on the operational matrix for fractional integration and ... · results as compared to chebyshev...

$: On the operational matrix for fractional integration and ... · results as compared to Chebyshev wavelets (all four kinds) and Legendre wavelets. Keywords and Phrases. Abel integral$
Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 15, Number 1 (2019), pp 81 - 101c© Research India Publications

http://www.ripublication.com

On the operational matrix for fractionalintegration and its application for solving Abel

integral equation using Bernoulli wavelets

Kailash Yadav1 and Jai Prakesh Jaiswal 2,∗

1 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal,M.P. India-462003.

Email: [email protected] Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal,

M.P. India-462003.Email: [email protected].

Abstract

The objective of the present article to derive the operational matrices for fractionalintegration based on different wavelets. By using this approach, we find theoperational matrix for Haar wavelets, which was also calculated earlier but theprocedure was complicated. In addition, operational matrices for Chebyshev wavelets,Legendre wavelets and Bernoulli wavelets are also derived and applied for solvingAbel integral equations. Numerical results show that Bernoulli wavelets extract betterresults as compared to Chebyshev wavelets (all four kinds) and Legendre wavelets.Keywords and Phrases. Abel integral equation, Haar wavelets, Chebyshev wavelets,Legendre wavelets, Block pulse function, Bernoulli wavelets, operational matrix.

1. INTRODUCTION

In this paper, we discuss about the solution of Abel integral equation of the following form

f(x)− λ∫ x

0

f(t)

(x− t)1−αdt = g(x), 0 < α < 1, (1)

where the function g(x) and the parameter λ are known, f(t) is unknown function, α > 0is a real number. Recently, fractional calculus has become the focus of interest formany researchers in different disciplines of science and technology, because of the factthat, the evaluation of the theory of fractional integrals and derivatives start with Euler,

∗Corresponding Author

server

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Text Box

82 Kailash Yadav and Jai Prakesh Jaiswal

Liouville and Abel (1823). But physicists and mathematicians paid more attention towardsfractional calculus during last ten years. In various field such as biology, mechanics,physics and chemistry actual problems is transformed into partial differential equationsor integral equations. Several authors have established applications of fractional calculusto viscoelastic model [1], continuum and statistical mechanics [2], direct current and whitenoise [3], econometrics [4], anomalous diffusion modeling [5], the dynamics of interfacesbetween soft-nanoparticles and rough substrates [6], the viscoelasticity [7], rheologicalmodels [8] and others. There are also several methods for solving fractional integralequations like He’s homotopy [9], Adomian decomposition [10], Haar wavelet method[11], Block-pulse functions [12], Taylor expansion [13] and power spectral density [14]etc.

In the current year, several numerical method for approximating the solution of Abelintegral equation (1) are known. Among these methods, the methods based on the waveletsare more attractive and considerable. The wavelets technique allows the creation of veryfast algorithms when compared with known algorithms. Various wavelets techniques areapplied in order to solve problem (1) namely, Legendre wavelets [15], B-polynomialmultiwavelets [16], Haar wavelets [[17]-[19]], quintic B-spline collocation [20], first kindChebyshev wavelets [21], second kind Chebyshev wavelets [22] etc.

Rest of the discussion is summarized as follows: In section 2, we mention the shortintroduction and some properties of Haar wavelets, four kinds of Chebyshev wavelets,Legendre wavelets and lastly Bernoulli wavelets. In Section 3, we include some necessarydefinitions and derived operational matrices of the fractional integration for above namedwavelets. In section 4, general formulation for solving Abel integral equations based onthe operational matrix is discussed. Before the final section, three numerical exampleshave been presented to compare the numerical results obtained by Chebyshev wavelets,Legendre wavelets and Bernoulli wavelets. At the end, we give the concluding remarks.

2. WAVELETS

Wavelets are a family of functions constructed from dilation and translation of a singlefunction called the mother wavelets. On changing dilation parameter a and the translationparameter b continuously we get the family of continuous wavelets defined as [23] :

ψa,b(x) =1

| a |1/2ψ

(x− ba

), a, b ∈ R, a 6= 0, (2)

If we restrict the parameters a and b to be discrete values as a = a−k0 , b = nb0a−a0 , a0 > 0,

b0 > 0, then we get the family of discrete wavelets as follows:

ψn,k(x) =| a0 |k/2 ψ(ak0x− nb0), (3)

where ψk,n(x) forms a wavelets basis for L2(R). Now are going to give brief review ofHaar, Chebyshev, Legendre and Bernoulli wavelets denoted by HWs, CWs LWs and BWs,respectively which will be used in the later discussions.

