Solution of System of Linear Fractional Differential ...

American Journal of Mathematical Analysis, 2015, Vol. 3, No. 3, 72-84 Available online at http://pubs.sciepub.com/ajma/3/3/3 © Science and Education Publishing DOI:10.12691/ajma-3-3-3

Solution of System of Linear Fractional Differential Equations with Modified Derivative of Jumarie Type Uttam Ghosh1, Susmita Sarkar2, Shantanu Das3,4,5,* 1

Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India 2 Department of Applied Mathematics, University of Calcutta, Kolkata, India 3 Reactor Control Systems Design Section E & I Group B.A.R.C Mumbai India 4 Department of Physics, Jadavpur University Kolkata 5 Department of Appl. Mathematics, University of Calcutta *Corresponding author: [email protected]

Abstract Solution of fractional differential equations is an emerging area of present day research because such equations arise in various applied fields. In this paper we have developed analytical method to solve the system of fractional differential equations in-terms of Mittag-Leffler function and generalized Sine and Cosine functions, where the fractional derivative operator is of Jumarie type. The use of Jumarie type fractional derivative, which is modified Rieman-Liouvellie fractional derivative, eases the solution to such fractional order systems. The use of this type of Jumarie fractional derivative gives a conjugation with classical methods of solution of system of linear integer order differential equations, by usage of Mittag-Leffler and generalized trigonometric functions. The ease of this method and its conjugation to classical method to solve system of linear fractional differential equation is appealing to researchers in fractional dynamic systems. Here after developing the method, the algorithm is applied in physical system of fractional differential equation. The analytical results obtained are then graphically plotted for several examples for system of linear fractional differential equation.

Keywords: fractional calculus, Jumarie fractional derivative, Mittag-Leffler function, generalized sine and cosine function, fractional differential equations Cite This Article: Uttam Ghosh, Susmita Sarkar, and Shantanu Das, “Solution of System of Linear Fractional Differential Equations with Modified Derivative of Jumarie Type.” American Journal of Mathematical Analysis, vol. 3, no. 3 (2015): 72-84. doi: 10.12691/ajma-3-3-3.

1. Introduction

Method [12] Variational Iteration Method [13], Differential transform method [14]. Recently in [15] Ghosh et al developed analytical method for solution of linear fractional differential equations with Jumarie type derivative [7] in terms of Mittag-Leffler functions and generalized sine and cosine functions. This new finding of [15] has been extended in this paper to get analytical solution of system of linear fractional differential equations. In section 1.0 we have defined some important definitions of fractional derivative that is basic Riemann-Liouvellie (RL) fractional derivative, the Caputo fractional derivative, the Jumarie fractional derivative, the Mittag-Leffler function and generalized Sine and Cosine functions. In section 2.0 solution of system of fractional differential equations has been described and in section 3.0 an application of this method to physical system has been discussed.

The fractional calculus is a current research topic in applied sciences such as applied mathematics, physics, mathematical biology and engineering. The rule of fractional derivative is not unique till date. The definition of fractional derivative is given by many authors. The commonly used definition is the Riemann-Liouvellie (R-L) definition [1,2,3,4,5]. Other useful definition includes Caputo definition of fractional derivative (1967) [1,2,3,4,5]. Jumarie’s left handed modification of R-L fractional derivative is useful to avoid nonzero fractional derivative of a constant functions [7]. Recently in the paper [8] Ghosh et al proposed a theory of characterization of non-differentiable points with Jumarie type fractional derivative with right handed modification of R-L fractional 1.1. The Basic Definitions of Fractional derivative. The differential equations in different form of Derivatives and Some Higher Transcendental fractional derivatives give different type of solutions [1-5]. Functions: Therefore, there is no standard algorithm to solve a) Basic definitions of fractional derivative: fractional differential equations. Thus the solution and its i) Riemann- Liouvellie (R-L) definition interpretation of the fractional differential equations is a The R-L definition of the left fractional derivative is, rising field of Applied Mathematics. To solve the linear m +1 x and non-linear differential equations recently used 1 d  α m −α D f ( x ) =   a x ∫ ( x − τ ) f (τ )dτ (1.1) methods are Predictor-Corrector method [9], Adomain Γ( m + 1 − α )  dx  a decomposition method [2,10,11], Homotopy Perturbation

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Where: m ≤ α < m + 1, m : is positive integer . In particular when 0 ≤ α < 1 then x

1 d ( x − τ )−α f (τ )dτ . Γ(1 − α ) dx ∫

α = a Dx f ( x )

(1.2)

a

The definition (1.1) is known as the left R-L definition of the fractional derivative. The corresponding right R-L definition is α

x Db

f ( x )=

 d  −  Γ( m + 1 − α )  dx  1

m +1 b

∫ (τ − x)

m −α

f (τ ) dτ (1.3)

x

where: m ≤ α < m + 1. The derivative of a constant is obtained as non-zero using the above definitions (1.1)-(1.3) which contradicts the classical derivative of the constant, which is zero. In 1967 Prof. M. Caputo proposed a modification of the R-L definition of fractional derivative which can overcome this shortcoming of the R-L definition. ii) Caputo definition M. Caputo defines the fractional derivative in the following form [6] x

