Research Article Haar Wavelet Operational Matrix Method ...downloads.hindawi.com/journals/ijmms/2014/174819.pdf · Haar wavelet is the lowest member of Daubechies family of wavelets

Research ArticleHaar Wavelet Operational Matrix Method forFractional Oscillation Equations

Umer Saeed and Mujeeb ur Rehman

School of Natural Sciences National University of Sciences and Technology Sector H-12 Islamabad Pakistan

Correspondence should be addressed to Umer Saeed umermathgmailcom

Received 25 February 2014 Accepted 29 June 2014 Published 15 July 2014

Academic Editor Marianna A Shubov

Copyright copy 2014 U Saeed and M ur RehmanThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Weutilized theHaarwavelet operationalmatrixmethod for fractional order nonlinear oscillation equations andfind the solutions offractional order force-free and forcedDuffing-Van der Pol oscillator and higher order fractional Duffing equation on large intervalsThe results are compared with the results obtained by the other technique and with exact solution

1 Introduction

Haar wavelet is the lowest member of Daubechies familyof wavelets and is convenient for computer implementa-tions due to availability of explicit expression for the Haarscaling and wavelet functions [1] Operational approachis pioneered by Chen and Hsiao [2] for uniform gridsThe basic idea of Haar wavelet technique is to convertdifferential equations into a system of algebraic equationsof finite variables The Haar wavelet technique for solvinglinear homogeneousinhomogeneous constant and variablecoefficients has been discussed in [3]

The fractional order forced Duffing-Van der Pol oscillatoris given by the following second order differential equation[4]

119888

119863120572

119910 (119905) minus 120583 (1 minus 1199102

(119905)) 1199101015840

(119905) + 119886119910 (119905) + 1198871199103

(119905)

= 119892 (119891 120596 119905) 1 lt 120572 le 2

(1)

where 119888119863120572 is the Caputo derivative 119892(119891 120596 119905) = 119891 cos(120596119905)represents the periodic driving function of time with period119879 = 2120587120596 where 120596 is the angular frequency of the drivingforce 119891 is the forcing strength and 120583 gt 0 is the dampingparameter of the system Duffing-Van der Pol oscillatorequation can be expressed in three physical situations

(1) single-well 119886 gt 0 119887 gt 0(2) double-well 119886 lt 0 119887 gt 0(3) double-hump 119886 gt 0 119887 lt 0The quasilinearization approach was introduced by Bell-

man and Kalaba [5 6] as a generalization of the Newton-Raphson method [7] to solve the individual or systems ofnonlinear ordinary and partial differential equations Thequasilinearization approach is suitable to general nonlinearordinary or partial differential equations of any order

The Haar wavelets with quasilinearization technique [8ndash10] are applied for the approximate solution of integerorder nonlinear differential equations In [11] we extend theHaar wavelet - quasilinearization technique for fractionalnonlinear differential equations

The aim of the present work is to investigate the solutionof the higher order fractional Duffing equation fractionalorder force-free and forced Duffing-Van der pol (DVP) oscil-lator using Haar wavelet-quasilinearization technique Wehave discussed the three special situations of DVP oscillatorequation such as single-well double-well and double- hump

2 Preliminaries

In this section we review basic definitions of fractionaldifferentiation and fractional integration [12]

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014 Article ID 174819 8 pageshttpdxdoiorg1011552014174819

2 International Journal of Mathematics and Mathematical Sciences

(1) Riemann-Liouville fractional integral operator oforder 120572 is as followsthe Riemann-Liouville fractional order integral oforder 120572 isin R+ is defined as

119868120572

119909119910 (119909) =

1

Γ (120572)int

119909

119886

(119909 minus 119905)120572minus1

119910 (119905) 119889119905 (2)

for 119886 lt 119909 le 119887(2) Riemann-Liouville and Caputo fractional derivative

operators of order 120572 are as followsthe Riemann-Liouville fractional order derivative oforder 120572 isin R+ is defined as

119903

119863120572

119909119910 (119909) =

1

Γ (119899 minus 120572)(119889

119889119909)

