An Application of Legendre Wavelet in Fractional …Ritu Arora* and N. S. Chauhan** *Department of Mathematics, Kanya Gurukul (Haridwar Campus) Gurukula Kangri University, Haridwar-249404

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 2 (2017), pp. 183-202

© Research India Publications

http://www.ripublication.com

An Application of Legendre Wavelet in Fractional

Electrical Circuits

Ritu Arora* and N. S. Chauhan**

*Department of Mathematics, Kanya Gurukul (Haridwar Campus)

Gurukula Kangri University, Haridwar-249404 (UK), India.

**Department of Mathematics and Statistics, Faculty of Science Gurukula Kangri University, Haridwar-249404 (UK), India.

**Corresponding Author

Abstract

In this paper, Legendre wavelet method is presented for solving fractional

electrical circuits. We first derive the operational matrix of fractional order

integration. The fractional integration is described in the Riemann-Liouville

sense. The operational matrix of fractional order integration is used to

transform fractional electrical circuit’s equation into the system of algebraic

equations. The uniform convergence theorem and accuracy estimation are also

derive for proposed method. In addition, using plotting tools, we compare

approximate solutions of each equation with its classical solution.

Keywords: Fractional Electrical circuit models, Legendre wavelets,

Operational matrix of fractional integration, Block pulse function.

Subject Classification AMS: 65T99, 47N70, 34A08, 65L03, 26A33.

INTRODUCTION

In recent years, wavelets tools are one of the growing and predominantly a new

technique. Today many classical physics models are being analysed using wavelet

methods. Here, we discuss the basic equations of electric circuits involving resistor

with a resistance R measured on ohms, an inductor with inductance L measured in

mailto:[email protected]

184 Ritu Arora and N. S. Chauhan

henries, and a capacitor with capacitance C measured in farads. We investigate the

following equations for four different types of circuits

' ' 1( ) ( ) 0J t J t

LC (1)

' 1( ) ( ) 0CV t V t

R (2)

' ' ' 1( ) ( ) ( ) 0,LQ t RQ t Q t

C (3)

' ( ) ( )LJ t RJ t V (4)

where ( ), ( )J t V t and ( )Q t are the current, voltage and electric charge in the shell

of capacitor with respect to time t respectively. Equation (1) represents the LC(inductor-capacitor) circuit, equation (2) represents the RC (resistor-capacitor) circuit,

equation (3) represents the LR (inductor-resistor) circuit and equation (4) represents

the RLC (resistor- inductor -capacitor) circuit. Some numerical results of consider

circuits are presented in [1-5].

The mathematical foundation of fractional calculus (FC) was established over 200

years ago, although the application of FC has attracted interest in recent decades. FC,

involving derivatives and integrals of non-integer order, is the natural generalization

of the classical calculus [6-9]. FC is a highly efficient and useful tool in the modelling

of many sorts of scientific phenomena including image processing earthquake,

engineering, physics etc [10].

Wavelets are used in signal analysis, medical science, approximation theory,

technology and many other areas. Wavelets analysis possesses several useful

properties such as compact support, orthogonality, exact representation of

polynomials to a certain degree, and ability to represent functions at different levels of

resolution [11, 12].

The outline of this paper is as follows: In section second, we discuss some notations,

definitions and preliminary facts of the fractional calculus theory. In section third, we

present wavelet and Legendre wavelet and their properties. In section four, we show

that the function approximation and collocation points. Section fifth is devoted to the

operational matrix of integration. In section six, we give convergence and mean

square error theorems for our method. Finally, we solve fractional electric circuit’s

equations given in section seven, to illustrate the performance of our method.

OVERVIEW ON FRACTIONAL CALCULUS

In this section, we present some notations, definitions and preliminary facts of the

fractional calculus theory. Using fractional calculus, one has many choices for

definitions of fractional derivative as well as fractional integral. Since, we want to

transform the electric circuit differential equations in to the algebraic system of

An Application of Legendre Wavelet in Fractional Electrical Circuits 185

equations, and consider the fractional integral and fractional derivative in the

Riemann-Liouville sense.

