Solution of System of Linear Fractional Differential Equations with Modified Derivative of Jumarie Type

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1 American ournal of Mathematical Analysis, 015, Vol 3, No 3, 7-84 Available online at Science and Education Publishing DOI:101691/ajma Solution of System of Linear Fractional Differential Equations with Modified Derivative of umarie Type Uttam Ghosh 1,*, Susmita Sarkar, Shantanu Das 3,4,5 1 Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India Department of Applied Mathematics, University of Calcutta, Kolkata, India 3 Reactor Control Systems Design Section E & I Group BARC Mumbai India 4 Department of Physics, adavpur University Kolkata 5 Department of Appl Mathematics, University of Calcutta *Corresponding author: uttam_math@yahoocoin 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 generalied Sine and Cosine functions, where the fractional derivative operator is of umarie type The use of umarie type fractional derivative, which is modified Rieman-Liouvellie fractional derivative, eases the solution to such fractional order systems The use of this type of umarie fractional derivative gives a conjugation with classical methods of solution of system of linear integer order differential equations, by usage of Mittag-Leffler and generalied 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 eamples for system of linear fractional differential equation Keywords: fractional calculus, umarie fractional derivative, Mittag-Leffler function, generalied 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 umarie Type American ournal of Mathematical Analysis, vol 3, no 3 (015: 7-84 doi: /ajma Introduction 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,,3,4,5] Other useful definition includes Caputo definition of fractional derivative (1967 [1,,3,4,5] umarie s left handed modification of R-L fractional derivative is useful to avoid nonero fractional derivative of a constant functions [7] Recently in the paper [8] Ghosh et al proposed a theory of characteriation of non-differentiable points with umarie type fractional derivative with right handed modification of R-L fractional derivative The differential equations in different form of fractional derivatives give different type of solutions [1-5] Therefore, there is no standard algorithm to solve fractional differential equations Thus the solution and its interpretation of the fractional differential equations is a rising field of Applied Mathematics To solve the linear and non-linear differential equations recently used methods are Predictor-Corrector method [9], Adomain decomposition method [,10,11], Homotopy Perturbation Method [1] 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 umarie type derivative [7] in terms of Mittag-Leffler functions and generalied sine and cosine functions This new finding of [15] has been etended in this paper to get analytical solution of system of linear fractional differential equations In section 10 we have defined some important definitions of fractional derivative that is basic Riemann-Liouvellie (RL fractional derivative, the Caputo fractional derivative, the umarie fractional derivative, the Mittag-Leffler function and generalied Sine and Cosine functions In section 0 solution of system of fractional differential equations has been described and in section 30 an application of this method to physical system has been discussed 11 The Basic Definitions of Fractional Derivatives and Some Higher Transcendental Functions: a Basic definitions of fractional derivative: i Riemann- Liouvellie (R-L definition The R-L definition of the left fractional derivative is, m+ 1 1 d m ad f( = ( f( d Γ ( m + 1 d a τ τ τ (11