119899

int

119909

119886

(119909 minus 119905)119899minus120572minus1

119910 (119905) 119889119905 (3)

for 119886 lt 119909 le 119887 where 119899 minus 1 lt 120572 lt 119899 119899 isin N and119899 = lceil120572rceil

The Caputo fractional order derivative of order 120572 isin R+ isdefined as

119888

119863120572

119909119910 (119909) =

1

Γ (119899 minus 120572)int

119909

119886

(119909 minus 119905)119899minus120572minus1

(119889

119889119905)

119899

119910 (119905) 119889119905 (4)

for 119886 lt 119909 le 119887 where 119899 minus 1 lt 120572 lt 119899 119899 isin N and 119899 = lceil120572rceil

3 The Haar Wavelets

The Haar functions contain just one wavelet during somesubinterval of time and remain zero elsewhere and areorthogonal The 119894th uniform Haar wavelet ℎ

119894(119909) 119909 isin [119886 119887]

is defined as [2]

ℎ119894(119909) =

1 119886 + (119887 minus 119886)119896

119898le 119909 lt 119886 + (119887 minus 119886)

119896 + 05

119898

minus1 119886 + (119887 minus 119886)119896 + 05

119898le 119909 lt 119886 + (119887 minus 119886)

119896 + 1

119898

0 otherwise(5)

where 119894 = 2119895 + 119896 + 1 119895 = 0 1 2 119869 is dilation parameterwhere 119898 = 2

119895 and 119896 = 0 1 2 2119895

minus 1 is translationparameter 119869 is maximal level of resolution and the maximalvalue of 119894 is 2119872where119872 = 2

119869 In particular ℎ1(119909) = 120594

[119886119887](119909)

where 120594[119886119887](119909) is characteristic function on interval [119886 119887]

is the Haar scaling function For the uniform Haar waveletthe wavelet-collocation method is applied The collocationpoints for the uniform Haar wavelets are usually taken as119909119895= 119886 + (119887 minus 119886)((119895 minus 05)2119872) where 119895 = 1 2 2119872

31 Fractional Integral of the Haar Wavelets Any function119910 isin 119871

2[119886 119887] can be represented in terms of the uniform Haar

series

119910 (119909) =

2119872

sum

119897=1

119887119897ℎ119897(119909) 119897 = 2

119895

+ 119896 + 1

119895 = 0 1 2 119869 119896 = 0 1 2 2119895

minus 1

(6)

where 119887119897are the Haar wavelet coefficients given as 119887

119897=

intinfin

minusinfin

119910(119909)ℎ119897(119909)119889119909

Any function of two variables 119906(119909 119905) isin 1198712[119886 119887] times [119886 119887]

can be approximated as

119906 (119909 119905) asymp

2119872

sum

119897=1

2119872

sum

119894=1

119888119897119894ℎ119897(119909) ℎ119894(119905) = 119867

119879

(119909) 119862119867 (119905) (7)

where 119862 is 2119872 times 2119872 coefficient matrix which can bedetermined by the inner product 119888

119897119894= ⟨ℎ119897(119909) ⟨119906(119909 119905) ℎ

119894(119905)⟩⟩

The Riemann-Liouville fractional integral of the uniformHaar wavelets is given as

119868120572

119909ℎ1(119909) =

(119909 minus 119886)120572

Γ (120572 + 1) (8)

119875120572119897(119909) = 119868

120572

119909ℎ119897(119909) =

1

Γ (120572)int

119909

119886

(119909 minus 119904)120572minus1

ℎ119897(119904) 119889119904

=1

Γ (120572 + 1)

(119909 minus 119886 (119897))120572

119886 (119897) le 119909 lt 119887 (119897)

(119909 minus 119886 (119897))120572

minus 2(119909 minus 119887 (119897))120572

119887 (119897) le 119909 lt 119888 (119897)

(119909minus119886 (119897))120572

minus 2(119909minus119887 (119897))120572

+(119909minus119888 (119897))120572

119909 ge 119888 (119897)

(9)

where 119886(119897) = 119886 + (119887 minus 119886)(119896119898) 119887(119897) = 119886 + (119887 minus 119886)((119896 + 05)119898)and 119888(119897) = 119886 + (119887 minus 119886)((119896 + 1)119898)