Definition 2.1 The Riemann-Liouville fractional integral operator I of order on the

usual Lebesgue space 1[ , ]L a b is given [13] by

1

0

1( ) , 0

( ), 0.

t

t s f s dsI f t

f t

(5)

It has the following properties:

(i) ,I I I (ii) ,I I I I

(iii) ( ) ( ),I I f t I I f t (iv) ( 1)

,( 1)

v vvI t a t av

(6)

where 1[ , ], , 0f L a b and 1.v

Definition 2.2 For a function f given on interval [ , ],a b the Caputo definition of

fractional order derivative of order 1n n of f is defined [13] by

11( ) ( ) ( ) ,

( )

n tn

aa

dD f t t s f s dsn dt

(7)

where 0,t n is a integer. It has the following two basic properties for 1n n and

1[ , ].f L a b

( ) ( )D I f t f t

and 1

( )

0

( )( ) ( ) (0 ) , 0.

!

knk

k

t aI D f t f t f tk

(8)

For more details on the mathematical properties of fractional derivatives and integrals

see [5-7].

WAVELETS AND LEGENDRE WAVELETS (LW)

Wavelets constitute a family of functions constructed by performing translation and

dilation on a single function , where is a mother wavelet. We define family of

continuous wavelets [7] by

1/2

, ( )a bt bt a

a

, , ; 0,a b R a

where a is called scaling parameter and b is translation parameter. If we

restricted the parameters a and b to discrete values as 0 0 0, ,k ka a b nb a where


0 01, 0,a b and k , n are positive integers. We have the following family of discrete

wavelets:

/2

0 0 0( ) , , ;k k

k n t a a t nb k n Z

where the function ( )k n t form a wavelet basis for 2 ( ).L R In particular, when

0 2a and 0 1,b the function ( )k n t form an orthonormal basis.

The Legendre wavelets ˆ( ) ( , , , )nm t k n m t have four arguments; 1ˆ 2 1, 1,2,3,...,2 ,kn n n k can assume any positive integer, m is the order for Legendre

polynomials and t is the normalized time. The Legendre wavelets are defined on the

interval [0,1]as [14-16]

/2 ˆ ˆ1 1 1ˆ.2 2 , ,

( ) 2 2 2

0, ,

k km k k

n m

n nm P t n for ttotherwise

(9)

where 0,1,2,..., 1,m M M is a fixed positive integer and 11,2,3,...,2 .kn The

coefficient 1/ 2m is for orthnormality, the dilation parameter is 2 ka and

translation parameter is ˆ2 .kb n ( )mP t is the well-known Legendre polynomial of order

m defined on the interval [1, 1] and can be determined with the help of the following

recurrence formulae:

0 1 1 1

2 1( ) 1, ( ) , ( ) ( ) ( ),

1 1m m m

m mP t P t t P t tP t P tm m

(10)

where 1,2,3,...m .

FUNCTION’S APPROXIMATION

A function ( )f t defined over [0,1] and approximated as

1 0

( ) ( )n m n mn m

f x g x

(11)

where ( ), ( )n m n mg f t t . If the infinite series in equation (11) is truncated

then equation (11) can be written as

2 1 1

0 0

( ) ( ) ( ),

k MT

n m n mn m

f t g t G t

(12)

where T indicates transposition and, G and ( )t are 1ˆ 2 1km M column

vectors given by


and 1 1

1 1

10 11 1( 1) 20 2( 1) 2 0 2 ( 1)

10 11 1( 1) 20 2( 1) 2 0 2 ( 1)

[ , ,..., , ,..., ,..., ,..., ] ;.

( ) [ , ,..., , ,..., ,..., ,..., ]

k k

k k

TM M M

TM M M

G g g g g g g g

t

(13)

Taking the collocation points as following:

12 1

, 1,2,..., 2 .2

ki k

it i M

M

(14)

Now, taking define the Legendre wavelet matrix ˆ ˆm m [17] as:

ˆ ˆ

1 3 5 2 1...

2 2 2 2m m

mm m m m

(15)

For example, when 3M and 2k the Legendre wavelet is expressed as

ˆ ˆ

1.4142 1.4142 1.4142 0 0 0

1.6329 0 1.6329 0 0 0

0.5270 1.5811 0.5270 0 0 0

0 0 0 1.4142 1.4142 1.4142

0 0 0 1.6329 0 1.6329

0 0 0 0.5270 1.5811 0.5270

m m

The Legendre matrix ˆ ˆm m is an invertible matrix [18], the coefficient vector TG is

obtained by

1

ˆ ˆˆ .T

m mG f

(16)