2 73 American ournal of Mathematical Analysis Where: m < m+ 1, m: is positiveinteger In particular when 0 < 1 then 1 d ad f( = ( τ f( τ dτ Γ(1 d (1 a The definition (11 is known as the left R-L definition of the fractional derivative The corresponding right R-L definition is m+ 1 b 1 d m Db f( = ( f( d Γ ( m + 1 d τ τ τ (13 where: m < m+ 1 The derivative of a constant is obtained as non-ero using the above definitions (11-(13 which contradicts the classical derivative of the constant, which is ero 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] C 1 n 1 ( n ad f( = ( τ f ( τ dτ Γ( n (14 a where: n 1 < n In this definition first differentiate f(, n times then integrate n times The disadvantage of this method is that f(, must be differentiable n-times then the -th order derivative will eist, where n 1 < 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 ero (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 ie lim 1 ad [ f( ] = b1 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 [] iii Modified definitions of fractional derivative: To overcome the fractional derivative of a constant, non-ero, another modification of the definition of left R- L type fractional derivative of the function f(, in the interval[ ab, ] was proposed by umarie [7] in the form, that is following ( L f ( = D f( a 1 1 ( τ f( τ dτ, < 0 Γ ( a 1 d (15 = ( τ ( f( τ f( a dτ,0< < 1 Γ(1 d a ( m ( m ( f (, m < m+ 1 We consider that f( f( a = 0 for < a In (15, the first epression is just fractional integration; the second line is RL derivative of order 0< < 1 of offset function that is f( f( a For > 1, we use the third line; that is first we differentiate the offset function with order 0 < ( 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 ; that is m < m+ 1 The logic of umarie 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 ero, which is non-ero by RL fractional derivative definition We have recently modified the right R-L definition of fractional derivative of the function f(, in the interval [ ab, ] in the following form [8], ( R f ( = D f( b b 1 1 ( τ f( τ dτ, < 0 Γ ( b 1 d (16 = ( τ ( f( b f( τ dτ,0 < < 1 Γ(1 d ( m f (, m < m+ 1 ( m ( In the same paper [8], we have shown that both the modifications (15 and (16 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 umarie 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 umarie 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 characteriing 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 D υ will represent umarie fractional derivative b Mittag-Leffler function and the generalied Sine and Cosine functions The Mittag-Leffler function was introduced by the Swedish mathematician Gösta Mittag-Leffler [17,18,19,0] in 1903 It is the direct generaliation of eponential functions

3 American ournal of Mathematical Analysis 74 The one parameter Mittag-Leffler function is defined (in series form as: def k E ( =,, Re( > 0 k= 0 Γ (1 + k 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 def 3 3 E1 ( = Γ (1+ 1 Γ (1+ Γ (1+ 3 = !! 3! = e def 3 3 E ( = Γ (1 + Γ (1 + 4 Γ (1 + 6 = ! 4! 