A05330107

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IOSR Journal of Engineering (IOSRJEN) ISSN (e): 2250-3021, ISSN (p): 2278-8719 Vol. 05, Issue 03 (March. 2015), ||V3|| PP 01-07

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wavelet collocation method for solving integro-differential equation. Asmaa Abdalelah Abdalrehman University of Technology Applied Science Department Abstract: - Wavelet collocation method for numerical solution nth order Volterra integro diferential equations (VIDE) by expanding the unknown functions, as series in terms of chebyshev wavelets second kind with unknown coefficients. The aim of this paper is to state and prove the uniform convergence theorem and accuracy estimation for series above. Finally, some illustrative examples are given to demonstrate the validity and applicability of the proposed method. Keywords: chebyshev wavelets second kind; integro-differential equation; operational matrix of integrations; uniform convergence; accuracy estimation.

I.

INTRODUCTION

Basic wavelet theory is a natural topic. By name, wavelets date back only to the 1980s. on the boundary between mathematics and engineering, wavelet theory shows students that mathematics research is still thriving, with important applications in areas such as image compression and the numerical solution of differential equations [1], integral equation [2],and integro differential equations [3,8]. The author believes that the essentials of wavelet theory are sufficiently elementary to be taught successfully to advanced undergraduates [4]. Orthgonal functions and polynomials series have received considerable attention in dealing with various problems wavelets permit the accurate representation of a functions and operators. Special attention has been given to application of the Legendre wavelets [5], Harr wavelets [6] and Sine-Cosine wavelets [7]. The solution of integro-differential equations have a major role in the fields of science and ingineering when aphysical system is modeled under the differential sense, it finally gives a differential equation, an integral equation or an integro-differential equations mostly appear in the last equation [8]. In this paper the operational matrix of integration for Hermite wavelets is derived and used it for obtaining approximate solution of the following nth order VIDE. đ?‘Ľ u(n)(x)=g(x)+ 0 đ?‘˜ đ?‘Ľ, đ?‘Ą đ?‘˘(đ?‘ ) đ?‘Ą đ?‘‘đ?‘Ą (1) where đ?‘› ≼ đ?‘ k(x,t) and g(x) are known functions, and u(x) is an unknown function.

II.SOME PROPERTIES OF SECOND CHEBYSHEV WAVELETS Wavelets constitute a family of functions constructed from dilation and traslation of a single function ď ˇ t called the mother wavelet.When the dilation parameter a and the translation parameter b vary continuously we have the following family of continuous wavelets as [9].

ď ˇđ?‘Ž ,đ?‘? t = a

−1 2

t−b

ď ˇ a , a, b ∈ R, a ≠0. The second chebyshev wavelets ď ˇ2 đ?‘›,đ?‘š t = ď ˇ(đ?‘˜, đ?‘›, đ?‘š, đ?‘Ą) involve four arguments,

đ?‘› = 1, ‌ , 2đ?‘˜ −1 , k is assumed any positive integer, đ?‘š is the degree of the second chebyshev polynomials and t is the normalized time. They are defined on the interval 0,1)as đ?‘˜

ď ˇ

2

đ?‘› ,đ?‘š

where

2 t = 2 0

đ?‘ˆđ?‘š 2đ?‘˜ đ?‘Ą − 2đ?‘› + 1 ,

đ?‘ˆđ?‘š đ?‘Ą =

đ?‘›âˆ’1 2 đ?‘˜âˆ’1

≤�<

đ?‘› 2 đ?‘˜âˆ’1

(2)

đ?‘œđ?‘Ąđ?‘•đ?‘’đ?‘&#x;đ?‘¤đ?‘–đ?‘ đ?‘’ 2 đ?œ‹

đ?‘ˆđ?‘š đ?‘Ą

đ?‘š = 0,1, ‌ , đ?‘€ − 1

(3)

here đ?‘ˆđ?‘š đ?‘Ą are the second chebyshev polynomialsof degree m with respect to the weight function đ?‘¤ đ?‘Ą = 1 − đ?‘Ą 2 on the interval [-1,1] and satisfy the following recursive formula đ?‘ˆ0 đ?‘Ą = 1, đ?‘ˆ1 đ?‘Ą = 2đ?‘Ą, đ?‘ˆđ?‘š +1 đ?‘Ą = 2đ?‘Ąđ?‘ˆđ?‘š đ?‘Ą − đ?‘ˆđ?‘š −1 đ?‘Ą , đ?‘š = 1,2, ‌ The set of chebyshev wavelets are an orthonirmal set with respect to the weight function đ?‘¤đ?‘› đ?‘Ą = đ?‘¤đ?‘› (2đ?‘˜ đ?‘Ą − 2đ?‘› + 1) .