1 ( x − τ ) n −α −1 f ( n) (τ )dτ Γ(n − α ) ∫

C α = a Dx f ( x )

(1.4)

a

where: n − 1 ≤ α < n. In this definition first differentiate f ( x), n − times then integrate n − α times. The disadvantage of this method is that f ( x), must be differentiable n-times then the α -th order derivative will exist, where n − 1 ≤ α < n . If the function is non-differentiable then this definition is not applicable. Two main advantages of this method are (i) fractional derivative of a constant is zero (ii) the fractional differential equation of Caputo type has initial conditions of classical derivative type but the R-L type differential equations has initial conditions fractional type i.e. lim a Dαx −1 [ f ( x) ] = b1 .

x →a

This means that a fractional differential equation composed with RL fractional derivatives require concept of fractional initial states, sometimes they are hard to interpret physically [2]. iii) Modified definitions of fractional derivative: To overcome the fractional derivative of a constant, non-zero, another modification of the definition of left RL type fractional derivative of the function f ( x), in the interval[a,b] was proposed by Jumarie [7] in the form, that is following. (α )

fL

α

J ( x ) = a Dx f ( x )

 1 x −α −1  ∫ ( x − τ ) f (τ )dτ , α < 0.  Γ( −α ) a  x (1.5) d  1 ( x − τ ) −α ( f (τ ) − f ( a ) )dτ , 0 < α < 1 =  ∫  Γ(1 − α ) dx a  (m)  f (α − m ) ( x) , m ≤ α < m + 1.  

(

)

We consider that f ( x) − f (a ) = 0 for x < a . In (1.5), the first expression is just fractional integration; the second line is RL derivative of order 0 < α < 1 of offset function that is f ( x) − f (a ) . For α > 1 , we use the third line; that is first we differentiate the offset function with order 0 < (α − m) < 1 , by the formula of second line, and then apply whole m order differentiation to it. Here we chose integer m , just less than the real number α ; that is m ≤ α < m + 1 . The logic of Jumarie fractional derivative is that, we do RL fractional derivative operation on a new function by forming that new function from a given function by offsetting the value of the function at the start point. Here the differentiability requirement as demanded by Caputo definition is not there. Also the fractional derivative of constant function is zero, which is non-zero by RL fractional derivative definition. We have recently modified the right R-L definition of fractional derivative of the function f ( x), in the interval

[a,b] in the following form [8], (α )

α

f R ( x ) = Jx Db f ( x ) b  1 −α −1 − ∫ (τ − x) f (τ )dτ , α < 0.  Γ( −α ) x  b (1.6) 1 d  (τ − x ) −α ( f (b) − f (τ ) )dτ , 0 < α < 1 = − ∫  Γ(1 − α ) dx x  (m)  f (α − m ) ( x) , m ≤ α < m + 1.  

(

)

In the same paper [8], we have shown that both the modifications (1.5) and (1.6) give fractional derivatives of non-differentiable points their values are different, at that point, but we get finite values there, of fractional derivatives. Whereas in classical integer order calculus, where we have different values of right and left derivatives at non differentiable points in approach limit from left side or right side, but infinity (or minus infinity) at that point, where function is non-differentiable. But in case of Jumarie fractional derivative and right modified RL fractional derivative [8], there is no approach limit at the non-differentiable points, but a finite value is obtained at that non differentiable point of the function. The difference is that integer order calculus returns infinity or minus infinity at non-differentiable points, where as the Jumarie fractional derivative returns a finite number indicating the character of otherwise non-differentiable points in a function, in left sense or right sense. This has a significant application in characterizing otherwise nondifferentiable but continuous points in the function. However, the finite value of the non differentiable point after fractional differentiation depends on the interval length. In the rest of the paper J Dυ will represent Jumarie fractional derivative. b) Mittag-Leffler function and the generalized Sine and Cosine functions The Mittag-Leffler function was introduced by the Swedish mathematician Gösta Mittag-Leffler [17,18,19,20] in 1903. It is the direct generalization of exponential functions.

American Journal of Mathematical Analysis

The one parameter Mittag-Leffler function is defined (in series form) as: def ∞

zk

∑ Γ(1 + α k ) ,

Eα= ( z)

k =0

z ∈ , Re(α ) > 0.