6! = cosh( The integral representation of the Mittag-Leffler function [17,18,19,0] is, 1 t 1 t e E ( = dt,, Re( 0 π > t C Here the path of the integral C is a loop which starts 1/ and ends at and encloses the circles of disk t in positive sense : arg( t π on C [17,18,19,0] The two parameter Mittag-Leffler function (in series form and its relation with few transcendental functions are as following def k Eβ, ( =, k= 0 Γ ( β + k, β, Re( > 0 E,1( = E( = e, Re( > 0 e 1 E1,( = sinh( E,( = The corresponding integral representation [17,18,19,0] of the two parameter Mittag-Leffler function is, β t 1 t e Eβ, ( = dt,, Re( 0 π > t C where the contour C is already defined, in the above paragraph Using the modified definition, of fractional derivative of umarrie type, [7,8] we get [ ] D 1 = 0, 0 < < 1 The umarie fractional derivative of any constant function is ero, unlike a non-ero value of fractional RL derivative of a constant We now find umarie fractional derivative of Mittag- Leffler function E ( at def 33 at a t a t E ( at = Γ (1 + Γ (1 + Γ (1 + 3 Using umarie derivative of order, with 0 < 1 n with start point as a = 0 for f( t = t, [15] that is for n = 1,,3, ( 1 ( 1 0 n n D t t Γ + ( t n = Γ + constant as ero [ 1] 0 useful identity ( ( n 1 1 ; and also using umarie derivative of D =, we get the following very at a t 1+ + (1 (1 0 D t E ( at Γ + Γ + = D 33 at + + (1 3 Γ + Γ (1 + a Γ (1+ at = Γ(1 Γ (1 + Γ (1+ Γ (1 + 3 Γ (1+ 3 at + + Γ (1+ 3 Γ (1+ 33 at a t a t = a Γ (1 + Γ (1 + (1 3 Γ + = ae ( at Thus 0 D t E ( at = ae( at This shows that AE ( at is a solution is a solution of the fractional differential equation [15] 0 D y = ay Where A is arbitrary constant Therefore with y (0 = 1has solution 0 D y = ay y = E ( at The fractional Sine and Cosine functions are epressed as following [16], def E( i = cos ( + isin ( def k k cos ( = ( 1 k= 0 Γ (1+ k def (k+ 1 k sin ( = ( 1 k= 0 Γ + + ( 1 (1 k The above series form of fractional Sine and Cosine are used to plots, in solutions It can be easily shown that [15,16]

4 75 American ournal of Mathematical Analysis D sin ( = cos ( D cos ( = sin ( This has been proved by the following term by term differentiation The series presentation of cos ( is [15], 4 6 cos ( = Γ (1 + Γ (1 + 4 Γ (1 + 6 Taking its term by term umarie fractional derivative of order we get, 0 D cos ( 4 Γ (1 + Γ (1 + 4 = 0 + Γ (1 + Γ (1 + Γ (1 + 4 Γ ( Γ ( Γ (1 + 6 Γ ( = + Γ (1 + Γ (1 + 3 = sin ( Similarly we can get the epression for fractional derivative of umarie type of order for sin ( 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 umarie derivative; we will be using the following theorems Theorem 1: The fractional differential equation has solution of the form ( D a( D b yt ( = 0 y = AE( at + BE( bt where A and B are constants Theorem : The fractional differential equation ( [ ] [ ] D y a D y + a y = 0 has solution of the form y = ( At + B E ( at where A and B are constants Theorem 3: Solution of the fractional differential equation is of the form ( [ ] [ ] D y a D y + ( a + b y = 0 ( ( ( cos ( sin ( y = E at A bt + B bt where A and B are constants Consider the system of linear fractional differential equations [ ] [ ] D = a + by (1 D y = c + dy Here a, b, c and d are constants, the operator D is the umarie fractional derivative operator, call it for convenience D d, and and y are functions of t dt In matri form we write the (1 in following way D a b d a b = also D y c d y = dt y c d y The above system (1 can be written in the following form ( D a by ( = 0 ( c + D d y = 0 Operating ( D a on both sides of the second equation of ( we get the following steps c( D a + ( D a( D d y = 0 cby + ( D a( D d y = 0 ( [ ] [ ] D y ( a + d D y + ( ad bc y = 0 (3 Equation (3 is a linear fractional order differential equation (with order, with umarie derivative operator Let a + d = λ1+ λ and ad bc = λλ 1 then the equation (3 can be re-written as, ( [ ] [ ] ( λ1+ λ + λλ 1 = 0 (4 D y D y y For λ1, λ real and distinct, solution of the equation (4 can be written in the form (from Theorem 1 y= AE 1 ( λ1t + BE 1 ( λt (5 Again from second equation of (1 we get after putting the value of y the following, [ ] D y = c + dy D AE 1 ( λ1t + BE 1 ( λt = c + d A 1E( λ1t + B1E ( λt c = A 1λ1E ( λ1t + B1λE ( λt d AE 1 ( λ1t + BE 1 ( λt c= A1( λ1 de ( λ1t + B1( λ de ( λt From above we obtain the following