On the operational matrix for fractional integration and its application for solving... 83

2.1. Haar Wavelets (HWs)

Group of square waves is known as orthogonal set for HWs denoted byHq(t) and is definedas [19]

H0(t) =

{1, 0 ≤ t < 1,

0, otherwise,

H1(t) =

1, 0 ≤ t < 1

2,

−1, 12≤ t < 1,

0, otherwise,

Hq(t) = H1(2st− l),

q = 2s + l, s, l ∈ N ∪ {0}, 0 ≤ l < 2s, (4)

such that ∫ 1

0

Hq(t)Hp(t)dt = 2−sδqp,

where δqp is the Kronecker delta. For more explanation, refer [[25]-[27]]. The Heavisidestep function is defined as

v(t) =

{1, t ≥ 0,

0, t < 0.

An important property of this function is

v(t− µ)v(t− ν) = v(t−max{µ, ν}), µ, ν ∈ R. (5)

Rewrite the equation (4) with the help heaviside step function as

H0(t) = v(t)− v(t− 1),

Hq(t) = v

(t− l

2s

)− 2v

(t−

l + 12

2s

)+ v

(t− l + 1

2s

), (6)

q = 2s + l, s, l ∈ N ∪ {0} , 0 ≤ l < 2s.

It was established that each square integrable function f(t) in the interval [0, 1) can beexpanded into a Haar series of infinite terms [19]:

f(t) = C0H0(t) +∞∑s=0

2s−1∑l=0

C2s+lH2s+l(t), t ∈ [0, 1], (7)

where the Haar coefficients are determined as

Cr = 2s∫ 1

0

f(t)Hr(t)dt, (8)

r = 0, 2s + l, s, l ∈ N ∪ {0} , 0 ≤ l < 2s,


such that the following integral square error εp is minimized:

εp =

∫ 1

0

[f(t)−

p−1∑r=0

CrHr(t)

]2

dt,

p = 2S+1, S ∈ N ∪ {0} .

By using (6), the above Haar coefficients can be rewritten as

Cr = 2s

∫ l+12

2s

l2s

f(t)dt−∫ l+1

2s

l+12

2s

f(t)dt

,r = 2s + l, s, l ∈ N ∪ {0} , 0 ≤ l < 2s. (9)

If f(t) is piecewise constant or can be approximated by a piecewise constant functionduring each subinterval, the series sum in (7) can be truncated after p terms (p = 2S+1, S ≥0 being the resolution level of the wavelets), that is

f(t) ∼= C0H0(t) +S∑s=0

2s−1∑l=0

C2s+lH2s+l(t)

= CTH(t) = fp(t), t ∈ [0, 1], (10)

where C = Cp×1 = [C0, C1, . . . , Cp−1]T , H(t) = Hp×1 = [H0(t), H1(t), ..., Hp−1(t)]T

and p = 2S+1. After considering the collocation points as follows

ts =s− 0.5

2P, s = 1, 2, ..., 2P ,

H(t) can be reduced into the following matrix (called Haar) form

Hp×p =

H0(t1) . . . H2P−1(t1). . . . . . . . .

H0(t2P ) . . . H2P−1(t2P )

.For example, when S = 2 the Haar matrix is expressed as

H8×8 =

1 1 1 1 1 1 1 11 1 1 1 −1 −1 −1 −11 1 −1 −1 0 0 0 00 0 0 0 1 1 −1 −11 −1 0 0 0 0 0 00 0 1 −1 0 0 0 00 0 0 0 1 −1 0 00 0 0 0 0 0 1 −1

.


2.2. Chebyshev Wavelets (CWs)

Here we describe the four kinds of Chebyshev wavelets which are named accordingly theChebyshev polynomials.a. First Kind Chebyshev Wavelets (FSTCWs): The FSTCWs ψp,q = ψ(l, p, q, x) has thefour arguments, q = 1, 2, . . . , 2l−1, l positive integer and defined as [[21], [24]]

ψqp(x) =

{2l/2Xp(2

lx− 2q + 1), q−12l−1 ≤ x < q

2l−1 ,

0, elsewhere,(11)

where

Xp =

1√π, p = 0,√2πXp, p > 0,

(12)

and p = 0, 1, . . . , P − 1 , p is the degree of chebyshev polynomials of the first kind and tdenotes the time. In equation (12), Xp(x) are Chebyshev polynomials of the first kind withdegree p. Based on the definition, Chebyshev polynomials are orthogonal with respectto the weight function w(x) = (1− x2)−1/2 within the interval [−1, 1]. The recursiveequations are:

X0(x) = 1, X1(x) = x, Xp+1(x) = 2xXp(x)−Xp−1, p = 1, 2, 3, .... (13)

It needs to be noticed that in Chebyshev wavelets, in order to obtain the orthogonalwavelets, the weight functions have to be dilated and translated as:

wq(x) = w(2lx− 2q + 1) = (1− (2lx− 2q + 1)2)−1/2. (14)

b. Second Kind Chebyshev Wavelets (SNDCWs) : The SNDCWs has the following form[[22], [28]-[31]]

ψpq(x) =

{2l/2Yp(2

lx− 2q + 1), q−12l−1 ≤ x < q

2l−1 ,

0, elsewhere,(15)

where

Yp =

√2

πYp, p ≥ 0 (16)

In the equation (16), the coefficients are used for the orthonormality, Yp(x) is the secondkind Chebyshev polynomials of degree p which respect to weight function w(x) =√

1− x2 in the interval [−1, 1] and satisfy the following recursive formula:

Y0(x) = 1, Y1(x) = 2x, Yp+1(x) = 2xYp(x)− Yp−1, p = 1, 2, 3, ... (17)

c. Third Kind Chebyshev Wavelets (TRDCWs): The TRDCWs expression is given by [[24],[32]]

ψqp(t) =

{2l/2Zp(2

lx− 2q + 1), q−12l−1 ≤ x < q

2l−1 ,

0, elsewhere,(18)


where

Zp =

√1

πZp, p ≥ 0, (19)

The coefficient in the expression (19) is for the orthonormality and third kind Chebyshevpolynomials of degree p i.e. Zp(x) are orthogonal with respect to the weight functionw(x) =

√1+x√1−x in the interval [−1, 1] and satisfy the following recursive formula

Z0(x) = 1, Z1(x) = 2x− 1, Zp+1(x) = 2xZp(x)− Zp−1, p = 1, 2, 3, .... (20)

d. Fourth Kind Chebyshev Wavelets (FTHCWs) : The FTHCWs is given by [24]

ψqp(x) =

{2l/2χp(2

lx− 2q + 1), q−12l−1 ≤ x < q

2l−1 ,

0, elsewhere,(21)

where

χp =

√1

πχp, p ≥ 0 (22)

The coefficient in the relation (22) is used for the orthonormality,Wp(x) are the fourth kindChebyshev polynomials of degree p, which are orthogonal with respect to weight functionw(x) =

√1−x√1+x

on the interval [−1, 1] and satisfy the following recursive formula:

χ0(x) = 1, χ1(x) = 2x+ 1, χp+1(x) = 2xχp(x)− χp−1, p = 1, 2, ... (23)

2.3. Legendre Wavelets (LWs)

The LWs ψqp(x) = ψ(l, q, p, x) has four arguments q = 1, 2, ..., 2k−1, where k is positiveinteger, x denotes time and p is the degree of the Legendre polynomials. They are definedon the interval [0, 1) as [[33], [34]],

ψqp(x) =

{√p+ 1

22

k2 ζp(2

lx− 2q + 1), q−12l−1 ≤ x < q

2l−1 ,

0, elsewhere,(24)

p = 0, 1, ..., P − 1 and ζp(x) are the Legendre polynomials of degree p with respect tothe weight function w(x) = 1 on the interval the interval [−1, 1] and satisfy the followingrecurrence formulae:

ζ0(x) = 1, ζ1(x) = x

ζp+1(x) =

(2p+ 1

p+ 1

)xζp(x)−

(p

p+ 1

)ζp−1(x), p = 1, 2, ....

A function f(x) defined over [0, 1], may be expressed in terms of the CWs and LWs as

f(x) =∞∑q=1

∞∑p=0

dqpψqp(x), (25)


where

dqp = (f(x), ψqp(x))wq =

∫ 1

0

wq(x)f(x)ψqp(x)dx, (26)

in which (., .) denotes the inner product. If the infinite series in the relation (25) istruncated, then we have

f(x) ≈2l−1∑q=1

P−1∑p=0

dqpψqp(x) = DTψ(x), (27)

where D and ψ(t) are 2l−1P × 1 matrices given by:

D = [d10, d11, . . . , d1(P−1), d20, . . . , d2(P−1), . . . , d2l−10, . . . , d2l−1(P−1)]T ,

ψ(x) = [ψ(x)10, ..., ψ(x)1(P−1), ..., ψ(x)2(P−1), ..., ψ(x)2l−10, ..., ψ(x)2l−1(P−1)]T .