In the solutions of FDE we use this series definition in MATLAB plots. One parameter Mittag-Leffler function in relation to few transcendental functions is as follows

We now find Jumarie fractional derivative of MittagLeffler function Eα (atα ) def atα a 2t 2α a3t 3α Eα (atα ) = + + + .......... 1+ Γ(1 + α ) Γ(1 + 2α ) Γ(1 + 3α )

Using Jumarie derivative of order α , with 0 ≤ α < 1 with start point as a = 0 for f (t ) = t nα , [15] that is

def z z2 z3 E1 ( z ) = + + + ................. 1+ Γ(1 + 1) Γ(1 + 2) Γ(1 + 3)

=1 + =e

z z 2 z3 + + + ................. 1! 2! 3!

useful identity.

z z 2 z3 + + + ................. 2! 4! 6! = cosh( z ) =1 +

The integral representation of the Mittag-Leffler function [17,18,19,20] is,

1 2π

t

e

∫ tα − z dt ,

z ∈ , Re(α ) > 0

C

and ends at −∞ and encloses the circles of disk | t |≤| z |1/α in positive sense :| arg(t ) |≤ π on C [17,18,19,20]. The two parameter Mittag-Leffler function (in series form) and its relation with few transcendental functions are as following def ∞

zk Eα , β ( z ) , = ∑ k =0 Γ( β + α k )

z , β ∈ ,

Eα ,1 ( z ) == Eα ( z ) e z z ∈ , ez −1 z sinh( z ) z

 atα a 2t 2α  + 1 +  Γ(1 + α ) Γ(1 + 2α )  J α  α  α 0 Dt  Eα ( at )  = D   3 3α + a t  ................. +  Γ(1 + 3α )    Γ(1 + α )a Γ(1 + 2α )a 2tα 0+ = + Γ(1)Γ(1 + α ) Γ(1 + 2α )Γ(1 + α ) +

α −1 t

Here the path of the integral C is a loop which starts

E2,2 ( z ) =

Γ(nα + 1) (t )α ( n −1) Dtα t nα  =   Γ (α (n − 1) + 1)

constant as zero J Dα [1] = 0 , we get the following very

def z z2 z3 E2 ( z ) = + + + ................. 1+ Γ(1 + 2) Γ(1 + 4) Γ(1 + 6)

E1,2 ( z ) =

J 0

for n = 1, 2,3,... ; and also using Jumarie derivative of

z

Eα ( z ) =

74

Re(α ) > 0

Γ(1 + 3α )a3t 2α + .......... Γ(1 + 3α )Γ(1 + 2α )

  atα a 2t 2α a3t 3α =+ + + a 1 + ..............   Γ(1 + α ) Γ(1 + 2α ) Γ(1 + 3α )    = aEα (atα ) Thus J α 0 Dt

 Eα (atα )  = aEα (atα )  

This shows that AEα (atα ) is a solution is a solution of the fractional differential equation [15] J α 0D

Re(α ) > 0

Where A is arbitrary constant. Therefore J α 0D

.

1 2π

∫ C

t α − β et tα − z

dt ,

z ∈ ,

Re(α ) > 0

α D= [1] 0,

y = Eα (atα ). The fractional Sine and Cosine functions are expressed as following [16], def

where the contour C is already defined, in the above paragraph. Using the modified definition, of fractional derivative of Jumarrie type, [7,8] we get J

y = ay

with y (0) = 1 has solution

The corresponding integral representation [17,18,19,20] of the two parameter Mittag-Leffler function is,

Eα= ,β ( z )

y = ay

0 < α < 1.

The Jumarie fractional derivative of any constant function is zero, unlike a non-zero value of fractional RL derivative of a constant.

= Eα (ixα ) cosα ( xα ) + i sinα ( xα ) def ∞

x 2kα

α cosα ( x= )

∑ (−1)k Γ(1 + 2α k )

α sinα ( x= )

∑ (−1)k

k =0 def ∞ k =0

x (2k +1)α Γ (1 + (1 + 2k )α )

The above series form of fractional Sine and Cosine are used to plots, in solutions. It can be easily shown that [15,16]

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American Journal of Mathematical Analysis

Dα sinα ( xα )  = cosα ( xα )   J α  α  D cosα ( x ) = − sinα ( xα ).  

where A and B are constants. Consider the system of linear fractional differential equations

J

Dα [ x= ] ax + by  . J α D [ y= ] cx + dy  J

This has been proved by the following term by term α

differentiation. The series presentation of cosα ( x ) is [15],

x 2α x 4α x 6α cosα ( xα ) = 1− + − + ..... Γ(1 + 2α ) Γ(1 + 4α ) Γ(1 + 6α ) Taking its term by term Jumarie fractional derivative of order α we get, J α 0 Dx

cosα ( xα )   

Γ(1 + 2α ) x 2α −α Γ(1 + 4α ) x 4α −α = + 0− Γ(1 + 2α )Γ(1 + α ) Γ(1 + 4α )Γ(1 + 3α ) −

Here a , b , c and d are constants, the operator J Dα is the Jumarie fractional derivative operator, call it for α convenience J Dα ≡ d α , and x and y are functions of t . dt

In matrix form we write the (2.1) in following way

 J Dα x   a b   x  d α  x  a b   x    = = also        .  J Dα y   c d   y  dtα  y   c d   y 

6α −α

Γ(1 + 6α ) x + ..... Γ(1 + 6α )Γ(1 + 5α )

The above system (2.1) can be written in the following form

0  ( J Dα − a ) x − by =  . J α −cx + ( D − d ) y = 0 

 xα  x3α = − − + .......  Γ(1 + α ) Γ(1 + 3α )  = − sinα ( xα ) Similarly we can get the expression for fractional derivative of Jumarie type of order α for sinα ( xα ) .