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wavelet collocation method for solving integro-differential equation. III.FUNCTION APPROXIMAT A function đ?‘“(đ?‘Ą) defined over [0, 1) may be expanded as đ?‘“ đ?‘Ą = ∞đ?‘›=1 ∞đ?‘š=0 đ??śđ?‘›đ?‘š ď ˇ2đ?‘›đ?‘š (đ?‘Ą) (4) đ?‘¤đ?‘•đ?‘’đ?‘&#x;đ?‘’ đ??śđ?‘›đ?‘š = (đ?‘“ đ?‘Ą , ď ˇ2đ?‘›đ?‘š đ?‘Ą ) In which . , . denoted the inner product in đ??ż2đ?‘¤ đ?‘› [0,1). If the infinite series equation (2.34) is truncated, then it can be written đ?‘˜âˆ’1 2 đ?‘‡ 2 đ?‘“ đ?‘Ą = 2đ?‘›=1 đ?‘€âˆ’1 (5) đ?‘š =0 đ??śđ?‘›đ?‘š ď ˇđ?‘›đ?‘š đ?‘Ą = đ??ś ď ˇ (đ?‘Ą) đ?‘‡ đ??ś = đ??ś10 , đ??ś11 , ‌ , đ??ś1(đ?‘€âˆ’1) , đ??ś20 , ‌ , đ??ś2(đ?‘€âˆ’1) , ‌ , đ??ś2đ?‘˜âˆ’1 , ‌ , đ??ś2đ?‘˜âˆ’1 đ?‘€âˆ’1 ď ˇ2 (t) = 2 2 2 ď ˇ10 (đ?‘Ą), ď ˇ11 (đ?‘Ą), ‌ , ď ˇ1đ?‘€âˆ’1 (đ?‘Ą), ď ˇ220 (đ?‘Ą), ‌ , ď ˇ22đ?‘˜âˆ’1 đ?‘€âˆ’1 (đ?‘Ą), ‌ ď ˇ22đ?‘˜âˆ’1 0 (đ?‘Ą), ‌ ď ˇ22đ?‘˜âˆ’1 đ?‘€âˆ’1 (đ?‘Ą)

�

Convergence Analysis for Chebyshev wavelets of Second kind. A function đ?‘“ đ?‘Ą defined over 0,1 may be expanded as: đ?‘“ đ?‘Ą = ∞đ?‘›=1 ∞đ?‘š=0 đ?‘“đ?‘›đ?‘š ď ˇ2đ?‘›đ?‘š đ?‘Ą

(6)

where

đ?‘“đ?‘›đ?‘š = đ?‘“ đ?‘Ą , ď ˇ2đ?‘›đ?‘š đ?‘Ą In eq.(5). . , . denotes the inner product with weight function đ?‘¤đ?‘› đ?‘Ą . If the infinite series in eq.(4) is trancated then eq.(4) can be written as: đ?‘˜âˆ’1 2 đ?‘“ đ?‘Ą −đ?‘“2đ?‘˜âˆ’1 đ?‘€âˆ’1 đ?‘Ą = 2đ?‘›=1 đ?‘€âˆ’1 đ?‘š=0 đ?‘“đ?‘›đ?‘š ď ˇđ?‘›đ?‘š đ?‘‡ 2 =đ??š ď ˇđ?‘›đ?‘š where F and ď ˇ2đ?‘›đ?‘š are 2đ?‘˜ đ?‘€ Ă— 1 matrices given by đ?‘‡ đ??š = đ?‘“10 , đ?‘“11 , ‌ , đ?‘“1đ?‘€ , đ?‘“20 , ‌ , đ?‘“2đ?‘€ −1 , ‌ , đ?‘“2đ?‘˜ 0 , ‌ , đ?‘“2đ?‘˜âˆ’1 đ?‘€âˆ’1

ď ˇ (t)= 2

ď ˇ210

đ?‘Ą

2 , ď ˇ11

đ?‘Ą

2 , ‌ , ď ˇ1đ?‘€âˆ’1

đ?‘Ą

, ď ˇ220

đ?‘Ą

, ‌ , ď ˇ22đ?‘€âˆ’1

đ?‘Ą

, ď ˇ22đ?‘˜âˆ’1

đ?‘Ą

, ‌ , ď ˇ22đ?‘˜âˆ’1 đ?‘€âˆ’1

(7)

(8) đ?‘Ą

�

Theorem(1): (Convergence Analysis theorm) Assume that a function f(t) ∈ đ??ż2đ?‘¤ ∗ 0,1 , đ?‘¤ ∗ = 1 − đ?‘Ą 2 with đ?‘“" đ?‘Ą ≤ đ??ż, can be expanded as infinite series of second kind chebyshev wavelets, then the series converges uniformly to f(t). Proof: since đ?‘“đ?‘›đ?‘š = đ?‘“ đ?‘Ą , ď ˇ2đ?‘›đ?‘š (đ?‘Ą) then 1

��� = �

=

đ?‘˜+1

2 đ?‘˜âˆ’1 2 2 đ?‘› −1 đ?œ‹ 2 đ?‘˜âˆ’1

đ?‘“(đ?‘Ą) ď ˇ2đ?‘›đ?‘š (đ?‘Ą)đ?‘¤đ?‘› đ?‘Ą đ?‘‘đ?‘Ą

0

đ?‘“(đ?‘Ą)đ?‘ˆđ?‘š 2đ?‘˜ đ?‘Ą − 2đ?‘› + 1 đ?‘¤ 2đ?‘˜ đ?‘Ą − 2đ?‘› + 1 đ?‘‘đ?‘Ą

(9)