5 American ournal of Mathematical Analysis 76 1 = A 1( λ1 de ( 1 1( ( c λ t + B λ de λ t = AE ( λ1t + BE ( λt (6 A1( λ1 d B1( λ d A = ; B= c c A1, B 1 are arbitrary constants in above derivation, and λ1+ λ = a+ d, λλ 1 = ad bc Thus the solution can be written in the form A B E( λ1t E( λt y = + A 1 B 1 (7 Again to solve the system of fractional differential equation (1 we use the method similar to as used in classical differential equations D t ( = λ t ( has solution in the form Since [ ] ( t = AE ( λt, A is arbitary constant [15], putting = AE( λt and y = BE( λt in (1 we get the following D a by = 0 D AE( λt aae ( λt bbe( λt = 0 AλE( λt AaE( λt bbe( λt = 0 A( λ a Bb = 0 D y c dy = 0 D BE( λt cae ( λt dbe( λt = 0 BλE( λt AcE( λt dbe( λt = 0 Ac + B( λ d = 0 A( λ a bb = 0 (8 ca + ( λ d B = 0 Eliminating A and B from (8 we get, a λ b = 0 giving λ ( a + d λ+ ( ad bc = 0 (9 c d λ which is known as the characteristic equation with roots λ1and λ, also termed as eigen-values Three cases may arises i The roots λ1and λ are real and distinct ii The roots are real and equal ie λ1= λ = λ (say iii The roots are comple ie of the form λ1, λ = p ± iq (say Case I For real and distinct roots λ1 and λ, we write 1 = A1E( λ1t, y1 = B1E( λ1t and = AE( λt, y = BE( λt The solution of (1 is = 1+ = AE 1 ( λ1t + AE( λt y= y1+ y = BE 1 ( λ1t + BE( λt Thus the solution of the system of fractional differential equation can be written as, = c1ae ( λ1t + cbe( λt y A1 A Where A= and B = are the arbitry constants B 1 B Eample: 1 Let D = + y, D y = + y 0 < 1 with (0 =, y(0 = 1 = AE( λt and y = BE( λt be the solution of the above differential equation Substituting this in the above equations we get, A( λ + B = 0 A+ ( λ B = 0 The corresponding characteristic equation is, λ 1 = 0 giving λ = 1, 3 1 λ (10 Putting λ = 1 in (8 we get A+ B = 0, taking A = 1 we get B = 1 and putting λ = 3 in (8 we get A= B, taking A = 1we get B = 1 Hence the solutions are, 1 = E( t, y1 = E( t and = E(3 t, y = E(3 t Thus the general solution is = ce 1 ( t + ce(3 t y= ce 1 ( t + ce(3 t Where c 1, c are arbitrary constants Using the initial condition (0 =, y(0 = 0 we get c 1 = c = 1 Thus the required solution is, = E( t + E(3 t y = E( t + E(3 t Figure 1 represents the graphical presentation of t ( and yt ( when the eigen-values of the system of differential equations are positive Numerical simulation shows that t ( and yt ( both grow rapidly with decrease of order of derivative ie as decreases from 1 towards 0 Eample: D = + y, 0 < 1 with (0 =, y(0 = 1 D y = y Let = AE( λt and y = BE( λt

6 77 American ournal of Mathematical Analysis be the solution of the above differential equation Substituting this in the above equations we get, A( λ + B = 0 A+ ( λ B = 0 Figure 1 Numerical simulation of the solutions of fractional differential equation in Eample-1 for (t and y(t for different values of = 0, 04, 06, 08 and 10 Figure (i & (k and figure (j & (l represents the same figure only length of -ais is changed here to represent the prominent initial values The corresponding characteristic equation is, λ 1 = 0 giving λ = 1, 3 1 λ Putting λ = 1 in (8 we get A B = 0, taking A = 1 we get B = 1 and putting λ = 3 in (8 we get A= B, taking A = 1we get B = 1 Hence the solutions are, 1 = E( t, y1 = E( t and = E( 3 t, y = E( 3 t Thus the general solution is = ce 1 ( t + ce( 3 t y= ce 1 ( t ce( 3 t Where c 1, c are arbitrary constants Using the initial condition (0 =, y (0 = 0 we get c1 = c = 1