4 Convergence Analysis

Our work is based on quasilinearization technique and Haarwavelet method first we analyze the convergence of bothschemes and then describe the role of their convergenceaccording to present work

41 Convergence of Quasilinearization Technique [6] Con-sider the nonlinear second order differential equation

11991010158401015840

(119909) = 119891 (119910) 119910 (0) = 119910 (119887) = 0 (10)

Applying quasilinearization technique to (10) yields

11991010158401015840

119899+1(119909) = 119891 (119910

119899) + (119910

119899+1minus 119910119899) 1198911015840

(119910119899)

119910119899+1(0) = 119910

119899+1(119887) = 0

(11)

Let 1199100(119909) be some initial approximation Each function

119910119899+1(119909) is a solution of a linear equation (11) where 119910

119899is

always considered to be known and is obtained from theprevious iteration

According to [6] and letting max119910(|119891(119910)| |119891

1015840

(119910)|) = 119898 lt

infin and 119896 = max119906|11989110158401015840

(119906)| we have

max119909

1003816100381610038161003816119910119899+1 minus 1199101198991003816100381610038161003816 le

1198872

(1198968)

1 minus (11988721198984)(max119909

(1003816100381610038161003816119910119899 minus 119910119899minus1

1003816100381610038161003816)2

(12)

This shows that quasilinearization technique has quadraticconvergence if there is convergence at all

International Journal of Mathematics and Mathematical Sciences 3

42 Convergence of Haar Wavelet Method [15] Let 119910(119909) be adifferentiable function and assume that 119910(119909) have boundedfirst derivative on (0 1) that is there exist 119870 gt 0 for all 119909 isin(0 1)

100381610038161003816100381610038161199101015840

(119909)10038161003816100381610038161003816le 119870 (13)

Haar wavelet approximation for the function 119910(119909) is given by

119910119872(119909) =

2119872

sum

119897=1

119887119897ℎ119897(119909) (14)

Babolian and Shahsavaran [15] gave 1198712-error norm for Haar

wavelet approximation which is

1003817100381710038171003817119910 (119909) minus 119910119872 (119909)1003817100381710038171003817

2

le1198702

3sdot1

(2119872)2

(15)

or1003817100381710038171003817119910 (119909) minus 119910119872 (119909)

1003817100381710038171003817 le 119874(1

119872) (16)

As119872 = 2119869 and 119869 is the maximal level of resolution according

to (16) we conclude that error is inversely proportional to thelevel of resolution Equation (16) ensures the convergence ofHaar wavelet approximation at higher level of resolution thatis when119872 is increased

Each iteration of quasilinearization technique gives lin-ear differential equation in 119910

119899+1(119909) which is solved to get

approximate value of 119910119899+1(119909) 119910

119899+1119872(119909) by Haar wavelet

method Since solution of our problems has bounded firstderivatives over (0 1) according to (16) 119910

119899+1119872(119909) converges

fast to 119910119899+1(119909) if we consider the higher level of resolution

119869 that is we get more accurate results while increasing 119869and at the same time quasilinearization technique worksthat is given an initial approximation 119910

0(119909) we get solution

1199101(119909) of linear differential equation (11) by Haar wavelet

method and at next iteration we get 1199102(119909) and so on Since

quasilinearization technique is second order accurate so itgives rapid convergence if there is convergence at all Weconclude that solution by Haar wavelet quasilinearizationtechnique 119910

119899+1119872(119909) converges to exact solution 119910(119909) when

both 119869 and 119899 approachinfin

5 Applications

In this section we solve force-free Duffing-Van der Poloscillator of fractional order forced Duffing-Van der Poloscillator of fractional order and higher order fractionalDuffing equation by the Haar wavelet-quasilinearizationtechnique and compare the results with those obtained byother methods and exact solution

51 Forced Duffing-Van Der Pol Oscillator Equation [4]

Example 1 Consider the 120572th order fractional forced DVPoscillator equation119888

119863120572

119910 (119909) minus 120583 (1 minus 1199102

(119909)) 1199101015840

(119909) + 119886119910 (119909) + 1198871199103

(119909)