After substituting the collocation points as following

xj =2j − 1

2lP, j = 1, 2, ..., 2l−1P , (28)

the CWs and LWs matrix φip′×p′ , i = 1, 2, 3, 4, 5, assumes the form

φip′×p′ =

[ψ

(1

2p′

), ψ

(3

2p′

), ..., ψ

(2p′ − 1

2p′

)], i = 1, 2, 3, 4, 5. (29)

Where p′ = 2l−1P . For example, when P = 2 and l = 2, theFSTCWs matrix (φ1) can be expressed as

φ14×4 =

1.1284 1.1284 0 0−0.7979 0.7979 0 0

0 0 1.1284 1.12840 0 −0.7979 0.7979

. (30)

SNDCWs matrix (φ2) may be written as

φ24×4 =

1.5958 1.5958 0 0−1.5958 1.5958 0 0

0 0 1.59578 1.59580 0 −1.5958 1.5958

(31)

and (for p′ = 6, which was also calculated in [[22], [28]-[31]] )

φ26×6 =

1.5958 1.5958 1.5958 0 0 0−2.1277 0 2.1277 0 0 01.2412 −1.5958 1.2412 0 0 0

0 0 0 1.59578 1.5958 1.59580 0 0 −2.1277 0 2.12770 0 0 1.2412 −1.5958 1.2412

. (32)


TRDCWs matrix (φ3) becomes

φ34×4 =

1.1284 1.1284 0 0−2.2568 0 0 0

0 0 1.1284 1.12840 0 −2.2568 0

(33)

FTHCWs matrix is given by

φ44×4 =

1.1284 1.1284 0 0

0 2.2568 0 00 0 1.1284 1.12840 0 0 2.2568

, (34)

and LWs matrix is given by

φ54×4 =

1.4142 1.4142 0 0−1.2247 1.2247 0 0

0 0 1.4142 1.41420 0 −1.2247 1.2247

. (35)

2.4. Bernoulli Wavelets (BWs)

The BWs ψqp(x) = ψ(l, q, p, x) has four arguments and is given by [[35]-[37]]

ψqp(x) =

{2

l−12 Bp(2

l−1x− q + 1), q−12l−1 ≤ x < q

2l−1 ,

0, otherwise,(36)

where

Bp(x) =

1, p = 0,1√

(−1)p−1(p!)2

(2p)!β2p

Bp(x), p > 0. (37)

Here p = 0, 1, 2, ...P − 1, q = 1, 2, . . . , 2l−1; l is positive integer, x denotes the time andp is the degree of Bernoulli polynomials. For normality the coefficient is 1√

(−1)p−1(p!)2

(2p)!β2p

,

the dilation parameter is a = 2−(l−1), the translation parameter b = (q − 1)2−(l−1) and

Bm(x) =∑p

j=0

(pj

)βp−jx

j are the popular Bernoulli polynomial of order p, where

βj, j = 0, 1, ..., p are Bernoulli numbers. These numbers are a sequence of signed rationalnumbers which arise in the series expansion of trigonometric functions and can be definedby the identity [38],

x

ex − 1=∞∑j=0

βjxj

j!.

The first few Bernoulli numbers are

β0 = 1, β1 = −1

2, β2 =

1

6, β4 = − 1

30, β6 =

1

42, β8 = − 1

30, β10 =

5

66, ..,


with β2j+1 = 0, j = 1, 2, 3, ... and few Bernoulli polynomials are

B0 = 1, B1 = x− 1

2, B2 = x2 − x+

1

6, B3 = x3 − 3

2x2 +

1

2x,

B4 = x4 − 2x3 + x2 − 1

30, B5 = x5 − 5

2x4 +

5

3x3 − 1

6x, ...