LEGENDRE WAVELET OPERATIONAL MATRIX OF FRACTIONAL

INTEGRAL

The fractional integral of order in the Riemann-Liouville sense of the vector ( )t ,

defined in equation (5), can be approximated by Legendre wavelet series with

Legendre wavelet coefficient matrix .P

ˆ ˆ

( ) ( ),m m

I t P t

(17)

where the ˆ ˆm m

P

is called ˆ ˆm m Legendre wavelet operational matrix of integral of

order . It show that the operational matrix ˆ ˆm m

P

can be approximate as [19]:

ˆ ˆ ˆ ˆ

1

ˆ ˆ ,m m m mm mP F

(18)

where ˆ ˆm m

F

is the operational matrix of fractional integration of order of the

block-pulse function (BPFs), which given in [20]


ˆ ˆ

1 2 3 1

1 2 2

1 3

4

1

0 1

0 0 1,

0 0 0 1

0 0 0 0

0 0 0 0 1 1

m m

m

m

m

m

F

where 1 1 1( 1) 2 ( 1) ,j j j j [16].

CONVERGENCE AND MEAN SQUARE ERROR THEOREMS FOR

METHOD

In this section we discus convergence and mean square error of Legendre wavelet

method.

Theorem (Convergence theorem): Let the function 2( ) [0,1]D f x C bounded and

( )D f x exists and can be expressed as in equation (11) and the truncated series given

in equation (12) converges towards the exact solution.

Proof: Let ( )D f x be a function in the interval [0,1]such that

( ) ,D f x K (19)

where K is a positive constant.

From equation (12), we approximate ( )D f x as

2 1 1

0 0

( ) ( ),

k M

nm nmn m

D f x g x

(20)

where ( ), ( )nm nmng D f x x and /2 ˆ ˆ1 1 1ˆ( ) 2 (2 ),

2 2 2

k knm m k k

n nx m P x n for x

and mP are well known Legendre polynomials.

Now, we consider

1

0

( ), ( ) ( ) ( )nm nm nmg D f x x D f x x dx (21)

ˆ ˆ( 1)/2 ( 1)/2 1

0 ˆ ˆ( 1)/2 ( 1)/2

( ) ( ) ( ) ( ) ( ) ( )

k k

k k

n n

nm nm nm nmn n

g D f x x dx D f x x dx D f x x dx

ˆ ˆ( 1)/2 ( 1)/2

/2

ˆ ˆ( 1)/2 ( 1)/2

1ˆ( ) ( ) ( ) 2 2 ,

2

k k

k k

n nk k

nm nm mn n

g D f x x dx D f x m P x n dx


Let ˆ2k x n t

1 1

/2 /2

1 1

ˆ ˆ1 12 2 ,

2 22 2 2

k knm m mk k k

t n dt t ng m D f P t m D f P t dt

(22)

We consider right hand side of equation (22) and integrate by part two times with

respect to ,t yield

1

3 /2 1 2 2

1

1

3 /2 2 2 2

1

ˆ ( ) ( ) ( ) ( )1 12

2 2 1 2 3 2 12

ˆ ( ) ( ) ( ) ( )1 1 12

2 2 1 2 3 2 12 2

k m m m mnm k

k m m m mk k

P t P t P t P tt ng m D fm m m

P t P t P t P tt nm D f dtm m m

1

3 /2 2 2 2

1

ˆ ( ) ( ) ( ) ( )1 1 12

2 2 1 2 3 2 12 2

k m m m mnm k k

P t P t P t P tt ng m D f dtm m m

(23)

Now,

1

3 /2 2 2 2

1

ˆ ( ) ( ) ( ) ( )1 1 12

2 2 1 2 3 2 12 2

k m m m mnm k k

P t P t P t P tt ng m D f dtm m m

1

5 /2 22 2

1

ˆ( ) ( ) ( ) ( )1 12

2 2 1 2 3 2 1 2

k m m m mnm k

P t P t P t P t t ng m dt D fm m m

(24)

5 /2

1

2

2 2

1

1 12

2 2 1 2 1 2 3

ˆ2 1 ( ) 2 1 ( ) 2 3 ( ) 2 3 ( )

2

knm

m m m m k

g mm m m

t nm P t m P t m P t m P t dt D f

5 1 /2

1

2 2

1

2 1 2

2 1 2 1 2 3

2 1 ( ) 2 1 ( ) 2 3 ( ) 2 3 ( )

k

nm

m m m m

m Kg

m m m

m P t m P t m P t m P t dt

5 1 /2

1

2 2

1

2

2 1 2 3 2 1

2 1 ( ) 2 1 ( ) 2 3 ( ) 2 3 ( )

k

nm

m m m m

Kgm m m

m P t m P t m P t m P t dt

(25)

Let 2 22 1 ( ) 4 2 ( ) 2 3 ( )m m mR t m P t m P t m P t

15 1 /2

1

2.