If we make use of the substitution 2đ?‘˜ đ?‘Ą − 2đ?‘› + 1 = cos đ?œƒ in (5), yields 2 đ?œ‹ cos đ?œƒ + 2đ?‘› − 1 đ?‘“đ?‘›đ?‘š = đ?‘˜ đ?‘“ sin đ?œƒ sin đ?‘š + 1 đ?œƒ đ?‘‘đ?œƒ 2đ?‘˜ 22 đ?œ‹ 0 (10) By using the integration by parts, Then eq.(10) becomes đ?œ‹ 1 cos đ?œƒ + 2đ?‘› − 1 sin đ?‘šđ?œƒ sin đ?‘š + 2 đ?œƒ đ?‘“đ?‘›đ?‘š = đ?‘“ − đ?‘˜ 2đ?‘˜ đ?‘š đ?‘š+2 0 2 22 đ?œ‹ đ?œ‹ 1 cos đ?œƒ + 2đ?‘› − 1 sin đ?‘šđ?œƒ sin đ?‘š + 2 đ?œƒ + đ?‘“′ sin đ?œƒ − đ?‘‘đ?œƒ 3đ?‘˜ 2đ?‘˜ đ?‘š đ?‘š+2 222 đ?œ‹ 0 đ?œ‹ 1 cos đ?œƒ + 2đ?‘› − 1 đ?‘“đ?‘›đ?‘š = đ?‘“′ sin đ?œƒ sin đ?‘šđ?œƒđ?‘‘đ?œƒ 3đ?‘˜ 2đ?‘˜ đ?‘š 222 đ?œ‹ 0 đ?œ‹ 1 cos đ?œƒ + 2đ?‘› − 1 − đ?‘“′ sin đ?œƒ sin đ?‘š + 2 đ?œƒđ?‘‘đ?œƒ 3đ?‘˜ 2đ?‘˜ đ?‘š+2 222 đ?œ‹ 0 (11) After performing integration by parts again yields,

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wavelet collocation method for solving integro-differential equation. ��� =

đ?œ‹

1 3 đ?‘š22

5đ?‘˜ 22

đ?‘“" đ?œ‹ −

0

cos đ?œƒ + 2đ?‘› − 1 2đ?‘˜ đ?œ‹

1 3 22

cos đ?‘š − 2 đ?œƒ − cos đ?‘šđ?œƒ cos đ?‘šđ?œƒ − cos đ?‘š + 2 đ?œƒ − đ?‘‘đ?œƒ đ?‘šâˆ’1 đ?‘š+1

5đ?‘˜ 22

đ?‘“"

cos đ?œƒ + 2đ?‘› − 1 2đ?‘˜

đ?‘š+2 đ?œ‹ 0 cos đ?‘š + 2 đ?œƒ − cos đ?‘š + 4 đ?œƒ − đ?‘‘đ?œƒ đ?‘š+3

cos đ?‘š đ?œƒ − cos đ?‘š + 2 đ?œƒ đ?‘š+1

(12) Consider: đ?œ‹

đ?‘“" 0

cos đ?œƒ + 2đ?‘› − 1 2đ?‘˜

cos đ?‘š − 2 đ?œƒ − cos đ?‘šđ?œƒ cos đ?‘šđ?œƒ − cos đ?‘š + 2 đ?œƒ − đ?‘‘đ?œƒ đ?‘šâˆ’1 đ?‘š+1