We can exand any function f(x) ∈ L2[0, 1] into truncated BWs series as [39]

f(x) ≈2l−1∑q=1

P−1∑p=0

dqpψqp(x) = DTψ(x), (38)

where D and ψ(t) are 2l−1P × 1 matrices, given by:

D = [d10, d11, . . . , d1(P−1), d20, . . . , d2(P−1), . . . , d2l−10, . . . , d2l−1(P−1)]T ,

ψ(x) = [ψ(x)10, ..., ψ(x)1(P−1), ..., ψ(x)2(P−1), ..., ψ(x)2l−10, ..., ψ(x)2l−1(P−1)]T .

To get D, we have

fij =

∫ 1

0

f(x)ψij(x)dx,

using (38) we get

fij =2l−1∑q=1

P−1∑p=0

dqp

∫ 1

0

ψqp(x)ψij(x)dx

=2l−1∑q=1

P−1∑p=0

dqpuijqp, j = 0, 1, ..., P − 1, i = 1, 2, ..., 2l−1,

where

uijqp =

∫ 1

0

ψqp(x)ψij(x)dx.

Therefore,

fij = DT[uij10, u

ij11, ..., u

ij1P−1, u

ij20, u

ij21, ..., u

ij2P−1, ..., u

ij2l−10

, ..., uij2l−1P−1

]T,

p = 0, 1, ..., P − 1, q = 1, 2, ..., 2l−1,

or

F T = DTU,


with

U =[uijqp],

and

F = [f10, f11, ..., f1P−1, f20, ..., f2P−1, ..., f2l−10, ..., f2l−1P−1]T ,

where U is a matrix of order 2l−1P − 1× 2l−1P − 1 and is given by

U =

∫ 1

0

ψ(x)ψT (x)dx.

Hence, DT is (38) is given by

DT = F TU−1.

After substituting the collocation points as following

xj =2j − 1

2lP, j = 1, 2, ..., 2l−1P , (39)

For P = 2 and l = 2, the BWs (φ5) matrix is written as

φ64×4 =

1.4142 1.4142 0 0−1.2247 1.2247 0 0

0 0 1.4142 1.41420 0 −1.2247 1.2247

. (40)

and (when p′ = 6, that was also derived in [[39], [38]])1.4142 1.4142 1.4142 0 0 0−1.6330 0 1.6330 0 00.5270 −1.5811 0.5270 0 0 0

0 0 0 1.4142 1.4142 1.41420 0 0 −1.6330 0 1.63300 0 0 0.5270 −1.5811 0.5270

. (41)

3. OPERATIONAL MATRIX FOR THE FRACTIONAL INTEGRATION

First of all we mention some basic definitions related to fractional calculus [[22], [40],[41]], which are required for establishing our results.

Definition 1. A real function f(x), x > 0, is said to be in the space Cu, u ∈ R if thereexists a real number l (l > u), such that f(x) = xlf1(x) where f1(x) ∈ C[0,∞), and it issaid to be in the space Cp

u if f (p) ∈ Cu, p ∈ N .


Definition 2. The Riemann-Liouville fractional integral operator Iα of order α, of afunction f ∈ Cu, u ≥ −1, is defined as:

(Iαf)(x) =

{1

Γ(α)

∫ t0(x− η)α−1f(η) dη, α > 0, x > 0,

f(x), α = 0.(42)

Now describe the definition of block pulse function (BPFs): an p-set of BPFs on [0, 1) isdefined as

bj(x) =

{1, j

p≤ x < j+1

p,

0, elsewhere,(43)

where j = 0, 1, 2, . . . , p− 1. The BPFs have disjointness and orthogonality as following:

bj(x)bk(x) =

{0, j 6= k,

bj(x), j = k,. (44)

and ∫ 1

0

bj(η)bk(η) =

{1p, j = k,

0, j 6= k.(45)

Every square integrable function f(x) in the interval [0, 1) has expansion in terms of BPFsseries as

f(x) ≈p−1∑j=0

fjbj(x) = F TBp(x), (46)

where F = [f0, f1, ..., fp−1]T , Bp(x) = [b0(x), b1(x), . . . , bp−1(x)]T . By using theorthogonality of BPFs, for j = 0, 1, . . . , p − 1 the coefficients fj can be obtained by theformula

fj = p

∫ 1

1

bj(x)f(x)dx.

By using the disjointness of BPFs and the evaluation of Bp(x), we have

Bp(x)BTp (x) =

b0(x) 0

b1(x). . .