2 1 2 3 2 1

k

nmKg R t dt

m m m

(26)


1 1

2 2

1 1

2 1 ( ) 4 2 ( ) 2 3 ( ) .m m mR t dt m P t m P t m P t dt

Using Cauchy-Schwarz’s inequality, we get

2

1 1 12

2 2

1 1 1

2 1 ( ) 4 2 ( ) 2 3 ( ) ,m m mR t dt dt m P t m P t m P t dt

1

2 2 22 2 2

2 2

0

4 2 1 ( ) 4 2 ( ) 2 3 ( ) ,m m mm P t m P t m P t dt

22 3

4.3 ,2 3

mm

1

1

2 3 2 3.

2 3

mR t dt

m

(27)

Using equation (27) in equation (26), we get

5 1 /2 2 3 2 32,

2 32 1 2 3 2 1

k

nm

mKgmm m m

5 /22 6,

2 1 2 1 2 3

k Km m m

5 /2

6.

2 2 1 2 1 2 3nm k

Kgm m m

(28)

Hence desire.

Theorem (Mean square error): Let the function 2( ) [0,1],D f x C and ( )D f x exists

bounded second derivative then we have the following accuracy estimation.

5/2

2

6.

2 1 2 1 2 3kn

m Mn

Kn m m m

Where

1 2 1 1

0 0 0 00

( ) ( ) .

k M

n nm nm nm nmn m n m

g x g x dx

Proof: Let us consider the quantity

21 2 1 1

2

0 0 0 00

( ) ( ) ,

k M

nm nm nm nmn m n m

g x g x dx

(29)


21 1

2 2 2

2 20 0

( ) ( ) ,k k

nm nm nm nmm M m Mn n

g x dx g x dx

(30)

Using the property of orthonormal wavelets

1

0

( ) ( ) .Tnm nmx x dx I (31)

2 2

2

,k

nmm Mn

g

From equation (28), we have

5/2

2

6.

2 1 2 1 2 3kn

m Mn

Kn m m m

Hence desire.

ELECTRICAL CIRCUITS

In this section, we apply Legendre wavelet collocation method and find the

approximate solution of fractional circuits and comparing their solutions with the

corresponding classical solutions.

Solution of LC circuit

Consider the LC circuit, only charged capacitor and inductor are present in the circuit

and its differential equation is given as follows.

' ' 1( ) ( ) 0.J t J t

LC (32)

with 0(0)J J and ' (0) 0.J

The classical solution of equation (32) is

0 0( ) ( ),LCJ t J Cos t (33)

where 2

0 1/ .LC

Now, we analyse equation (32) using fractional calculus, we replace '' ( )J t by ( )D J t ,

where (1,2).

In the sense of Riemann-Liouville derivative, we get the fractional order LC circuit

and its differential equation as

2

0( ) ( ) 0.D J t J t (34)


with 0(0)J J and (0) 0.D J (35)

We use equation (12) to approximate ( )D J t as

12 1

1 0

( ) ( ) ( ),

k MT

nm nmn m

D J t u t U t

(36)

Integrating equation (36) with respect to t over the interval [0, ]t , we get

1

ˆ ˆ( ) (0) ( ),Tm mDJ t DJ U P t

Using condition (35), yield

ˆ ˆ0( ) ( ).Tm mJ t J U P t

(37)

Substituting equations (35-36) in equation (33), we obtain

2

ˆ ˆ0 0( ) ( ) 0.T Tm mU t J U P t (38)

From equation (16), we can approximate 0J as:

1

ˆ ˆ0 0 0 0 ˆ1, ,..., ( ).m mm

J J J J t

(39)

Substituting equation (39) in equation (38), we have

2 2 1

ˆ ˆ ˆ ˆ0 0 0 0 0( ) ( ) [ , ,..., ] ( ),T Tm m m mU t U P t J J J t

(40)

2 2 1

ˆ ˆ ˆ ˆ0 0 0 0 0[ , ,..., ] ,Tm m m mU I P J J J

1

2 1 2

ˆ ˆ ˆ ˆ0 0 0 0 0[ , ,..., ] .Tm m m mU J J J I P

(41)