= 119891 cos (120596119909) 1 lt 120572 le 2

(17)

subject to the initial conditions 119910(0) = 1 and 1199101015840(0) = 0

Applying the quasilinearization technique to (17) weobtain119888

119863120572

119910119899+1(119909) minus 120583 (1 minus 119910

2

119899(119909)) 119910

1015840

119899+1(119909)

+ (119886 + 2120583119910119899(119909) 1199101015840

119899(119909) + 3119887119910

2

119899(119909)) 119910

119899+1(119905)

= 119891 cos (120596119909) + 21205831199102119899(119909) 1199101015840

119899(119909) + 2119887119910

3

119899(119909)

(18)

with the initial conditions 119910119899+1(0) = 1 and 1199101015840

119899+1(0) = 0

Now we apply the Haar wavelet method to (18) andapproximate the higher order derivative term by the Haarwavelet series as

119888

119863120572

119910119899+1(119909) =

2119872

sum

119897=1

119887119897ℎ119897(119909) (19)

Lower order derivatives are obtained by integrating (19)and use the initial condition

119910119899+1(119909) =

2119872

sum

119897=1

119887119897119901120572119897(119909) + 1 119910

1015840

119899+1(119909) =

2119872

sum

119897=1

119887119897119901120572minus1119897

(119909)

(20)

Substitute (19) and (20) into (18) to get

2119872

sum

119897=1

119887119897[ℎ119897(119909) minus 120583 (1 minus 119910

2

119899(119909)) 119901

120572minus1119897(119909)

+ (119886 + 2120583119910119899(119909) 1199101015840

119899(119909) + 3119887119910

2

119899(119909)) 119901

120572119897(119909)]

= 119891 cos (120596119909) + 21205831199102119899(119909) 1199101015840

119899(119909) + 2119887119910

3

119899(119909)

minus (119886 + 2120583119910119899(119909) 1199101015840

119899(119909) + 3119887119910

2

119899(119909))

(21)

with the initial approximations 1199100(119909) = 1 and 1199101015840

0(119909) = 0

(1) (Single-well 119886 gt 0 119887 gt 0) 119886 = 05 119887 = 05 120583 = 01119891 = 05 120596 = 079

(2) (Double-well 119886 lt 0 119887 gt 0) 119886 = minus05 119887 = 05 120583 = 01119891 = 05 120596 = 079

(3) (Double-hump 119886 gt 0 119887 lt 0) 119886 = 05 119887 = minus05120583 = 01 119891 = 05 120596 = 079

The results obtained using the Haar wavelet quasilin-earization technique at fifth iteration for the three situationssingle-well double-well and double-hump are given inTables 1 2 and 3 respectively Here we fix the order ofequation 120572 = 2 and level of resolution 119869 = 9 We comparedthe obtained solution with variational iteration method [13]homotopy perturbation method [13] and numerical solutionbased on the fourth-order Runge-Kutta (RK) method Alsothe absolute error relative to RK method is shown in Tables1 2 and 3 It shows that obtained results are more accurateas compared to variational iteration method and homotopyperturbation method Figures 1 2 and 3 showed the solution

4 International Journal of Mathematics and Mathematical Sciences

Table 1 Single-well situation comparison of solutions by the Haarwavelet-quasilinearization technique 119910HAAR at 5th iteration and levelof resolutions 119869 = 9 with numerical methods [13] and numericalsolution based on the fourth-order Runge-Kutta

120572 = 2

5th iteration 119869 = 9

119909 119910RK 119910HPM [13] 119910VIM [13] 119910HAARAbsoluteError

02 09900451 099004 099004 09900451 31e minus 804 09607026 096075 09607 09607024 15e minus 706 09134154 091383 091341 09134150 35e minus 708 08502496 085216 085025 08502491 58e minus 710 0773523 077973 077353 0773522 80e minus 7

Table 2 Double-well situation comparison of solutions by theHaarwavelet-quasilinearization technique 119910HAAR at 5th iteration and levelof resolutions 119869 = 9 with numerical methods [13] and numericalsolution based on the fourth-order Runge-Kutta