0 bp−1(x)

. (47)

The equation (46) implies that HWs, CWs, LWs and BWs can also be expanded into anp′-term BPFs as

ψp′(x) = φp′×p′Bp′(x). (48)

The block pulse operational matrix of the fractional integration Fα is given by [42],

1

Γ(α)

∫ x

0

(x− η)α−1Bp′(η)dη ≈ FαBp′(x), (49)


where

Fα =1

p′α1

Γ(α + 2)

1 ξ1 ξ2 ξ3 . . . ξp′−1

0 1 ξ1 ξ2 . . . ξp′−2

0 0 1 ξ1 . . . ξp′−3

. . . . . .

. . . . . .

. . . . . .0 0 . . . 0 1 ξ1

0 0 0 . . . 0 1

(50)

andξl = (l + 1)α+1 − 2lα+1 + (l − 1)α+1.

We observed that for α = 1, Fα is BPFs operational matrix of integration. Let

(Iαψp′(x)) ≈ Pαp′×p′ψp′(x), (51)

where matrix Pαp′×p′ is called the HWs, CWs, LWs and BWs operational matrix of fractional

integration integration [[19], [22]]. Using (48) and (49), we get

(Iαψp′)(x) ≈ (Iαφp′×p′Bp′)(x) = φp′×p′(IαBp′)(x) ≈ φp′×p′F

αBp′(x). (52)

From the relations (51) and (52), we get

Pαp′×p′ψp′(x) = Pα

p′×p′φp′×p′Bp′(x) = φp′×p′FαBp′(x). (53)

Hence, the matrix Pαp′×p′ is given by

Pαp′×p′ = φp′×p′F

αφ−1p′×p′ . (54)

For example for HWs, when α = 13, S = 0 and 1 the operational matrix of fractional

integration has the following form (which is also derived in [19] but in a very complicatedway)

P1/32 =

[0.8399 −0.17330.1733 0.4933

],

P1/34 =

0.8399 −0.1733 −0.1375 −0.05710.1733 0.4933 −0.1375 0.21790.0286 0.1090 0.3916 −0.04280.0688 −0.0688 0 0.3916

. (55)

For α = 12, p′ = 4 , the operational matrix of fractional integration for CWs have the

following form:a. for FSTCWs

P1/24×4 =

0.5319 0.2203 0.4407 −0.1043−0.1102 0.2203 0.0522 −0.0350

0 0 0.5319 0.22030 0 −0.1102 0.2203

.


b. for SNDCWs

P1/24×4 =

0.5319 0.1558 0.4407 −0.0738−0.1558 0.2203 0.0738 −0.0350

0 0 0.5319 0.15580 0 −0.1558 0.2203

.and (for p′ = 6 is also calculated in [[28], [29], [31]])

P1/26×6 =

0.5319 0.1485 −0.0222 0.4343 −0.0711 0.0191−0.2385 0.2243 0.1213 0.0888 −0.0449 0.01810.1050 −0.0314 0.1577 0.1597 −0.0358 0.0124

0 0 0 0.5319 0.1485 −0.02220 0 0 −0.2385 0.2243 0.12130 0 0 0.1050 −0.0314 0.1577

.c. for TRDCWs

P1/24×4 =

0.6877 0.1558 0.3669 −0.0738−0.6232 0.0645 −0.3281 0.0388

0 0 0.6877 0.15580 0 −0.6232 0.0645

.d. for FTHCWs

P1/24×4 =

0.3761 0.1558 0.5144 −0.0738

0 0.3761 0.6232 −0.10870 0 0.3761 0.15580 0 0 0.3761

.for LWs

P1/24×4 =

0.5319 0.1799 0.4407 −0.0852−0.1349 0.2213 0.0642 −0.0349

0 0 0.5341 0.18120 0 −0.1349 0.2203

.and finally for BWs

P1/24×4 =

0.5319 0.1799 0.4407 −0.0852−0.1349 0.2203 0.0639 −0.0350

0 0 0.5319 0.17990 0 −0.1349 0.2203

.and (when p′ = 6, it was also calculated in [32])

P1/26×6 =

0.528223 0.181881 −0.029782 0.443844 −0.087099 0.025638−0.145160 0.224295 0.132924 0.079882 −0.044905 0.019811−0.059817 −0.096441 0.168799 −0.041724 −0.000186 0.002868

0 0 0 0.528223 0.181881 −0.0297820 0 0 −0.145160 0.224295 0.1329240 0 0 −0.059817 −0.096441 0.168799

.