−c( J Dα − a ) x + ( J Dα − a )( J Dα − d ) y = 0

2. System of Linear Fractional Differential Equations Before considering the system of fractional differential equations we state the results [15] which arises in solving a single linear fractional differential equations composed by Jumarie derivative; we will be using the following theorems. Theorem 1: The fractional differential equation

(

α

D −a

)(

J

)

α

α

α

= y AEα (at ) + BEα (bt ) where A and B are constants. Theorem 2: The fractional differential equation J

J

(

)

D 2α [ y ] − 2a J Dα [ y ] + a 2 y = 0

D

2α

[ y ] − (a + d ) (

J

α

D

0 [ y ]) + (ad − bc) y =

a + d = λ1 + λ2

and

ad − bc = λ1λ2

then the equation (2.3) can be re-written as, J

(

)

D 2α [ y ] − (λ1 + λ2 ) J Dα [ y ] + λ1λ2 y = 0 (2.4)

For λ1 , λ2 real and distinct, solution of the equation (2.4) can be written in the form (from Theorem 1)

= y A1Eα (λ1tα ) + B1Eα (λ2tα )

J

Dα [ y= ] cx + dy

Dα  A1Eα (λ1tα ) + B1Eα (λ2tα )    α cx + d  A1Eα (λ1t ) + B1Eα (λ2tα )  = where A and B are constants.   Theorem 3: Solution of the fractional differential α α cx  A1λ1Eα (λ1t ) + B1λ2 Eα (λ2t  = equation   α α  J 2α J α 2 α  − d A1Eα (λ1t ) + B1Eα (λ2t ) D [ y ] − 2a D [ y ] + ( a + b ) y = 0  

(

y

J

)

is of the form

( Eα (atα ) )( A cosα (btα ) + B sinα (btα ) )

(2.5)

Again from second equation of (2.1) we get after putting the value of y the following,

has solution of the form

= y ( Atα + B ) Eα (atα )

(2.3)

Equation (2.3) is a linear fractional order differential equation (with order 2α ), with Jumarie derivative operator. Let

D − b y (t ) = 0

has solution of the form

(2.2)

Operating ( J Dα − a ) on both sides of the second equation of (2.2) we get the following steps.

− cby + ( J Dα − a )( J Dα − d ) y = 0

J

(2.1)

cx = A1 (λ1 − d ) Eα (λ1tα ) + B1 (λ2 − d ) Eα (λ2tα ) From above we obtain the following

American Journal of Mathematical Analysis

x=

1  A (λ − d ) E (λ t α ) + B (λ 1 2 α 1 c 1 1

− d ) Eα (λ2tα )  

= A2 Eα (λ1tα ) + B2 Eα (λ2tα ) A1 (λ1 − d ) B1 (λ2 − d ) A2 = B2 ; = c c

Thus the solution of the system of fractional differential equation can be written as, (2.6)

A1 , B1 are arbitrary constants in above derivation, and λ1 + λ2 =a + d , λ1λ= 2 ad − bc . Thus the solution can be written in the form  B2   x   A2  α α =  y   A  Eα (λ1t ) +  B  Eα (λ2t )    1  1

76

 x α α = α (λ1t ) + c2 BEα (λ2 t )  y  c1 AE      A1  A  Where A = and B  2  are the arbitry constants. =  B   1   B2  Example: 1

Dα= x 2 x + y  , J α D y= x + 2 y  0 < α ≤ 1 with x= (0) 2, y= (0) 1. J

(2.7)

Again to solve the system of fractional differential equation (2.1) we use the method similar to as used in classical differential equations.

Let

α α x AE = α (λ t ) and y BEα (λ t ). Since J Dα [ x(t ) ] = λ x(t ) has solution in the form = be the solution of the above differential equation. x(t ) = AEα (λ tα ), A is arbitary constant. [15], putting Substituting this in the above equations we get,

α = x AE = and y BEα (λ tα ) α (λ t )

A(2 − λ ) + B = 0  A + (2 − λ ) B = 0

in (2.1) we get the following J

Dα x − ax − by = 0

J

Dα  AEα (λ tα )  − aAEα (λ tα ) − bBEα (λ tα ) = 0  

(2.10)

The corresponding characteristic equation is,

0 Aλ Eα (λ tα ) − AaEα (λ tα ) − bBEα (λ tα ) = A(λ − a) − Bb = 0

2−λ 1 giving λ 1,3. = 0= 1 2−λ

Putting λ = 1 in (2.8) we get A + B = 0 , taking A = 1 we get B = −1 and putting λ = 3 in (2.8) we get A = B , taking A = 1 we get B = 1 . Hence the solutions are,

J

Dα y − cx − dy = 0

J

Dα  BEα (λ tα )  − cAEα (λ tα ) − dBEα (λ tα ) = 0  

x1 = Eα (tα ) , y1 = − Eα (tα )