120572 = 2

5th iteration 119869 = 9

119909 119910RK 119910VIM [13] 119910HPM [13] 119910HAARAbsoluteError

02 1009945 100994 100994 1009945 98e minus 904 1039114 103911 103918 1039114 67e minus 806 1085448 108544 108621 1085448 19e minus 708 1145384 114539 114937 1145384 39e minus 710 1213777 121382 122785 1213778 64e minus 7

Table 3 Double-hump situation comparison of solutions by theHaar wavelet-quasilinearization technique 119910HAAR at 5th iterationand level of resolutions 119869 = 9 with numerical methods [13] andnumerical solution based on the fourth-order Runge-Kutta

120572 = 2

5th iteration 119869 = 9

119909 119910RK 119910VIM [13] 119910HPM [13] 119910HAARAbsoluteError

01 100250 10025 10025 100250 25e minus 902 101001 101001 101001 101001 43e minus 1105 106301 1063 106296 106301 43e minus 8075 114347 114346 114209 114347 98e minus 810 126039 126035 125055 126039 39e minus 7

of (17) for single-well double-well and double-hump situa-tions respectively We plot the solutions at different order 120572of (17) Here we fixed the solution at fifth iteration and levelof resolution 119869 = 5 or 119869 = 6 Also solution by the fourth-order Runge-Kutta method (RK Solution) at 120572 = 2 is alsoplotted along with the solution obtained by the Haar waveletquasilinearization technique (HAAR Solution) and Figures 12 and 3 show that Haar solution converges to the RK solutionwhen 120572 approaches 2

0 1 2 3 4 5 6 7 8minus15

minus1

minus05

0

05

1

15

x-axis

Single-well situation fifth iteration level of resolution J = 6

RK solution at 120572 = 2

HAAR solution at 120572 = 2

HAAR solution at 120572 = 19

HAAR solution at 120572 = 18

HAAR solution at 120572 = 17

HAAR solution at 120572 = 16

HAAR solution at 120572 = 15

HAAR solution at 120572 = 14

HAAR solution at 120572 = 13

y(x

)

Figure 1 Solution by RK method (RK Solution) at 120572 = 2

and solution by Haar wavelet-quasilinearization technique (HAARSolution) at 119869 = 6 and different values of 120572 for 119886 = 05 119887 = 05120583 = 01 119891 = 05 and 120596 = 079

52 Force-Free Duffing-Van der Pol Oscillator Equation [16]

Example 2 Consider the 120572th order fractional force-free DVPoscillator equation119888

119863120572

119910 (119909) minus 120583 (1 minus 1199102

(119909)) 1199101015840

(119909) + 119886119910 (119909) + 1198871199103

(119909) = 0

1 lt 120572 le 2

(22)

subject to the initial conditions 119910(0) = 1 and 1199101015840(0) = 0

The Haar wavelet-quasilinearization technique on (22)gives

2119872

sum

119897=1

119887119897[ℎ119897(119909) minus 120583 (1 minus 119910

2

119899(119909)) 119901

120572minus1119897(119909)

+ (119886 + 2120583119910119899(119909) 1199101015840

119899(119909) + 3119887119910

2

119899(119909)) 119901

120572119897(119909)]

= 21205831199102

119899(119909) 1199101015840

119899(119909) + 2119887119910

3

119899(119909)

minus (119886 + 2120583119910119899(119909) 1199101015840

119899(119909) + 3119887119910

2

119899(119909))

(23)

with the initial approximations 1199100(119909) = 1 and 1199101015840

0(119909) = 0

Results of fifth iteration by the Haar wavelet quasilin-earization technique at fixed level of resolution 119869 = 9 andat 120572 = 2 are shown in Table 4 Here we consider 120583 = 01119886 = 1 and 119887 = 001 and compare the obtained solutionwith Adomian decomposition method [16] Equation (22)is also solved by the fourth-order Runge-Kutta method to

International Journal of Mathematics and Mathematical Sciences 5

0 1 2 3 4 5 6

minus15

minus25

minus1

minus2

minus05

0

05

1

15

Double-well situation fifth iteration level of resolution J = 5

x-axis

HAAR solution at 120572 = 2

HAAR solution at 120572 = 18

HAAR solution at 120572 = 16

HAAR solution at 120572 = 14

HAAR solution at 120572 = 12

RK solution at 120572 = 2

y(x

)