Hence required

1

1 2 1 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ0 0 0 0 0 0 0 0( ) [ , ,..., ] ( ) [ , ,..., ] ( ).m m m m m m m mJ t J J J t J J J I P P t

(42)

By solving the above system (41) of linear equations, we can find the value of vector

.U substituting value of U in equation (37), hence we required the numerical results of

the LC circuit for different value of ,k M and . Here, solution obtained by the

proposed LWM approach for 1.50,1.75,1.999 and 2k and 3M is graphically show

in figure 1. As it can be clearly seen, for 1.999 and 2k and 3,M the fractional

RC circuit graph behave similar to the classical solution graph for 1 . It is show that

the proposed LWM approach is more close to the exact solution. Table 1 describes the

efficiency of the proposed method by comparing with the classical solution at 1 .

Table 1 show that very high accuracies are obtained for 2k and 3M by the present

method.


Figure 1.Current versus time graph 01, 1, 0.01 1.50,1.75 1.999 .L C J and and

Table 1. Numerical results of LC circuit for

01, 1, 0.01 1.5,1.75 1.999 .L C J and and

t 1.50 1.75 1.999 2

LW LW LW CS

0.1 39.7394 10 39.8706 10 39.9382 10 39.9500 10

0.2 39.3239 10 39.6153 10

39.7888 10 39.8006 10

0.3 38.8129 10 39.2517 10

39.5430 10 39.5533 10

0.4 38.2064 10 38.7801 10 39.2007 10 39.2106 10

0.5 37.5620 10 38.2393 10

38.7757 10 38.7758 10

0.6 36.8577 10 37.5956 10

38.2444 10 38.2534 10

0.7 36.1371 10 36.9084 10

37.6406 10 37.6484 10

0.8 35.4179 10 36.1776 10

36.9640 10 36.9671 10

0.9 34.6945 10 35.4032 10

36.2147 10 36.2161 10

Solution of RC circuit

Consider the RC circuit differential equation given in equation (2), only charged

capacitor and resistor are present to the circuit and its differential equation is given as

follows

' 1( ) ( ) 0.CV t V t

R (43)

with condition '

0(0) (0) 0.V V andV (44)

0.2 0.4 0.6 0.8 1.0t

0.005

0.006

0.007

0.008

0.009

0.010

Current

0.2 0.4 0.6 0.8 1.0t

0.006

0.007

0.008

0.009

0.010

Current

0.2 0.4 0.6 0.8 1.0t

0.007

0.008

0.009

0.010

Current

0.2 0.4 0.6 0.8 1.0t

0.007

0.008

0.009

0.010

Current



1

0( ) .t

RCRCV t V e

(45)

Now, we consider equation (43) using fractional calculus, we replace ' ( )V t by ( )D V t ,

where (0,1). In the sense of Riemann-Liouville derivative, we get the fractional

order RC circuit and its differential equation as

1

( ) ( ) 0D V t V tRC

(46)

with condition 0(0) (0) 0.V V and D V (47)

We use equation (12) to approximate ( )D V t as

12 1

1 0

( ) ( ) ( ),

k MT

nm nmn m

D V t w t W t

(48)

Integrating equation (48) with respect to t, over [0, ]t , we get

ˆ ˆ( ) (0) ( ),Tm mV t V W P t

(49)

Similarly equation (39), we can approximate 0V as

1

ˆ ˆ0 0 0 0 ˆ1(0) , ,..., ( ).m mm

V V V V V t

(50)

So, equation (49) become

1

ˆ ˆ ˆ ˆ0 0 0 ˆ1( ) , ,..., ( ) ( ),T

m m m mmV t V V V t W P t

(51)

Substituting equations (48) and equation (51) in equation (46), we obtain

1

ˆ ˆ ˆ ˆ0 0 0 ˆ1

1( ) , ,..., ( ) ( ) 0T T

m m m mmW t V V V t W P t

RC

1

ˆ ˆ ˆ ˆ0 0 0

1 1( ) ( ) [ , ,..., ] ( ),T T

m m m mW t W P t V V V tRC RC

1

ˆ ˆ ˆ ˆ0 0 0

1 1[ , ,..., ] ,T

m m m mW I P V V VRC RC

1

1

ˆ ˆ ˆ ˆ0 0 0

1 1[ , ,..., ] . .T

m m m mW V V V I PRC RC

(52)

Hence required

1

1 1

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ0 0 0 0 0 0

1 1( ) [ , ,..., ] ( ) [ , ,..., ] . ( ).m m m m m m m mV t V V V t V V V I P P t

RC RC

(53)

We solving the above system (52)of linear equations and obtain the value of vector w.