2

2

đ?œ‹

cos đ?œƒ + 2đ?‘› − 1 cos đ?‘š − 2 đ?œƒ − cos đ?‘šđ?œƒ cos đ?‘šđ?œƒ − cos đ?‘š + 2 đ?œƒ − đ?‘‘đ?œƒ 2đ?‘˜ đ?‘šâˆ’1 đ?‘š+1 0 2 đ?œ‹ đ?œ‹ cos đ?œƒ + 2đ?‘› − 1 2 đ?‘š − 1 cos đ?‘š + 2 đ?œƒ − 2đ?‘š cos đ?‘š đ?œƒ + đ?‘š + 1 cos đ?‘š − 2 đ?œƒ ≤ đ?‘“" đ?‘‘đ?œƒ Ă— đ?‘‘đ?œƒ 2đ?‘˜ đ?‘šâˆ’1 đ?‘š+1 0 0 đ?œ‹ 2 2 2 2 2 2 đ?‘š − 1 cos đ?‘š + 2 đ?œƒ + 4đ?‘š cos đ?‘šđ?œƒ + đ?‘š + 1 cos đ?‘š − 1 < đ?œ‹đ??ż2 đ?‘‘đ?œƒ đ?‘šâˆ’1 2 đ?‘š+1 2 0 đ?œ‹đ??ż2 đ?œ‹ đ?œ‹ đ?œ‹ = đ?‘š − 1 2 + 4đ?‘š2 + đ?‘š + 1 2 2 2 đ?‘šâˆ’1 đ?‘š+1 2 2 2 đ?œ‹ 2 đ??ż2 2 = 3đ?‘š + 1 đ?‘šâˆ’1 2 đ?‘š+1 2 Thus, we get đ?œ‹ cos đ?œƒ + 2đ?‘› − 1 cos đ?‘š − 2 đ?œƒ − cos đ?‘šđ?œƒ cos đ?‘šđ?œƒ − cos đ?‘š + 2 đ?œƒ đ?‘“" − đ?‘‘đ?œƒ 2đ?‘˜ đ?‘šâˆ’1 đ?‘š+1 0 đ?œ‹đ??ż 3đ?‘š2 + 1 1 2 < (13) đ?‘šâˆ’1 đ?‘š+1 Similarly, 2 đ?œ‹ cos đ?œƒ + 2đ?‘› − 1 cos đ?‘š đ?œƒ − cos đ?‘š + 2 đ?œƒ cos đ?‘š + 2 đ?œƒ − cos đ?‘š + 4 đ?œƒ đ?‘“" − đ?‘‘đ?œƒ 2đ?‘˜ đ?‘š+1 đ?‘š+3 0 đ?œ‹ cos đ?œƒ + 2đ?‘› − 1 = đ?‘“" 2đ?‘˜ 0 2 đ?‘š + 1 cos đ?‘š + 4 đ?œƒ − (đ?‘š + 1) cos đ?‘š + 2 đ?œƒ + đ?‘š + 3 cos đ?‘š + 2 đ?œƒ + đ?‘š + 3 cos đ?‘š đ?œƒ Ă— (đ?‘š + 1) đ?‘š + 3 đ?œ‹ cos đ?œƒ + 2đ?‘› − 1 2 ≤ đ?‘“" đ?‘‘đ?œƒ 2đ?‘˜ 0 2 đ?œ‹ đ?‘š cos đ?‘š + 3 đ?œƒ − 2đ?‘š − 2 cos đ?‘š + 1 đ?œƒ + đ?‘š + 2đ?œƒ cos đ?‘š − 1 đ?œƒ Ă— đ?‘š đ?‘š+2 0 đ?œ‹ 2 2 2 2 đ?‘š + 1 cos đ?‘š + 4 đ?œƒ + 2đ?‘š + 4 cos đ?‘š + 2 đ?œƒ + đ?‘š + 3 2 cos2 đ?‘š đ?œƒ < đ?œ‹đ??ż2 đ?‘š+1 2 đ?‘š+3 2 0 2 2 đ?œ‹ đ??ż = 6đ?‘š2 + 24đ?‘š + 26 2 đ?‘š+1 2 đ?‘š+3 2 đ?œ‹ 2 đ??ż2 = 3đ?‘š2 + 12đ?‘š + 13 đ?‘š+1 2 đ?‘š+3 2 Thus we get đ?œ‹ cos đ?œƒ + 2đ?‘› − 1 cos đ?‘š đ?œƒ − cos đ?‘š + 2 đ?œƒ cos đ?‘š + 2 đ?œƒ − cos đ?‘š + 4 đ?œƒ đ?‘“" − đ?‘‘đ?œƒ 2đ?‘˜ đ?‘š+1 đ?‘š+3 0 đ?œ‹đ??ż < 3đ?‘š2 + 12đ?‘š + 13 1 2 đ?‘š+1 đ?‘š+3 (14) Using eqs.(13)and(14), one can get =

đ?‘“"

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wavelet collocation method for solving integro-differential equation. đ?‘“đ?‘›đ?‘š < 5đ?‘˜ 1

��� <

− 2 2đ?œ‹ 2 3

2 2 đ?‘š +1

1

1

đ?œ‹đ??ż 3đ?‘š2 + 1 2 đ?œ‹đ??ż 3đ?‘š2 + 12đ?‘š + 13 2 − đ?‘š đ?‘šâˆ’1 đ?‘š+1 đ?‘š+2 đ?‘š+1 đ?‘š+3

