + 1) (10)


Γ (120572 + 1) = int

+infin

0

119909120572119890minus119909119889119909

(119909 minus 119896)120572

+= max (119909 minus 119896 0)120572

(11)


Δ120572

+119891 (119909) =

+infin

sum

0

(minus1)119896(120572

119896) 119891 (119909 minus 119896) (12)

where

(120572

119896) =

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (13)


1205731205721

+times 1205731205722

+= 1205731205721+1205722

+ (14)


200 220 240 260 280 300

2

1

0

minus1

minus2

1D

-sym

6sp

ectr

a

Wavelength (nm)


2594nm

2445nm

(a)

0

01

02

225 250 275

minus01

minus02

Wavelength (nm)


1D

-FW

T sp

ectr

a 2592nm

2722nm

(b)


0 1 2 3 4 5(min)

400350300250200150100

500

(mAU

)

0858

Para

ceta

mol

4328

Ibup

rofe

n



120573120572

lowast(119909) =

1

Γ (120572 + 1)sum

119896isin119885

(minus1)119896

10038161003816100381610038161003816100381610038161003816

120572 + 1

119896

10038161003816100381610038161003816100381610038161003816

|119909 minus 119896|120572

lowast (15)


|119909|120572

lowast=

|119909|120572


1199092119899 log119909(minus1)1+119899120587 120572 even

(16)


120595120572

+(119909

2) = sum

119896isin119885

(minus1)119896

2120572sum

1isin119885

(120572 + 1

1)

times 1205732120572+1

lowast(1 + 119896 minus 1) 120573

120572

+(119909 minus 119896)

(17)


int

+infin

minusinfin

119909119899120595120572

+(119909) 119889119909 = 0 (18)


120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

(19)









119878119887

119878119910sdot119909

1198772




















































Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789


Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124


IBI 1207II 0464III 1692

PAI 1377II 1597III 1067




IBI 6923II 6201III 6506

PAI 7352II 5810III 2870








5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


225 250 275

2690nm

2625nm

minus30

minus20

minus10

0

10

20

30

40

Sym6

-1D

spec

tra

Wavelength (nm)


(a)

0

10

20

30

40

50

225 250 275minus30

minus20

minus10

Wavelength (nm)


2725nm

2316nm

Haa

r-1D

spec

tra

(b)

0

20

40

225 250 275

minus40

minus60

minus20

2400nm

2599nm

Wavelength (nm)


Mex

h-1D

spec

tra

(c)



119905119898minus119901minus1


1198731198940(119905) =

1 if 119905119894le 119905 lt 119905

119894+1 119905119894lt 119905119894+1

0 otherwise

119873119894119901 (119905) =

119905 minus 119905119894

119905119894+119901minus 119905119894

119873119894119901minus1 (119905) +

119905119894+119901+1

minus 119905

119905119894+119901+1

minus 119905119894+1

119873119894+1119901minus1 (119905)

(8)


119862 (119905) =

119899

sum

119894=0

119875119894119873119894119901(119905) (9)

is a B-spline


120573120572

+(119909) =

Δ120572+1

+119909120572

+

Γ (120572 + 1)=sum+infin

119896=0(minus1)119896(120572+1

119896) (119909 minus 119896)

120572

+

Γ (120572 + 1) (10)


Γ (120572 + 1) = int

+infin

0

119909120572119890minus119909119889119909

(119909 minus 119896)120572

+= max (119909 minus 119896 0)120572

(11)


Δ120572

+119891 (119909) =

+infin

sum

0

(minus1)119896(120572

119896) 119891 (119909 minus 119896) (12)

where

(120572

119896) =

Γ (120572 + 1)

Γ (119896 + 1) Γ (120572 minus 119896 + 1) (13)


1205731205721

+times 1205731205722

+= 1205731205721+1205722

+ (14)


200 220 240 260 280 300

2

1

0

minus1

minus2

1D

-sym

6sp

ectr

a

Wavelength (nm)


2594nm

2445nm

(a)

0

01

02

225 250 275

minus01

minus02

Wavelength (nm)


1D

-FW

T sp

ectr

a 2592nm

2722nm

(b)


0 1 2 3 4 5(min)

400350300250200150100

500

(mAU

)

0858

Para

ceta

mol

4328

Ibup

rofe

n



120573120572

lowast(119909) =

1

Γ (120572 + 1)sum

119896isin119885

(minus1)119896

10038161003816100381610038161003816100381610038161003816

120572 + 1

119896

10038161003816100381610038161003816100381610038161003816

|119909 minus 119896|120572

lowast (15)


|119909|120572

lowast=

|119909|120572


1199092119899 log119909(minus1)1+119899120587 120572 even

(16)


120595120572

+(119909

2) = sum

119896isin119885

(minus1)119896

2120572sum

1isin119885

(120572 + 1

1)

times 1205732120572+1

lowast(1 + 119896 minus 1) 120573

120572

+(119909 minus 119896)

(17)


int

+infin

minusinfin

119909119899120595120572

+(119909) 119889119909 = 0 (18)


120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

(19)









119878119887

119878119910sdot119909

1198772




















































Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789


Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124


IBI 1207II 0464III 1692

PAI 1377II 1597III 1067




IBI 6923II 6201III 6506

PAI 7352II 5810III 2870








5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


200 220 240 260 280 300

2

1

0

minus1

minus2

1D

-sym

6sp

ectr

a

Wavelength (nm)


2594nm

2445nm

(a)

0

01

02

225 250 275

minus01

minus02

Wavelength (nm)


1D

-FW

T sp

ectr

a 2592nm

2722nm

(b)


0 1 2 3 4 5(min)

400350300250200150100

500

(mAU

)

0858

Para

ceta

mol

4328

Ibup

rofe

n



120573120572

lowast(119909) =

1

Γ (120572 + 1)sum

119896isin119885

(minus1)119896

10038161003816100381610038161003816100381610038161003816

120572 + 1

119896

10038161003816100381610038161003816100381610038161003816

|119909 minus 119896|120572

lowast (15)


|119909|120572

lowast=

|119909|120572


1199092119899 log119909(minus1)1+119899120587 120572 even

(16)


120595120572

+(119909

2) = sum

119896isin119885

(minus1)119896

2120572sum

1isin119885

(120572 + 1

1)

times 1205732120572+1

lowast(1 + 119896 minus 1) 120573

120572

+(119909 minus 119896)

(17)


int

+infin

minusinfin

119909119899120595120572

+(119909) 119889119909 = 0 (18)


120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

120572

lowast(120603) = 119862(119895120603)

120572+1 as 120596 997888rarr 0

(19)









119878119887

119878119910sdot119909

1198772




















































Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789


Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124


IBI 1207II 0464III 1692

PAI 1377II 1597III 1067




IBI 6923II 6201III 6506

PAI 7352II 5810III 2870








5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




119878119887

119878119910sdot119909

1198772




















































Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789


Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124


IBI 1207II 0464III 1692

PAI 1377II 1597III 1067




IBI 6923II 6201III 6506

PAI 7352II 5810III 2870








5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of






























Sum of squares

IBI 2511 11090 13601II 1270 14620 15890III 2856 9000 11856

PAI 2415 9355 11770II 3546 11845 15391III 1799 8990 10789


Mean of squares

IBI 1674 1386II 0847 1827III 1904 1125

PAI 1610 1169II 2364 1481III 1199 1124


IBI 1207II 0464III 1692

PAI 1377II 1597III 1067




IBI 6923II 6201III 6506

PAI 7352II 5810III 2870








5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



5 Conclusion




Acknowledgment


References


















































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

Research Article UV Spectrophotometric Simultaneous...

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