4. FORMULATION FOR THE SOLUTION OF ABEL INTEGRAL EQUATION

Consider the nonlinear Abel integral equation, given by

f(x)− λ∫ x

0

f(t)

(x− t)1−α = g(x), 0 ≤ x ≤ 1, (56)

where f(x) is unknown function, g(x) and the parameter λ are known. The functions f(x),g(x) may be approximated the HWs, CWs, LWs and BWs as following

f(x) = F Tψ(x), (57)

g(x) = GTψ(x). (58)

In view of (48), the relation (57) can be rewritten as

f(x) = F Tφp′×p′Bp′(x). (59)

For solving (56), we substitute (57), (58) and (59) into (56) and we get

F Tψ(x)− λ∫ x

0

(x− t)α−1F Tφp′×p′Bp′(t)dt = GTψ(x). (60)

where

λ

∫ x

0

(x− t)α−1F Tφp′×p′Bp′(t)dt = λF Tφp′×p′

∫ x

0

(x− t)α−1Bp′(t)dt, (61)

which becomes in view of (49) that

λF Tφp′×p′

∫ x

0

(x− t)α−1Bp′(t)dt = λΓ(α)F Tφp′×p′FαBp′(x). (62)

Thus the equation (60) becomes

F Tψ(x)− λΓ(α)F Tφp′×p′FαBp′(x)dt = GTψ(x). (63)

By making use of the expression (48) in the relation (63), we attain

F Tφp′×p′Bp′(x)− λΓ(α)F Tφp′×p′FαBp′(x) = GTφp′×p′Bp′(x). (64)

Whic is equivalent to the following system of linear equations

F Tφp′×p′ − λΓ(α)F Tφp′×p′Fα = GTφp′×p′ . (65)

φ−1p′×p′ to post multiply two side of the above equation, we can get the following system of

linear equationsF T − λΓ(α)F Tφp′×p′F

αφ−1p′×p′ = GT . (66)

orF T − λΓ(α)F TPα = GT . (67)


In [36], Sahu and Ray have shown that the error bound is uniformly convergent to the levelof resolution of BWs. This ensures the convergence of BWs approximation when p′ isincreased.

Lemma. Let f ∗(x) =∑2l−1

q=1

∑P−1p=0 dpqψpq(x) be truncated series, then the truncation error

Epq(x) can be defined as

‖ Epq ‖22≤

∞∑q=2l

∞∑p=P

(A

2l−12

16p!

(2π)p+1M

)2

Proof. Any function f(x) ∈ L2[0, 1] can be expressed by BWs as

f(x) =∞∑q=1

∞∑p=0

dpqψpq(x).

If f ∗(x) be truncated series, then the truncated error term can be calculated as

Epq = f(x)− f ∗(x) =∞∑q=2l

∞∑p=P

dpqψpq(x)

Now,

‖ Epq ‖22=

∞∑q=2l

∞∑p=P

‖ dpqψpq(x) ‖22

=

∫ 1

0

|∞∑q=2l

∞∑p=P

dpqψpq(x) |2 dx

≤∞∑q=2l

∞∑p=P

| dpq |2∫ 1

0

| ψpq(x) |2 dx

≤∞∑q=2l

∞∑p=P

| dpq |2∫ n−1

2k−1

n

2k−1

| 2k−12 Bp(2

k−1x− q + 1) |2 dx

≤∞∑q=2l

∞∑p=P

| dpq |2∫ 1

0

| Bp(t) |2 dt

≤∞∑q=2l

∞∑p=P

(A

2l−12

16p!

(2π)p+1M

)2

(by orthonormality of Bp)


5. NUMERICAL EXAMPLES

In this section, we applied the method presented in this paper for solving Abel integralequation (1)

Example 5.1: Consider the following Abel integral equation∫ x

0

f(t)√x− t

dt = x2, 0 ≤ x < 1, (68)

the exact solution is f(x) = 83πx

32 .

Example 5.2: Consider the following equation [19].∫ x

0

f(t)√x− t

dt = x, 0 ≤ x < 1, (69)

the exact solution f(x) = 2π

√x.

Example 5.3: Consider the following Abel integral equation∫ x

0

f(t)√x− t

dt = 1, 0 ≤ x ≤ 1. (70)

It has f(x) = 1π√x

as the exact solution.

Table 1, Table 2 and Table 3, shows the approximate norm-2 of the absolute error fork = 2, 3, 4 and M = 3, respectively for Examples 5.1-5.3. The comparison of exact andnumerical solutions when k = 4, M = 3 i.e. p′ = 24 are shown in Fig. 1-3, respectively.From the results, we can clearly say that numerical solutions become more accurate whenthe value of k is increasing.