5đ?‘˜âˆ’3 1 2− 2 đ?œ‹ 2 5đ?‘˜ 1

2 đ?‘š+1 đ?‘šâˆ’1 2

=

− 2 2 đ?œ‹ 2đ??ż 1

2 2 đ?‘š −1 2

đ?‘˜

Finally since đ?‘› ≤ 2 − 1,then 1

��� <

đ?œ‹ 2 đ??ż đ?‘› +1

−

5 2

2 đ?‘š −1 2

Accuracy Estimation of ď ˇđ?&#x;?đ?’?đ?’Ž (đ?’™): If the function f(x) is expanded interms of fourth kind chebyshev wavelets, đ?‘“ đ?‘Ľ = ∞đ?‘›=1 ∞đ?‘š=0 đ?‘“đ?‘›đ?‘š ď ˇ2đ?‘›đ?‘š (đ?‘Ľ) (15) It is not possible to perform computation an infinite number of terms, therfore we must truncate the series in (15). In place of (5), we take đ?‘˜âˆ’1 2 đ?‘“đ?‘€ đ?‘Ľ = 2đ?‘›=1 đ?‘€âˆ’1 (16) đ?‘š=0 đ?‘“đ?‘›đ?‘š ď ˇđ?‘›đ?‘š (đ?‘Ľ) so that đ?‘“ đ?‘Ľ = đ?‘“đ?‘€ đ?‘Ľ + ∞đ?‘›=2đ?‘˜âˆ’1 +1 ∞đ?‘š=đ?‘€ đ?‘“đ?‘›đ?‘š ď ˇ2đ?‘›đ?‘š (đ?‘Ľ) or đ?‘“ đ?‘Ľ − đ?‘“đ?‘€ đ?‘Ľ = đ?‘&#x;(đ?‘Ľ) (17) where r(x) is the residual function đ?‘&#x; đ?‘Ľ = ∞đ?‘›=2đ?‘˜âˆ’1 +1 ∞đ?‘š =đ?‘€ đ?‘“đ?‘›đ?‘š ď ˇ2đ?‘›đ?‘š (đ?‘Ľ) (18) we must select coefficients in eqs.(17) and (18) such that the norm of the residual function đ?‘&#x;(đ?‘Ľ) is less than some convergence criterion ∈,that is 1 2

1 2

đ?‘“ đ?‘Ľ − đ?‘“đ?‘€ đ?‘¤đ?‘› đ?‘Ľ đ?‘‘đ?‘Ľ

<∈

0

for all M greater than some value đ?‘€0 . Theorem (2) Let f(x) be a continuaus function defined on [0,1), and đ?‘“ ′′ (đ?‘Ľ) < đ??ż, then we have the following accuracy estimation đ?œ‹đ??ż 1 1 đ?‘?đ?‘˜,đ?‘€ < 2 2 ∞đ?‘›âˆ’2đ?‘˜âˆ’1 +1 ∞đ?‘š=đ?‘€ đ?‘› +1 5 đ?‘š −1 4 (19) where 1 0

đ?‘?đ?‘˜,đ?‘€ =

2

đ?‘&#x; đ?‘Ľ

�� � ��

1 2

Proof:đ?‘Şđ?’Œđ?’Ž =

Since

1 0

2 đ??śđ?‘˜đ?‘š =

=

1 0

∞ đ?‘›=2 đ?‘˜âˆ’1 +1

đ?‘&#x; đ?‘Ľ

đ?&#x;? đ?’“ đ?&#x;Ž 2

đ?’™

đ?&#x;?

đ?’˜đ?’? đ?’™ đ?’…đ?’™

�� � ��

2 ���

ď ˇ đ?‘¤đ?‘› đ?‘Ľ đ?‘‘đ?‘Ľ

2 đ??śđ?‘˜đ?‘š =

∞ đ?‘›=2 đ?‘˜âˆ’1 +1

∞ đ?‘š=đ?‘€

đ?&#x;? đ?&#x;?

2

∞ đ?‘›=2 đ?‘˜âˆ’1 +1

=

∞ 2 đ?‘š=đ?‘€ đ?‘“đ?‘›đ?‘š

1 0

ď ˇ2đ?‘›đ?‘š (đ?‘Ľ) 2 đ?‘¤đ?‘› đ?‘Ľ đ?‘‘đ?‘Ľ

or

2 đ??śđ?‘˜đ?‘š =

∞ đ?‘›=2 đ?‘˜âˆ’1 +1

∞ đ?‘š=đ?‘€

Let đ?‘Ą = 2đ?‘˜ đ?‘Ľ − 2đ?‘› + 1, 2 đ??śđ?‘˜đ?‘š = we have,

đ?‘˜

đ?‘› 2 đ?‘˜âˆ’1 đ?‘› −1 2 đ?‘˜âˆ’1

2

đ?‘“ 2 22

∞ đ?‘›=2 đ?‘˜âˆ’1 +1

1 2 � 0 ��

∞ đ?‘š=đ?‘€

∞ đ?‘š=đ?‘€

ď ˇ2 2 đ?‘¤đ?‘› đ?‘Ľ đ?‘‘đ?‘Ľ

đ?‘ˆđ?‘š 2đ?‘˜ đ?‘Ľ − 2đ?‘› + 1 đ?‘“2

1 đ?‘ˆ2 −1 đ?‘š

đ?‘Ą

2

1 − đ?‘Ą 2 đ?‘‘đ?‘Ľ

1

đ?‘ˆđ?‘š2 đ?‘Ą

1 − đ?‘Ą 2 đ?‘‘đ?‘Ľ =

−1

then 2 đ??śđ?‘˜đ?‘š = 2 đ??śđ?‘˜đ?‘š <

∞ đ?‘›=2 đ?‘˜âˆ’1 +1

∞ đ?‘›=2 đ?‘˜âˆ’1 +1

∞ đ?‘š=đ?‘€

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1 − 2đ?‘˜ đ?‘Ľ − 2đ?‘› + 1 2 đ?‘‘đ?‘Ľ

đ?œ‹ 2

∞ 2đ?œ‹ đ?‘š=đ?‘€ đ?‘“ 2 đ?œ‹ 2 đ??ż2 đ?‘š +1 −5 8 đ?‘š −1 4

.