α α and x2 E= = Bλ Eα (λ tα ) − AcEα (λ tα ) − dBEα (λ tα ) = 0 α (3t ) , y2 Eα (3t ) . Thus the general solution is 0 − Ac + B (λ − d ) =

0  A(λ − a ) − bB =  0 −cA + (λ − d ) B =

(2.8)

Eliminating A and B from (2.8) we get,

a−λ b = 0 giving λ 2 − (a + d )λ + (ad = − bc) 0 (2.9) c d −λ which is known as the characteristic equation with roots λ1 and λ2 , also termed as eigen-values. Three cases may arises i) The roots λ1 and λ2 are real and distinct. ii) The roots are real and equal i.e. λ= 1 λ= 2 λ (say) . iii) The roots are complex i.e. of the form λ1 , λ2= p ± iq (say) . Case –I For real and distinct roots λ1 and λ2 , we write α α x1 A= = 1 Eα (λ1t ) , y1 B1 Eα (λ1t ) α α and x2 A= = 2 Eα (λ2 t ) , y2 B2 Eα (λ2 t )

The solution of (2.1) is

x = x1 + x2 = A1Eα (λ1tα ) + A2 Eα (λ2tα ) y = y1 + y2 = B1Eα (λ1tα ) + B2 Eα (λ2 tα )

x c1Eα (tα ) + c2 Eα (3tα ) = y= −c1Eα (tα ) + c2 Eα (3tα ). Where c1, c2 are arbitrary constants. Using the initial condition

= x(0) 2,= y (0) 0. we get c= 1 c= 2 1. Thus the required solution is,

x Eα (tα ) + Eα (3tα ) = y= − Eα (tα ) + Eα (3tα ) Figure 1 represents the graphical presentation of x(t ) and y (t ) when the eigen-values of the system of differential equations are positive. Numerical simulation shows that x(t ) and y (t ) both grow rapidly with decrease of order of derivative i.e. as α decreases from 1 towards 0. Example: 2

Dα x = −2 x + y  (0) 2, y= (0) 1  , 0 < α ≤ 1 with x= J α D y= x − 2 y  J

α α Let x AE = = α (λ t ) and y BEα (λ t )

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American Journal of Mathematical Analysis

be the solution of the above differential equation. Substituting this in the above equations we get,

A(−2 − λ ) + B =0  . A + (−2 − λ ) B =0 

Figure 1. Numerical simulation of the solutions of fractional differential equation in Example-1 for x(t) and y(t) for different values of α = 0.2, 0.4, 0.6, 0.8 and 1.0. Figure (i) & (k) and figure (j) & (l) represents the same figure only length of x-axis is changed here to represent the prominent initial values

The corresponding characteristic equation is,

−2 − λ 1

1 =0 −2 − λ

giving

λ =−1, −3 .

Putting λ = −1 in (2.8) we get A − B = 0 , taking A = 1 we get B = 1 and putting λ = −3 in (2.8) we get A = − B , taking A = 1 we get B = −1 . Hence the solutions are,

x1 = Eα (−tα ) , y1 = Eα (−tα )

and x2 = Eα (−3tα ) , y2 = − Eα (−3tα ) . Thus the general solution is

x= c1Eα (−tα ) + c2 Eα (−3tα ) y= c1Eα (−tα ) − c2 Eα (−3tα ). Where c1, c2 are arbitrary constants. Using the initial condition x(0) = 2 , y (0) = 0 we get

c= 1 c= 2 1.

American Journal of Mathematical Analysis

78

x = Eα (−tα ) + Eα (−3tα ), y = Eα (−tα ) − Eα (−3tα ).

Thus the required solution is,

Figure 2. Numerical simulation of the solutions of fractional differential equation in Example-2 for and 1.0

Figure 2 represents the graphical presentation of x(t ) and y (t ) when the eigen-values of the system of differential equations are negative. Numerical simulation shows that x(t ) and y (t ) are both decaying rapidly with decrease of order of derivative i.e. as α decreases from 1 to 0.6. For negative eigen-values the solutions are decaying asymptotically to zero. Case –II The roots of the equation (2.3) are complex and are of the form λ1 , λ2= p ± iq then the solution x(t ) and y (t ) can be written in the form (from Theorem 3),

x= ( A1 + iA2 )  Eα ( p + iq )tα     and y = ( B1 + iB2 ) Eα ( p + iq)tα  .  

x(t ) and y (t ) for different Values of α = 0.6, 0.8

Similarly we get by repeating the above steps for y as follows

(

 B cos (qtα ) − B sin (qtα ) 1 2 α α

)

 . α α   +i B2 cosα (qt ) + B1 sinα (qt ) 

y =  Eα ( ptα )    

(

)

In above obtained expressions for x and y , we have complex quantity as u + iv . This may be also considered as linear combination of u and v considering i a constant. Therefore, we can say that x is linear combination of x1 (the real part of obtained complex x ), and x2 (the imaginary part of obtained complex x ). Similarly we have y as linear combination of y1 and y2 [21]. With this argument we write the following