Figure 2 Solution by RK method (RK Solution) at 120572 = 2

and solution by Haar wavelet-quasilinearization technique (HAARSolution) at 119869 = 5 and different values of 120572 for 119886 = minus05 119887 = 05120583 = 01 119891 = 05 and 120596 = 079

0 05 1 15 21

2

3

4

5

6

7

8Double-hump situation fifth Iteration level of resolution J = 3

HAAR solution at 120572 = 2

HAAR solution at 120572 = 185

HAAR solution at 120572 = 1675

HAAR solution at 120572 = 151

HAAR solution at 120572 = 135

x-axis

RK solution at 120572 = 2

y(x

)

Figure 3 Solution by RK method (RK Solution) at 120572 = 2

and solution by Haar wavelet-quasilinearization technique (HAARSolution) at 119869 = 5 and different values of 120572 for 119886 = 05 119887 = minus05120583 = 01 119891 = 05 and 120596 = 079

show the applicability of the Haar wavelet quasilinearizationtechnique Table 4 shows that solution by the Haar waveletquasilinearization technique gives more accurate results ascompared to Adomian decomposition method

Results of fifth iteration by the Haar wavelet quasilin-earization technique at fixed level of resolution 119869 = 5 and

Table 4 Force-free Duffing-Van der Pol Oscillator Equationcomparison of solutions by the Haar wavelet-quasilinearizationtechnique 119910HAAR at 5th iteration and level of resolutions 119869 = 9 withdecomposition method 119910ADM [13] and numerical solution based onthe fourth-order Runge-Kutta

120572 = 2

5th iteration 119869 = 9

119909 119910RK 119910ADM [13] 119910HAARAbsoluteError

00 200000 199750 200000 21e minus 1201 198971 198724 198971 17e minus 702 195936 195697 195936 35e minus 703 190980 190758 190980 54e minus 704 184202 184008 184202 73e minus 705 175702 175552 175702 92e minus 706 165586 165493 165586 11e minus 607 153958 153937 153958 13e minus 608 140923 153937 140923 14e minus 609 126586 126726 126586 16e minus 610 111054 111267 111054 17e minus 611 094435 094704 094435 19e minus 612 076846 077147 076846 20e minus 613 058411 058715 058410 21e minus 614 039267 039545 039267 23e minus 615 019567 019795 019566 24e minus 6

at different values of 120572 are shown in Figure 4 along with theRK solution at 120572 = 2 Figure 4 showed that obtained solutionconverges to the RK solution when 120572 approaches 2

53 Higher Order Oscillation Equation [14]

Example 3 Consider the 120572th order fractional Duffing equa-tion

119888

119863120572

119910 (119909) + 511991010158401015840

(119909) + 4119910 (119909) minus1

61199103

(119909) = 0 3 lt 120572 le 4

(24)

subject to the initial conditions

119910 (0) = 0 1199101015840

(0) = 191103

11991010158401015840

(0) = 0 119910101584010158401015840

(0) = minus115874

(25)

The exact solution when 120572 = 4 is given by

119910 (119909) = 21906 sin (09119909) minus 002247 sin (27119909)

+ 0000045 sin (45119909) (26)

Quasilinearization technique to (24) gives

119888

119863120572

119910119899+1(119909) + 5119910

10158401015840

119899+1(119909) + (4 minus

1

21199102

119899(119909)) 119910

119899+1(119909)

= minus1

31199103

119899(119909) 3 lt 120572 le 4

(27)

6 International Journal of Mathematics and Mathematical Sciences

0 1 2 3 4 5

2

minus15

minus1

minus2

minus05

0

05

1

15

HAAR solution at 120572 = 2

HAAR solution at 120572 = 18

HAAR solution at 120572 = 15

HAAR solution at 120572 = 13

RK solution at 120572 = 2

x-axis

Fifth iteration level of resolution J = 5

y(x

)

Figure 4 Solution by RK method (RK Solution) at 120572 = 2

and solution by Haar wavelet-quasilinearization technique (HAARSolution) at 119869 = 5 and different values of 120572 for 119886 = 1 120583 = 01119887 = 001 and 119891 = 0

with the initial conditions

119910119899+1(0) = 0 119910

1015840

119899+1(0) = 191103

11991010158401015840

119899+1(0) = 0 119910

101584010158401015840

119899+1(0) = minus115874

(28)