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wavelet collocation method for solving integro-differential equation. wavelet collocation method for VIDE with mth order: In this section the introduced wavelets collocation will be applied to solve VIDE with mth order, đ?‘Ľ (đ?‘› ) (đ?‘ ) đ?‘˘đ?‘– đ?‘Ľ = đ?‘”đ?‘– đ?‘Ľ + 0 đ??žđ?‘–,đ?‘— đ?‘Ľ, đ?‘Ą đ?‘˘đ?‘– đ?‘Ą đ?‘‘đ?‘Ą , đ?‘› ≼ đ?‘ (20) đ?‘ With the following conditions đ?‘˘đ?‘– 0 = đ?‘Žđ?‘–đ?‘ đ?‘– = 1,2, ‌ , đ?‘™ đ?‘ = 0,1,2, ‌ , đ?‘› − 1 Afunction đ?‘˘đ?‘–đ?‘› đ?‘Ľ which is defined on the interval đ?‘Ľ ∈ 0,1 can be expanded into the second chebyshev wavelet series đ?‘˘đ?‘–đ?‘› đ?‘Ľ = đ?‘€ (21) đ?‘–=1 đ?‘?đ?‘– ď ˇđ?‘– (đ?‘Ą) Where ci are the wavelet coefficients. Integrate eq.(21) m times,yields đ?‘Ľ đ?‘Ľ đ?‘€ đ?‘–=0 đ?‘?đ?‘– 0 ‌ 0

� � =

Using the following formula đ?‘Ľ

đ?‘Ľ

‌ 0

đ?‘—

đ?‘šâˆ’1 đ?‘Ľ đ?‘— =0 đ?‘— !

ď ˇđ?‘– đ?‘Ą đ?‘‘đ?‘Ą +

ď ˇđ?‘– đ?‘Ą đ?‘‘đ?‘Ą =

0

đ?‘Žđ?‘š −đ?‘—

(22)

đ?‘Ľ

1 (đ?‘› − 1)!

đ?‘Ľâˆ’đ?‘Ą

đ?‘›âˆ’1

ď ˇđ?‘– (đ?‘Ą)đ?‘‘đ?‘Ą

+

đ?‘›âˆ’1 đ?‘Ľ đ?‘— =0 đ?‘— !

0

therefore eq.(22) becomes đ?‘˘ đ?‘Ľ = Let đ??žđ?‘› đ?‘Ľ, đ?‘Ą =

đ?‘Ľâˆ’đ?‘Ą đ?‘› −1 đ?‘›âˆ’1 !

đ?‘Ľ 1 đ?‘€ đ?‘› −1 ď ˇđ?‘– (đ?‘Ą)đ?‘‘đ?‘Ą đ?‘–=0 đ?‘?đ?‘– (đ?‘›âˆ’1)! 0 đ?‘Ľ − đ?‘Ą đ?‘Ľ and đ??żđ?‘›đ?‘– = 0 đ??žđ?‘› đ?‘Ľ, đ?‘Ą đ?‘•đ?‘– đ?‘Ą đ?‘‘đ?‘Ą

đ?‘—

đ?‘Žđ?‘› −đ?‘—

(23)

i=0,1,‌,M

This leads to đ?‘€ đ?‘› đ?‘–=0 đ?‘?đ?‘– đ??żđ?‘–

� � =

đ?‘—

đ?‘›âˆ’1 đ?‘Ľ đ?‘— =0 đ?‘— !

+

đ?‘Žđ?‘›âˆ’đ?‘—

In similar way, we can get đ?‘˘(đ?‘ ) đ?‘Ľ =

đ?‘€ đ?‘›âˆ’đ?‘ đ?‘–=0 đ?‘?đ?‘– đ??żđ?‘–

+

đ?‘—

đ?‘›âˆ’đ?‘ −1 đ?‘Ľ đ?‘— =0 đ?‘—!

đ?‘Žđ?‘›âˆ’đ?‘ −đ?‘—

(24)

Substituting eqs (22) and (24) in (20), yield đ?‘€ đ?‘–=1 đ?‘?đ?‘– ď ˇđ?‘– (đ?‘Ą)

= �� � +

or where đ??´đ?‘– đ?‘Ľ = đ??ľđ?‘— đ?‘Ľ =

đ?‘Ľ 0

đ?‘€ đ?‘›âˆ’đ?‘ đ?‘–=0 đ?‘?đ?‘– đ??żđ?‘–

đ??žđ?‘–,đ?‘— đ?‘Ľ, đ?‘Ą

đ?‘€ đ?‘–=1 đ?‘?đ?‘– ď ˇđ?‘– (đ?‘Ą) − đ??´đ?‘– đ?‘Ľ đ?‘Ľ đ??žđ?‘› đ?‘Ľ, đ?‘Ą đ??żđ?‘›âˆ’đ?‘ đ?‘Ą đ?‘‘đ?‘Ą đ?‘– 0 đ?‘Ľ đ?‘— đ??ž đ?‘Ľ, đ?‘Ą đ?‘Ą đ?‘‘đ?‘Ą 0 đ?‘›