Using the definition of Mittag-Leffler = function and x1  Eα ( ptα )   A1 cosα (qtα ) − A2 sinα (qtα )     fractional cosine and sine functions, that is α α α x2  Eα ( pt )   A2 cosα (qt ) + A1 sinα (qt )  = def    α α α = Eα (ix ) cosα ( x ) + i sin( x ) α α α y1  Eα ( pt )   B1 cosα (qt ) − B2 sinα (qt )  =    we get, α  α α   y2 E ( pt ) B2 cosα (qt ) + B1 sinα (qt ) =  α   x= ( A1 + iA2 )  Eα ( p + iq )tα    It can be shown that ( x1 , y1 );( x2 , y2 ) are solutions of = ( A1 + iA2 )  Eα ( ptα )   Eα (iq )tα     the given equations (2.1). Thus the general solution in this case can be written in ( A1 + iA2 )  Eα ( ptα )  cosα (qtα ) + i sinα (qtα )  =    the form as in classical integer order differential equation [[21], pp.305].  A cos (qtα ) − A sin (qtα )  1 2 α α  α  The linear combination of x1 and x2 , gives x and  . = Eα ( pt )    α α  linear combination of y1 and y2 gives y , which is  +i A2 cosα (qt ) + A1 sinα (qt )  represented as following

(

(

)

)

79

American Journal of Mathematical Analysis

(

)  ( )  M B cos (qtα ) − B sin (qtα )  (1 α ) . 2 α α   y = Eα ( pt )    α α   + N ( B2 cosα (qt ) + B1 sinα (qt ) ) 

y = (−1 + 3i )  Eα (2 + 3i )tα    α = (−1 + 3i )  Eα (2t )  cosα (3tα ) + i sinα (3tα )      − cos (3tα ) − 3sin (3tα )  α α . =  Eα (2tα )     α α   +i 3cosα (3t ) − sinα (3t ) 

 M A cos (qtα ) − A sin (qtα ) 1 2 α α α   x = Eα ( pt )    α α  + N A2 cosα (qt ) + A1 sinα (qt )

(

) )

(

Hence With M , N as arbitrary constants, determined from initial states. We demonstrate by following examples. y1 =  Eα (2tα )   − cosα (3tα ) − 3sinα (3tα )     Example: 3 α  α  and y2 E (2t ) 3cosα (3t ) − sinα (3tα )  . = J α  α   D = x 3 x + 2 y  ,  J α Therefore the general solution is linear combination of D y= −5 x + y  x1 , x2 for x and linear combination of y1 , y2 for y , and 0 < α ≤ 1 with x= (0) 2, y= (0) 1. we write the following Let x  Eα (2tα )   2 M cosα (3tα ) + 2 N sinα (3tα )  =    α α = x AE = α (λ t ) and y BEα (λ t ) α α  M − cos (3t ) − 3sin (3t )  α α be the solution of the above differential equation, then . y =  Eα (2tα )     putting in the above equation we get, α α   + N 3cosα (3t ) − sinα (3t )  A(3 − λ ) + 2 B = 0  (2.11)  where M and N are arbitrary constants. Using initial 0 −5 A + (1 − λ ) B = conditions x(0) = 2, y (0) = 1 we get 2 M = 2, and The corresponding characteristic equation is, 3N − M = 1 giving = M 1,= N 2. 3−λ 2 Hence the required solution is, λ= 2 ± 3i . = 0 giving −5 1 − λ x  Eα (2tα )   2 cosα (3tα ) + 4sinα (3tα )  =    Putting λ= 2 + 3i in (2.8) by putting a = 3, b = 2, y  Eα (2tα )  cosα (3tα ) − 5sinα (3tα )  . = c = −5, d = 1 we get the following   

(

A(λ − a ) − 2 B = 0 A(2 + 3i − 3) + 2 B = 0 2 B = (−1 + 3i ) A −cA + (λ − d ) B = 0 5 A + (2 + 3i − 1) B = 0 5 A =−(1 + 3i ) B 5 A × (1 − 3i ) =−(1 + 3i ) × (1 − 3i ) B −10 B 5(1 − 3i ) A = 2 B = (−1 + 3i ) A

Numerical simulation in Figure 3 shows that for α = 0.6, 0.8 and 1.0 after the initiation (t = 0) of the system

x(t ) and y (t ) both oscillate, Period of oscillation changes with decrease of α . Case-III In this case roots of the equations (2.9) being equal, that is λ= 1 λ= 2 λ . Then one solution will be of the form α = x1 AE = and y1 BEα (λ tα ) α (λ t )

The (2.8) returns the same answer that is 2= B (3i − 1) A .We choose here A = 2 , so B= 3i − 1 , as we obtained one equation with two unknowns. Thus B= 3i − 1 and A = 2 can be taken as one of the trial solution of the above. Hence the solution is,

(

= x 2 Eα (2 + 3i )t

α

)