Substituting the value of vector W in equation (51), hence we require the approximate

results of the RC circuit model for different value of ,k M and . Here, we use the

proposed LWM approach for 0.25,0.75,0.999 and 2k and 3.M This has been seen


from figure 2 that the obtain solution for 0.999 and 2k and 3,M the fractional

RC circuit graph behave similar to the classical solution graph for 1 . It is show that

the proposed LWCM approach is more close to the exact solution. Table 2 describes

the good organization of the proposed method by comparing with the classical

solution at 1 . Table 2 also shows that, very high accuracies are obtained for 2k and 3M by the present method.

Figure 2.Voltage versus time graph 010, 1, 20 0.5,0.75 0.999 .R C V and and

Table 2.Numerical results of RC circuit for

010, 1, 20 0.5,0.75 0.999 .R C J and and

t 0.50 0.75 0.999 1

LW LW LW CS

0.1 19.3481 19.6340 19.8012 19.8010

0.2 19.0500 19.3717 19.6039 19.6040

0.3 18.8163 19.1413 19.4086 19.4089

0.4 18.6471 19.9428 19.2154 19.2158

0.5 18.4976 18.7573 19.0242 19.0246

0.6 18.3669 18.5854 18.8349 18.8353

0.7 18.2450 18.4207 18.6475 18.6479

0.8 18.1319 18.2630 18.4620 18.4623

0.9 18.0275 18.1124 18.2784 18.2786

0.2 0.4 0.6 0.8 1.0t

18.5

19.0

19.5

Voltage

0.2 0.4 0.6 0.8 1.0t

18.5

19.0

19.5

Voltage

0.2 0.4 0.6 0.8 1.0t

19.0

19.5

20.0

Voltage

0.2 0.4 0.6 0.8 1.0t

19.0

19.5

20.0

Voltage


Solution of RLC circuit

Here, we analyze RLC circuit differential equation given in equation (3). LCR circuit

consisting of three kinds of circuit elements: a resistor, an inductor and a capacitor its

differential equation is given as follows

' ' ' 1( ) ( ) ( ) 0.LQ t RQ t Q t

C (54)

With the condition '

0(0) , (0) 0.Q Q Q (55)


2

20 2

1( ) .

4

R tL

RLCRQ t Q e Cos t

LC L

(56)

Now, we analyse equation (54) using fractional calculus, we replace '' ( )Q t by 2 ( )D Q t ,


order RLC circuit and its differential equation as

2 1( ) ( ) ( ) 0.L D Q t R D Q t Q t

C (57)

With the condition 0(0) (0) 0.Q Q and D Q (58)

We use equation (12) to approximate 2 ( )D Q t as

12 1

2

1 0

( ) ( ) ( ),

k MT

nm nmn m

D Q t y t Y t

(59)

Integrating equation (59) with respect to t, over [0, ]t and using condition (58), we get

ˆ ˆ ˆ ˆ( ) (0) ( ) ( ),T Tm m m mD Q t D Q Y P t Y P t

(60)

and

2

ˆ ˆ( ) (0) ( ).Tm mQ t Q Y P t

(61)

Similarly equation (39), we can approximate 0Q as

1

ˆ ˆ0 0 0 0(0) [ , ,..., ] ( ).m mQ Q Q Q Q t

(62)

From equation (61) and equation (62), we have

1 2

ˆ ˆ ˆ ˆ0 0 0( ) [ , ,..., ] ( ) ( ).Tm m m mQ t Q Q Q t Y P t

(63)

Substituting equations (59,60) and equation (63) in equation (57), we obtain

1 2

ˆ ˆ ˆ ˆ ˆ ˆ0 0 0

1( ) ( ) [ , ,..., ] ( ) ( ) 0,T T T

m m m m m mRY t Y P t Q Q Q t Y P tL LC

(64)


Let 21

LC and 2 ,

RL

then equation (64) become

2 2 1 2 2

ˆ ˆ ˆ ˆ ˆ ˆ0 0 0( ) ( ) [ , ,..., ] ( ) ( ) 0T T Tm m m m m mY t Y P t Q Q Q t Y P t