+

= �� � +

đ?‘—

đ?‘›âˆ’đ?‘ −1 đ?‘Ľ đ?‘Ž đ?‘‘đ?‘Ą đ?‘— =0 đ?‘— ! đ?‘›âˆ’đ?‘ −đ?‘— đ?‘Ž đ?‘› −đ?‘ −đ?‘— đ?‘›âˆ’đ?‘ −1 đ??ľđ?‘— (đ?‘Ľ) đ?‘— =0 đ?‘—!

(25) (26)

i=0,1,2,‌,M j=0,1,2,‌,n-s-1

(27)

1

Next the interval đ?‘Ľ ∈ 0,1 is devided in to đ?‘™ ∆đ?‘Ľ = đ?‘™ and introduce the collocation points

đ?‘Ľđ?‘˜ =

đ?‘˜ −1

, k=1,2,‌,l

đ?‘™ M c [ i=1 i ď ˇi (x)

eq(21) is satisfied only at the collocation points we get asystem of linear equations

− Ai x ] = g i x +

n−s−1 a n −s−j j=0 j!

Bj (x)

(28)

The matrix form of this system a n −s−j is C F=G+ n−s−1 Bj (x) where F=ď ˇ (x), G=g(x) j=0 j! 1.Design of the matrix A:When chebyshev wavelets second kind are integrated m times, the following integral must be evaluated. đ?‘Ľ đ??żđ?‘›đ?‘– = 0 đ??žđ?‘› đ?‘Ľ, đ?‘Ą ď ˇđ?‘– đ?‘Ą đ?‘‘đ?‘Ą , i=0, 1, 2, ‌, M

1 −3

đ??żđ?‘›đ?‘– đ?‘Ľ =

đ?‘Ľâˆ’đ?‘Ą đ?‘› 2 đ?‘˜ đ?‘› −1 !

4 1

1

0 ‌

2

0 1

3

−6

â‹Ž

â‹Ž

0

⋎ 2 0 ‌ 0

‌

0

â‹Ž 0 0

0

0

0 ‌

0

â‹Ž 0 0

0

0

â‹Ž

â‹ą

â‹Ž

1 4

â‹Ž

â‹ą

1 2(đ?‘€âˆ’1)

â‹Ž

â‹Ž

đ?‘™âˆ’1 2đ?‘˜

đ?‘™

≤ đ?‘Ľ < 2đ?‘˜

−1đ?‘€ −2

0 0 ‌ 0 â‹Ž 0 0 ‌ 0 đ?‘€ Therefore the matrix Ai x can be constructed as follows đ?‘Ľ Since đ??´đ?‘– đ?‘Ľ = 0 đ??žđ?‘› đ?‘Ľ, đ?‘Ą đ??żđ?‘›âˆ’đ?‘ đ?‘Ą đ?‘‘đ?‘Ą i=0,1,2,‌,M đ?‘–

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wavelet collocation method for solving integro-differential equation. đ?‘Ľ0

đ??´đ?‘– đ?‘Ľ =

0 đ?‘Ľđ?‘›

đ??žđ?‘› đ?‘Ľ0 , đ?‘Ą đ??żđ?‘›âˆ’đ?‘ đ?‘Ą đ?‘‘đ?‘Ą đ?‘–

đ??žđ?‘› đ?‘Ľđ?‘– , đ?‘Ą đ??żđ?‘›âˆ’đ?‘ đ?‘Ą đ?‘‘đ?‘Ą đ?‘–

đ?‘–=0 đ?‘–>0

0

IV.

WAVELETS METHOD FOR VIDE WITH NTH ORDER

For solving VIDE with mth order the matrix đ??żđ?‘›đ?‘– đ?‘Ľ in section above will be followed to get M n−s−1 a n −s−j Bj (xL ) đ??ż ∈ đ?‘Ž, đ?‘? i=1 ci [ď ˇi (xL − A L )] = g xL + j=0 j!