2  Eα (2tα )  cosα (3tα ) + i sinα (3tα )    

(

and the other solution will be

x2 = ( A1tα + A2 ) Eα (λ tα ) and y2 = ( B1tα + B2 ) Eα (λ tα ). Hence the general solution is,

x c1 AEα (λ tα ) + c2 ( A1tα + A2 ) Eα (λ tα ) = y c1BEα (λ tα ) + c2 ( B1tα + B2 ) Eα (λ tα ) =

where A,A1 ,A2 ,B,B1 ,B2 ,c1 ,c2 are arbitry constants Example 4: Dα= x 4 x − y  (0) 2, y= (0) 1 (2.12)  , 0 < α ≤ 1 with x= J α D y= x + 2 y  J

Thus

x1 = 2  Eα (2tα )  cosα (3tα )     α and x2 = 2  Eα (2t )  sinα (3tα )    

) )

Let

α = x AE = and y BEα (λ tα ) α (λ t ) Similarly the solution for y can be written in the form

American Journal of Mathematical Analysis

80

A(4 − λ ) − B = 0 . A + (2 − λ ) B = 0

be the solutions of the above differential equation, then putting in the above equation we get

(2.13)

Figure 3. Numerical simulation of the solutions of fractional differential equation in Example-3 for x (t ) and y (t ) for different Values of α = 0.6, 0.8 and 1.0. Figure (e) & (f) and Figure (g) & (h) represents the same figure only length of x-axis is changed here to represent the prominent initial values

Eliminating A and B as in previous examples, we get,

4−λ 1 giving = 0= λ 3,3. 1 2−λ For λ = 3 from equation (2.4) we get A= B= 1 . Thus α

α

= x1 E= and y1 Eα (3t ) α (3t ) is one solution of the equation. The second solution is as in classical integer order differential equation [[21], pp.307]

x2 ( A1tα + A2 )  Eα (3tα )  =   α  and y2 ( B1t + B2 ) Eα (3tα )  =  

α -th order differentiating for above x and y we obtain the following

= Dα x 3( A1tα + A2 ) Eα (3tα ) + Γ(1 + α ) A1Eα (3tα )

J

and

= Dα y 3( B1tα + B2 ) Eα (3tα ) + Γ(1 + α ) B1Eα (3tα ).

J

Putting the above obtained result in the given equation (2.12) we get,

3( A1tα + A2 ) Eα (3tα ) + Γ(1 + α ) A1Eα (3tα ) = Eα (3tα )(4 A1 − B1 )tα + Eα (3tα )(4 A2 − B2 ) 3( B1tα + B2 ) Eα (3tα ) + Γ(1 + α ) B1Eα (3tα ) = Eα (3tα )( A1 + 2 B1 )tα + Eα (3tα )( A2 + 2 B2 ) 3( A1tα + A2 ) + Γ(1 + α )= A1 (4 A1 − B1 )tα + (4 A2 − B2 ) α 3( B1tα + B2 ) + Γ(1 + α ) B= 1 ( A1 + 2 B1 )t + ( A2 + 2 B2 ).

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American Journal of Mathematical Analysis

Comparing the coefficients and simplifying we get

A1 = B1 , A2 − B2 = A1Γ(1 + α ) = B1Γ(1 + α ) for simple non-zero values we take

A1 =B1 =1, A2 − B2 =Γ(1 + α ),

for simplicity take B2 = 0 then A2 =Γ(1 + α ). Thus the other solution is

x tα + Γ(1 + α )   Eα (3tα )  and y =tα  Eα (3tα )  . =     

Figure 4. Numerical simulation of the solutions of fractional differential equation in Example-3 for x (t ) and y (t ) for different Values of α = 0.2, 0.4, 0.6, 0.8 and 1.0. Figure (i) & (k) and Figure (j) & (l) represents the same figure only length of x-axis is changed here to represent the prominent initial values

Hence the general solution can be written in the form as in classical integer order differential equation [[21], pp.307]

x c1Eα (3tα ) + c2 tα + Γ(1 + α )  Eα (3tα ) =   and y c1Eα (3tα ) + c2tα Eα (3tα ). =

American Journal of Mathematical Analysis

Putting the initial condition.

= x(0) 2= and y (0) 1 and solving we get

1 = c1 1= and c2 . Γ(1 + α )

3. Application of the Above Formulation in Real Life Problem: Consider the following fractional damped oscillator, formulated by Jumarie fractional derivative J

Hence the solution is,

and y Eα (3t ) + =

1 tα E (3tα ). α Γ (1+α )

(

)

D 2α [ x ] + 2a J Dα [ x ] + bx = 0.