2 2 2 2 1

ˆ ˆ ˆ ˆ ˆ ˆ0 0 0[ , ,..., ] ,T T Tm m m m m mY Y P Y P Q Q Q

2 2 2 2 1

ˆ ˆ ˆ ˆ ˆ ˆ0 0 0[ , ,..., ] ,Tm m m m m mY I P P Q Q Q

1

2 1 2 2 2

ˆ ˆ ˆ ˆ ˆ ˆ0 0 0[ , ,..., ] . .Tm m m m m mY Q Q Q I P P

(65)

Hence required

1

1 2 1 2 2 2 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ0 0 0 0 0 0( ) [ , ,..., ] ( ) [ , ,..., ] ( ).m m m m m m m m m mQ t Q Q Q t Q Q Q I P P P t

(66)

We manipulate the above system (65) of linear equations and obtain the unknown

vectorY . Having these value of vector Y in equation (63), we get numerical results of

RLC circuit for different value of ,k M and . Here, solution obtained by the proposed

LWM approach for 0.25,0.75,0.999 and 2k and 3M is graphically shown in

figure 3. As it can be clearly seen, for 0.999 and 2k and 3,M the fractional RLC

circuit graph behave similar to the classical solution graph for 2 . It is show that the

proposed LWCM approach is more close to the exact solution. Table 3 describes the

effectiveness of the proposed method by comparing with the classical solution at 2

. Table 3 shows that very high accuracies are obtained for 2k and 3M by the

present method.

Figure 3.Charge ( )Q t versus time graph

010, 10, 10, 0.01 0.5,0.75 0.999 .R L C V and and

0.2 0.4 0.6 0.8 1.0t

0.00996

0.00997

0.00998

0.00999

0.01000

Charge Q t

0.2 0.4 0.6 0.8 1.0t

0.00997

0.00998

0.00999

0.01000

Charge Q t

0.2 0.4 0.6 0.8 1.0t

0.0099700.0099750.0099800.0099850.0099900.0099950.010000

Charge Q t

0.2 0.4 0.6 0.8 1.0t

0.00700.00750.00800.00850.00900.00950.0100

Charge Q t


Table 3. Numerical results of RC circuit for

010, 10, 10, 0.01 0.5,0.75 0.999 .R L C V and and

t 0.5 0.75 0.999 2

LW LW LW CS

0.1 39.9924 10 39.9977 10 39.9994 10 39.9928 10

0.2 39.9853 10 39.9941 10 39.9980 10 39.7523 10

0.3 39.9790 10 39.9899 10 39.9958 10 39.5205 10

0.4 39.9733 10 39.9851 10 39.9929 10 39.2197 10

0.5 39.9679 10 39.9800 10 39.9893 10 38.8529 10

0.6 39.9629 10 39.9746 10 39.9850 10 38.4235 10

0.7 39.9580 10 39.9691 10 39.9803 10 37.9354 10

0.8 39.9534 10 39.9635 10 39.9750 10 37.3927 10

0.9 39.9489 10 39.9578 10 39.9693 10 36.8001 10

Solution of RL circuit

Finally, we consider RL circuit differential equation given in equation (4). RL circuit

consists only resistor, inductor and a non-variant voltage source are present in the

circuit and its differential equation is given as follows

' ( ) ( ) .LJ t RJ t V (67)

with 0(0)J J and V is the constant voltage source.


0( ) .RtLVL VLJ t I e

R R

(68)

Now, we analyse equation (67) using fractional calculus, we replace ' ( )J t by ( )D J t ,


order RL circuit and its differential equation as

( ) ( ) .R VD J t J tL L

(69)

Let 2RL

and 2 ,VL

then equation (69) become

2 2( ) ( ) .D J t J t (70)


We use equation (12) to approximate ( )D J t as

12 1

1 0

( ) ( ) ( ),

k MT

nm nmn m

D J t z t Z t

(71)

Integrating equation (71) with respect to t , over [0, ]t , we get

ˆ ˆ( ) (0) ( ),Tm mJ t J Z P t

(72)

Similarly equation (39), we can approximate 0J and 2 as

1

ˆ ˆ ˆ0 0 0 0 1(0) [ , ,..., ] ( )m m mJ J J J J t

and 2 2 2 2 1

ˆ ˆ ˆ1[ , ,..., ] ( ).m m m t

(73)

Substituting equations (71-73) in equation (70), we obtain

2 1 2 2 2 1

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ0 0 0 1 1( ) [ , ,..., ] ( ) ( ) [ , ,..., ] ( ),T Tm m m m m m m mZ t J J J t Z P t t