đ?‘Ľ

But đ??´đ?‘– đ?‘Ľđ??ż = 0 đ??ż đ??žđ?‘› xL , đ?‘Ą đ??żđ?‘›âˆ’đ?‘ đ?‘Ą đ?‘‘đ?‘Ą where i=0,‌,M đ?‘– đ?‘Ľđ??ż đ?‘›âˆ’đ?‘ Bj (xL )= 0 đ??žđ?‘› xL , đ?‘Ą đ?‘Ą đ?‘‘đ?‘Ą đ?‘›âˆ’đ?‘ where đ??żđ?‘– đ?‘Ą as in eq(17),(18) that is đ??´đ?‘– đ?‘Ľđ??ż = đ??´đ??ż , đ??šđ?‘– đ?‘Ľđ??ż = ď ˇđ?‘– đ?‘Ľđ??ż − đ??´đ?‘– đ?‘Ľđ??ż = đ??šđ??ż Numerical Results: In this section VIDE is considered and solved by the introduced method. parameters k and M are considered to be 1 and 3 respectively. Example 1: Consider the following VIDE: đ?‘Ľ U′′ đ?‘Ľ = đ?‘’ 2đ?‘Ľ − 0 đ?‘’ 2(đ?‘Ľâˆ’đ?‘Ą) đ?‘ˆ ′ đ?‘Ą đ?‘‘đ?‘Ą Initial conditions đ?‘ˆ(0) = 0, đ?‘ˆâ€™(0) = 0. The exact solution đ?‘ˆ đ?‘Ľ = đ?‘Ľđ?‘’ đ?‘Ľ − đ?‘’ đ?‘Ľ + 1. Table 1 shows the numerical results for this example with k=2, M=3 with error =10-3 and k=2, M=4, with error =10-4 are compared with exact solution graphically in fig.

x 0 0.2 0.4 0.6 0.8 1

Table 1:some numerical results for example 1 Exact solution Approximat solution Approximat solution k=2,M=3 k=2,M=4 0.00000000 0.00000001 0.00000001 0.02287779 0.02280000 0.02287000 0.10940518 0.10945544 0.10940544 0.27115248 0.25826756 0.27826756 0.55489181 0.54330957 0.55330957 1.00000000 0.99999995 0.99999998 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig 1:Approximate solution for example 1 Example 2: Consider the following VIDE : đ?‘Ľ U(5) đ?‘Ľ = −2 sin đ?‘Ľ + 2 cos đ?‘Ľ − đ?‘Ľ + 0 (đ?‘Ľ − đ?‘Ą)đ?‘ˆ (3) đ?‘Ą đ?‘‘đ?‘Ą Initial conditions đ?‘ˆ(0) = 1, đ?‘ˆâ€™(0) = 0, đ?‘ˆ"(0) = −1, đ?‘ˆ 3 (0) = 0. The exact solution đ?‘ˆ đ?‘Ľ = cos đ?‘Ľ. Table 2 shows the numerical results for this example with k=2, M=3 with error=10-3 and k=2, M=4, with error =10-4 are compared with exact solution graphically in fig, 2.

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wavelet collocation method for solving integro-differential equation. x 0 0.2 0.4 0.6 0.8 1

Table 2:some numerical results for example 2 Exact solution Approximat solution Approximat solution k=2,M=3 k=2,M=4 1.00000000 0.99812235 0.99999875 0.98006658 0.98024711 0.98005541 0.92106099 0.92158990 0.92104326 0.82533561 0.82479820 0.82535367 0.69670671 0.69689632 0.69678976 0.54030231 0.54032879 0.54035879 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig 2:Approximate solution for example

V.

CONCLUSION

This work proposes a powerful technique for solving VIDE second kind using wavelet in collocation method comparison of the approximate solutions and the exact solutions shows that the proposed method is more faster algorithms than ordinary ones. The convergence and accuracy estimation of this method was examined for several numerical examples.

REFERENCES [1] [2] [3]

[4] [5] [6]

[7] [8] [9]

Asmaa A. A , 2014, Numerical, Solution of Optimal Control Problems Using New Third kind Chebyshev Wavelets Operational Matrix of Integration, Eng&Tech journal,Vol 32,part(B),1:145-156. Tao. X. and Yuan. L. 2012. Numerical Solution of Fredholm Integral Equation of Second kind by General Legendre Wavelets, Int. J. Inn. Comp and Cont. 8(1): 799-805. A.Barzkar, M.K.Oshagh, 2012, Numerical solution of the nonlinear Fredholm integro-differential equation of second kind using chebysheve wavelets, World Applied Scinces Journal (WASJ),Vol.18, N(12):1774-1782. E.Johansson, 2005, Wavelet Theory and some of its Applications. Jafari. H and Hosseinzadeh. H. 2010. Numerical Solution of System of Linear Integral Equations by using Legendre Wavelets, Int. J. Open Problems Compt. Math., 3(5): 1998-6262 . Shihab. S. N. and Mohammed. A. 2012. An Efficient Algorithm for n thOrder Integro- Differential Equations Using New Haar Wavelets Matrix Designation, International Journal of Emerging & Technologies in Computational and Applied Sciences (IJETCAS). 12(209): 32-35. M.Razzaghi & S.Yousefi. 2002. Sin-Cosine wavelets operational matrix of integration and it’s applications in the calculus of variations. Vol 33. No 10: 805-810. A.Arikoglu & I.Ozkol. 2008. Solution of integral integro-differential equation systems by using differential transform method. Vol 56,Issue 9.2411-2417. Arsalani. M. & Vali. M. A.,2011 Numerical Solution of Nonlinear Problems With Moving Boundary Conditions by Using Chebyshev Wavelets, Applied Mathematical Sciences,Vol.5(20): 947-964.

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