(3.1)

α Let J Dα [ x ] ≡ d αx = y then the given equation reduce

x Eα (3tα ) + Γ (11+α ) tα + Γ(1 + α )   Eα (3tα )  =    α

82

dt

to the following system of equation

 dα y J α D [ y] = = −2ay − bx  Numerical simulation in Figure 4 shows that x(t ) and  dtα (3.2) . y (t ) start from 2 and 1 respectively; and both of them α d x  J α D= y start to grow. This time interval to grow decreases as α [ x] =  α dt  increases. Moreover growth of x(t ) and y (t ) is higher for The above system of equation can be written in the is for lower α values. As α increases growth rate of the form solution decreases. This implies for lower values of α , x(t ) and y (t ) grow initially slowly. Once they start to  dα x  grow, their growth rate is very high whereas for higher  J Dα x   dtα   0 1   x    =  = α values x(t ) and y (t ) start to grow sooner but their   . α J α  D y   d y   −b −2a   y  growth rate is low.  dtα  From the above discussion of the three cases one can state a theorem in the following form Let Theorem 4: α α x AE = The solutions of the system of differential equations = α (λ t ) and y BEα (λ t ) a b  x J α D X AX , = = Where A = ,X    c d   y

be solutions of the differential equations. Then

A(0 − λ ) + B = 0  . −bA + (−2a − λ ) B =0 

are Case (i)

= X c1 AEα (λ1tα ) + c2 BEα (λ2tα ),   Where A, B, c1and c2 are the arbitry constants,   λ1 + λ2 =a + d ,

λ1λ= 2 ad − bc, λ1 ≠ λ2 . Case (ii)

= X c1 AEα (λ tα ) + c2 ( B1tα + B2 ) Eα (λ tα ),    Where A, B1 , B2 , c1and c2 are the arbitry constants,    2λ= a + d , 2 λ= ad − bc.

Case (iii)

(3.3)

For the above system of equation the auxiliary equation is,

0−λ 1 = 0 −b −2a − λ

giving = λ 2 + 2aλ + b 0.

Here the discriminant is 4( a 2 − b). We consider the case when a 2 − b < 0. then the eigen-values are

λ1 , λ2 = p ± iq where p = −a, q =− b a2 . λ= p + iq we get A( p + iq ) = B , we can take the solution in the form = A 1, B = ( p + iq ) . The general solution will be of the form, Then

from

(3.3)

putting

 A1 cosα (qtα ) − A2 sinα (qtα )  x Eα ( p + iq )tα )  =     α α   ( cos ( ) sin ( )) A qt A qt + + α 2 1 α α  Eα ( pt )  cosα (qtα ) + sinα (qtα )  = X =  Eα ( ptα )          cos ( α ) − sin ( α )  B1 α qt B2 α qt α  α     x1 = Eα ( pt ) cosα (qt )     +( B2 cosα (qtα ) + B1 sinα (qtα ))  α and x2 =  Eα ( pt )  sinα (qtα )  . Where A1 , A2 , B1 , B2 are the arbitry constants,    2 p= a + d , Thus the general solution can be written in the form as in classical integer order differential equation [[21], pp.305] p 2 + q 2 = ad − bc.

83

American Journal of Mathematical Analysis

x  Eα ( ptα )  C1 cosα (qtα ) + C2 sinα (qtα )     Where C1 and C2 are arbitrary constants.

Here x(0) = 2 and J Dα x(0) = 1 and solving we get C1= 1 andC2= (1 − 2 p) / q. Hence the solution is

1− 2 p = x  Eα ( ptα )  cosα (qtα ) + q sinα (qtα )  .   

Figure 5. Numerical simulation of the solutions of fractional differential equation in equation (3.1) for different Values α = 0.2, 0.4, 0.6, 0.8, and 1.0. for b=2 and for a=0.1 & a=0

The numerical simulation of the solutions of the differential equation (3.1) has shown in Figure 5, for b = 2 , left hand figures for a = 0.1 and right hand figures for a = 0 . Figures 4 (a) and (b) are drawn for α = 1 , it is clear from the figure in presence of damping the amplitude of the oscillation decreases with time. Figures 4 (c) and (d) are drawn for α = 0.8 , it is clear from the figure in both the cases the amplitude of the oscillation decreases with time and ultimately amplitude tends to zero. Figures 4 (e) & (f) and (g) & (h) and (i) & (j) are drawn for α = 0.6,

0.4 and 0.2 respectively, it is clear from the figures with decrease of order of derivative the oscillator losses the oscillating behavior.

4. Conclusions The system of fractional differential equation arises in different applications. Here we develop an algorithm to solve the system of fractional differential equations with

American Journal of Mathematical Analysis

modified fractional derivative (with Jumarie’s fractional derivative formulation) in terms of Mittag-Leffler function and the generalized Sine and Cosine functions. From the numerical simulations it is observed that the growing or decaying of the solutions is fast in fractional order derivative case compare to the integer order derivative. The use of this type of Jumarie fractional derivative gives a conjugation with classical methods of solution of system of linear integer order differential equations, by usage of Mittag-Leffler and generalized trigonometric functions that we have demonstrated here in this paper. The ease of this method and its conjugation to classical method to solve system of linear fractional differential equation is appealing to researchers in fractional dynamic systems.

[6] [7]

[8]

[9] [10] [11]

Acknowledgement Acknowledgments are to Board of Research in Nuclear Science (BRNS), Department of Atomic Energy Government of India for financial assistance received through BRNS research project no. 37(3)/14/46/2014BRNS with BSC BRNS, title “Characterization of unreachable (Holderian) functions via Local Fractional Derivative and Deviation Function”.

[12] [13]

[14] [15]

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