2 1 2 2 2 2 1

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ0 0 0 1 1( ) [ , ,..., ] ( ) ( ) [ , ,..., ] ( )T Tm m m m m m m mZ t J J J t Z P t t

2 1 2 2 2 2 1

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ0 0 0 1 1[ , ,..., ] [ , ,..., ]T Tm m m m m m m mZ J J J Z P

1

2 2 2 2 1 2

ˆ ˆ ˆ ˆ ˆ ˆ1 0 0 0 1[ , ,..., ] [ , ,..., ] . .Tm m m m m mZ J J J I P

(74)

Hence required

1

1 2 2 2 2 1 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ0 0 0 1 1 0 0 0 1( ) [ , ,..., ] [ , ,..., ] [ , ,..., ] . . ( ).m m m m m m m m m m mJ t J J J J J J I P P t

(75)

By solving the above system (74) of linear equations, we can find the value of

coefficient vector Z . Inserting the value of vector Z in equation (72), hence we obtain

the numerical results for different value of ,k M and . Here, solution obtained by the

proposed LWM approach for 0.25,0.75,0.999 and 2k and 3M is graphically show

in figure 4. As it can be clearly seen, for 0.999 and 2;k 3,M the fractional RL

circuit graph behave similar to the classical solution graph for 1 . It is show that the

proposed LWM approach is more close to the exact solution. Table 4 shows the

capability of the presented method and also shows that very high accuracies are

obtained for 2k and 3.M


Figure 4.Current versus time graph

010, 1, 10, 0.01 0.5,0.75 0.999 .R L V V and and

Table 4. Numerical results of RL circuit for

010, 1, 10, 0.01 0.5,0.75 0.999 .R L V V and and

x 0.5 0.75 0.999 1

LW LW LW CS

0.1 17.7867 10 16.6899 10 15.2978 10 16.3579 10

0.2 18.8787 10 18.8797 10 18.5076 10 18.6602 10

0.3 19.2552 10 19.7257 10 11.0111 10 19.5071 10

0.4 19.1906 10 19.4748 10 11.0108 10 19.8186 10

0.5 19.2404 10 19.4989 10 19.9895 10 19.9332 10

0.6 19.2929 10 19.5497 10 19.9938 10 19.9755 10

0.7 19.3397 10 19.5953 10 19.9967 10 19.9909 10

0.8 19.3808 10 19.6359 10 19.9983 10 19.9966 10

0.9 19.4162 10 19.6713 10 19.9986 10 19.9987 10

0.2 0.4 0.6 0.8 1.0t

0.80

0.85

0.90

Current

0.2 0.4 0.6 0.8 1.0t

0.7

0.8

0.9

Current

0.0 0.2 0.4 0.6 0.8 1.0t

0.6

0.7

0.8

0.9

1.0

Current

0.2 0.4 0.6 0.8 1.0t

0.7

0.8

0.9

1.0

Current


CONCLUSION

In this paper, the Legendre wavelet method (LWM) is applied to obtain approximate

analytical solutions of the fractional electrical circuit models. It can be concluded that,

LWM is very powerful and efficient technique for finding approximate solutions for

many real life problems [14-16]. The main advantage of the method is its fast

convergence to the solution. It has been shown in the theorem 6.1, by increasing k and

order m of Legendre polynomial, the Legendre wavelet series converges very fast.

See, in the figure 1, figure 2 and figure 3 approximate solutions graph behave as

similar to the classical solution but 1.999 the Caputo Fabrizio approach [1] shows

damping and behave very differently for LC circuit. As similar proposed method

present good approximated results for RC, LCR and RL at 0.999 but the Caputo

fractional derivative [1] graph for RL circuit coincide with the classical solution but

diverges to very large positive values as time progresses. The numerical results

obtained here, conform to its high degree of accuracy. Such analysis can be further

applied to other physical models to develop a better understanding of use of wavelets

in real life problems. The implementation of this method is a very easy acceptable and

valid. The solutions of the electrical circuit equations are presented graphically and in

tabular form.

ACKNOWLEDGMENT

The authors are very thankful to respected Dr. Sag Ram Verma, Department of

Mathematics and Statistics, Gurukula Kangri University, Haridwar, for

encouragement and support.

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Documents

An Application of Legendre Wavelet in Fractional …Ritu Arora* and N. S. Chauhan** *Department of Mathematics, Kanya Gurukul (Haridwar Campus) Gurukula Kangri University, Haridwar-249404