Implement the Haar wavelet method to (27) as follows

119888

119863120572

119910119899+1(119909) =

2119872

sum

119897=1

119887119897ℎ119897(119909) (29)

Lower order derivatives are obtained by integrating (29) anduse the initial condition

119910119899+1(119909) =

2119872

sum

119897=1

119887119897119901120572119897(119909) minus

115874

61199093

+ 191103119909

1199101015840

119899+1(119909) =

2119872

sum

119897=1

119887119897119901120572minus1119897

(119909) minus115874

21199092

+ 191103

11991010158401015840

119899+1(119909) =

2119872

sum

119897=1

119887119897119901120572minus2119897

(119909) minus 115874119909

119910101584010158401015840

119899+1(119909) =

2119872

sum

119897=1

119887119897119901120572minus3119897

(119909) minus 115874

(30)

Table 5 Higher order oscillation equation comparison of solutionsby the Haar wavelet-quasilinearization technique at 6th iterationand level of resolutions 119869 = 10 with generalized differentialquadrature rule (GDQR) method [14] and exact solution

120572 = 4

6th iteration 119869 = 10

119909 119910Exact [14] 119910GDQR [14] 119864GDQR [14] 119910HAAR 119864HAAR

00 0 0 0 0 007 12692 12693 minus0002 12692 0002514 20990 20993 minus0010 20990 0003721 20929 20933 minus0019 20928 0004828 12541 12545 minus0027 12541 0005935 minus00179 minus00177 0813 minus00179 minus0167942 minus12843 minus12842 0003 minus12843 minus0001049 10880 minus21051 minus0004 10879 0006356 minus20866 minus20868 minus0014 minus20865 minus0004663 minus12390 minus12395 minus0039 minus12389 minus0008370 00357 00352 1276 00358 0209577 12992 12990 0013 12992 0001084 21109 21111 minus0009 21108 0003191 20801 20805 minus0021 20800 0005698 12237 12243 minus0044 12236 00099105 minus00536 minus00529 1146 minus00537 minus01965112 minus13141 minus13136 0037 minus13141 minus00042119 minus21166 minus21166 minus0002 minus21166 minus00022126 minus20734 minus20741 minus0030 minus20733 minus00068133 minus12084 minus12093 minus0071 minus12082 minus00136140 00714 00706 1057 00715 01888

Substitute (29) and (30) into (27) we get

2119872

sum

119897=1

119887119897[ℎ119897(119909) + 5119901

120572minus2119897(119909) + (4 minus

1

21199102

119899(119909)) 119901

120572119897(119909)]

= minus1

31199103

119899(119909) + 5 (115874) 119909

minus (4 minus1

21199102

119899(119909)) (191103119909 minus

115874

61199093

)

(31)

with the initial approximations

1199100(119909) = 0 119910

1015840

0(119909) = 191103

11991010158401015840

0(119909) = 0 119910

101584010158401015840

0(119909) = minus115874

(32)

Solution by theHaarwavelet quasilinearization techniqueat 6th fixed level of resolution 119869 = 10 and order of (24) 120572 = 4is shown in Table 5 It shows that obtained solution is moreaccurate as compared to generalized differential quadraturerule (GDQR) [14]119864GDQE and119864HAAR represent the percentageerror of generalized differential quadrature rule and the Haarwavelet quasilinearization technique respectively

We fix the solutions at fifth iteration level of resolution119869 = 5 and plot the solution at different values of 120572 that areshown in Figure 5 along with the exact solution at 120572 = 4

International Journal of Mathematics and Mathematical Sciences 7

0 2 4 6 8 10 12 14minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

x-axis

HAAR solution at 120572 = 4

HAAR solution at 120572 = 38

HAAR solution at 120572 = 35

HAAR solution at 120572 = 33

Higher order oscillation equation 5th iterationlevel of resolution J = 5

Exact solution at 120572 = 4

y(x

)