Tabl

e1:

Com

pari

son

ofap

prox

imat

eno

rm-2

ofth

eab

solu

teer

ror(‖e p′ (x

)‖ 2

)for

Exa

mpl

e5.

1fo

rdiff

eren

tval

ues

ofp′

P’FS

TC

Ws

SND

CW

sT

RD

CW

sFT

HC

Ws

LWs

BW

s

68.

8022

1379

4727

4E-0

28.

8022

1379

4717

7E-0

28.

8022

1379

4699

5E-0

28.

8022

1379

4725

4E-0

28.

8022

1379

4729

8E-0

28.

8022

1379

4719

8E-0

212

3.28

8040

0138

691E

-02

3.28

8040

0138

692E

-02

3.58

2436

6705

575E

-02

7.48

4912

9569

128E

-02

3.87

3187

8233

650E

-02

7.62

9246

7621

974E

-05

246.

4100

5730

4426

3E-0

35.

6246

9494

1613

3E-0

35.

6246

9494

1613

4E-0

33.

3824

7012

4187

3E-0

36.

4100

5730

4426

8E-0

39.

3066

4443

6635

2E-0

6

Tabl

e2:

Com

pari

son

ofap

prox

imat

eno

rm-2

ofth

eab

solu

teer

ror(‖e p′ (x

)‖ 2

)for

Exa

mpl

e5.

2fo

rdiff

eren

tval

ues

ofp′

P’FS

TC

Ws

SND

CW

sT

RD

CW

sFT

HC

Ws

LWs

BW

s

63.

4010

6752

4601

9E-0

23.

4010

6752

4602

3E-0

23.

4010

6752

4602

8E-0

23.

4010

6752

4602

2E-0

23.

4010

6752

4482

0E-0

23.

4010

6752

4602

1E-0

212

3.28

8040

0138

690E

-03

2.40

4986

8369

642E

-02

2.40

4986

8369

643E

-02

2.43

7070

3985

536E

-02

2.54

8971

8752

980E

-02

7.96

3968

7906

751E

-05

241.

7005

9652

8503

5E-0

36.

7495

5148

2867

4E-0

35.

3402

9124

1030

4E-0

37.

0714

7827

1866

5E-0

31.

7005

9652

8503

5E-0

21.

1888

1078

8765

8E-0

5

Tabl

e3:

Com

pari

son

ofap

prox

imat

eno

rm-2

ofth

eab

solu

teer

ror(‖e p′ (x

)‖ 2

)for

Exa

mpl

e5.

3fo

rdiff

eren

tval

ues

ofp′

P’FS

TC

Ws

SND

CW

sT

RD

CW

sFT

HC

Ws

LWs

BW

s

68.

0783

9290

5270

6E-0

18.

0783

9290

5270

5E-0

18.

0783

9290

5270

4E-0

18.

0783

9290

5270

5E-0

18.

0783

9290

5270

5E-0

18.

0783

9290

5270

5E-0

112

5.04

5071

4730

665E

-02

5.04

5071

4730

666E

-02

5.04

5071

4730

664E

-02

1.28

8550

6036

106E

-02

5.04

5071

3941

042E

-02

2.86

0130

7731

945E

-04

241.

0050

6471

6286

2E-0

33.

5322

5214

9235

7E-0

33.

5911

6528

2146

0E-0

39.

1233

7360

1232

6E-0

39.

1233

7360

1235

1E-0

35.

4547

4193

7710

4E-0

5


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

x

fHxL

Numerical

Exact

(a) Example 5.1

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

x

fHxL

Numerical

Exact

(b) Example 5.2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

x

fHxL

Numerical

Exact

(c) Example 5.3

Figure 1: Physical behavior of exact and numerical solutions using Bernoulli waveletswith p′ = 24

6. CONCLUSION

In the present discussion, we have derived the operational matrix for fractional integrationbased on general wavelets and then calculated for Haar, which was also derived in earlierwell existing article but the approach was arduous. The discussed idea is also applied forgetting the operational matrix for Chebyshev, Legendre and Bernoulli wavelets. Theseoperational matrices have been used for solving Abel integral equations and it is observedthat Bernoulli wavelets shows better accuracy as compared to Chebyshev wavelets (all fourkinds) and Legendre wavelets.

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on the operational matrix for fractional integration and ... · results as compared to chebyshev...

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