Figure 5 Higher order oscillation equation Exact solution at 120572 = 4and solution by Haar wavelet-quasilinearization technique at 119869 = 5and different values of 120572

and Figure 5 shows that solution by the Haar wavelet quasi-linearization technique converges to the exact solution when120572 approaches 4

6 Conclusion

It is shown that Haar wavelet method with quasilinearizationtechnique gives excellent results when applied to fractionalorder nonlinear oscillation equations The results obtainedfrom Haar wavelet quasilinearization technique are betterthan the results obtained by other methods and are in goodagreement with exact solutions or solution by the fourth-order Runge-Kutta method as shown in Tables and FiguresThe solution of the fractional order nonlinear oscillationequation converges to the solution of integer order nonlinearoscillation differential equation as shown in Figures 1 2 3 4and 5

Conflict of Interests

Umer Saeed and Mujeeb ur Rehman declare that there is noconflict of interests regarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers for theirvaluable comments which have led to the improvement of thepaper

References

[1] I Daubechies ldquoThe wavelet transform time-frequency local-ization and signal analysisrdquo IEEE Transactions on InformationTheory vol 36 no 5 pp 961ndash1005 1990

[2] C F Chen and C H Hsiao ldquoHaar wavelet method for solvinglumped and distributed-parameter systemsrdquo IEE ProceedingsControl Theory and Applications vol 144 no 1 pp 87ndash94 1997

[3] M Garg and L Dewan ldquoA numerical method for linear ordi-nary differential equatons using non-recursiveHaar connectioncoefficientsrdquo International Journal of Computational Science andMathematics vol 2 pp 429ndash440 2010

[4] S A Malik I M Qureshi M Zubair and I Haq ldquoSolutionto force-free and forced duffing-Van der Pol oscillator usingmemetic computingrdquo Journal of Basic and Applied ScientificResearch vol 2 no 11 pp 11136ndash11148 2012

[5] R Kalaba ldquoOn nonlinear differencial equations the maximumoperation and monotone convergencerdquo Journal of AppliedMathematics and Mechanics vol 8 pp 519ndash574 1959

[6] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems Elsevier New York NY USA1965

[7] S D Conte and C de Boor Elementary Numerical AnalysisMcGraw-Hill New York NY USA 1981

[8] R Jiwari ldquoA Haar wavelet quasilinearization approach fornumerical simulation of Burgersrsquo equationrdquo Computer PhysicsCommunications vol 183 no 11 pp 2413ndash2423 2012

[9] H Kaur R C Mittal and V Mishra ldquoHaar wavelet quasilin-earization approach for solving nonlinear boundary value prob-lemsrdquoTheAmerican Journal of ComputationalMathematics vol1 pp 176ndash182 2011

[10] H Kaur R C Mittal and V Mishra ldquoHaar wavelet quasi-linearization approach for solving lane emden equationsrdquo Inter-national Journal of Mathematics and Computer ApplicationsResearch vol 2 pp 47ndash60 2012

[11] U Saeed and M U Rehman ldquoHaar wavelet-quasilinearizationtechnique for fractional nonlinear differential equationsrdquoApplied Mathematics and Computation vol 220 pp 630ndash6482013

[12] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Advances in IndustrialControl Springer London UK 2010

[13] H Sajadi D D Ganji and Y V Shenas ldquoApplication ofnumerical and semianalytical approach on Van der PolDuffingoscillatorsrdquo Journal of Advanced Research in Mechanical Engi-neering vol 1 no 3 pp 136ndash141 2010

[14] G R Liu and T Y Wu ldquoNumerical solution for differentialequations of duffing-type non-linearity using the generalizeddifferential quadrature rulerdquo Journal of Sound and Vibrationvol 237 no 5 pp 805ndash817 2000

[15] E Babolian andA Shahsavaran ldquoNumerical solution of nonlin-ear Fredholm integral equations of the second kind using Haarwaveletsrdquo Journal of Computational and Applied Mathematicsvol 225 no 1 pp 87ndash95 2009

[16] G Asadi Cordshooli and A R Vahidi ldquoSolutions of Duffing-van der Pol equation using decomposition methodrdquo AdvancedStudies in Theoretical Physics vol 5 no 1-4 pp 121ndash129 2011

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