7 American ournal of Mathematical Analysis 78 Thus the required solution is, = E( t + E( 3 t, y = E( t E( 3 t Figure Numerical simulation of the solutions of fractional differential equation in Eample- for t ( and yt ( for different Values of = 06, 08 and 10 Figure represents the graphical presentation of t ( and yt ( when the eigen-values of the system of differential equations are negative Numerical simulation shows that t (and yt ( are both decaying rapidly with decrease of order of derivative ie as decreases from 1 to 06 For negative eigen-values the solutions are decaying asymptotically to ero Case II The roots of the equation (3 are comple and are of the form λ1, λ = p ± iq then the solution t (and yt ( can be written in the form (from Theorem 3, = ( A1+ ia E ( p + iq t and y = ( B1+ ib E ( p + iq t Using the definition of Mittag-Leffler function and fractional cosine and sine functions, that is we get, def E( i = cos ( + i sin( = ( A1+ ia E ( p + iq t = ( A1+ ia E ( pt E ( iq t = ( A1+ ia E ( pt cos ( qt + i sin ( qt ( A1cos ( qt Asin ( qt = E ( pt i ( Acos ( qt A1sin ( qt + + Similarly we get by repeating the above steps for y as follows ( B1cos ( qt Bsin ( qt i ( Bcos ( qt B1sin ( qt y = E ( pt + + In above obtained epressions for and y, we have comple 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 is linear combination of 1 (the real part of obtained comple, and (the imaginary part of obtained comple Similarly we have y as linear combination of y 1 and y [1] With this argument we write the following 1 = E ( pt A1cos ( qt Asin ( qt = E ( pt Acos ( qt + A1sin ( qt y1 = E ( pt B1cos ( qt Bsin ( qt y = E ( pt B cos ( qt + B1sin ( qt It can be shown that ( 1, y 1 ;(, y are solutions of the given equations (1 Thus the general solution in this case can be written in the form as in classical integer order differential equation [[1], pp305] The linear combination of 1 and, gives and linear combination of y 1 and y gives y, which is represented as following

8 79 American ournal of Mathematical Analysis ( 1cos ( sin ( ( cos ( 1sin ( M A qt A qt = E ( pt + N A qt + A qt ( 1cos ( sin ( ( cos ( 1sin ( M B qt B qt y = E ( pt + N B qt + B qt With M, N as arbitrary constants, determined from initial states We demonstrate by following eamples Eample: 3 Let D = 3+ y, D y = 5+ y 0 < 1 with (0 =, y(0 = 1 = AE( λt and y = BE( λt be the solution of the above differential equation, then putting in the above equation we get, A(3 λ + B = 0 5 A+ (1 λ B = 0 The corresponding characteristic equation is, 3 λ = 0 giving λ = ± 3i 5 1 λ (11 Putting λ = + 3i in (8 by putting a = 3, b =, c = 5, d = 1 we get the following A( λ a B = 0 A( + 3i 3 + B = 0 B= ( 1+ 3 ia ca + ( λ d B = 0 5 A+ ( + 3i 1 B = 0 5 A= (1 + 3 ib 5 A (1 3 i = (1 + 3 i (1 3 ib 5(1 3 ia = 10B B= ( 1+ 3 ia The (8 returns the same answer that is B = (3i 1 AWe choose here A =, so B = 3i 1, as we obtained one equation with two unknowns Thus B = 3i 1 and A = can be taken as one of the trial solution of the above Hence the solution is, Thus ( = E ( + 3 it = E ( t cos (3 t + isin (3 t 1 = E ( t cos (3 t and = E ( t sin (3 t Similarly the solution for y can be written in the form y= ( 1+ 3 i E (+ 3 it = ( 1+ 3 i E ( t cos (3 t + isin (3 t ( cos (3 t 3sin (3 t = E ( t i( 3cos (3 t sin (3 t + Hence y1 = E ( t cos (3 t 3sin (3 t and y = E ( t 3cos (3 t sin (3 t Therefore the general solution is linear combination of 1, for and linear combination of y 1, y for y, and we write the following = E ( t M cos (3 t + Nsin (3 t M ( cos (3 t 3sin (3 t y = E ( t N( 3cos (3 t sin (3 t + where M and N are arbitrary constants Using initial conditions (0 =, y (0 = 1 we get M =, and 3N M = 1 giving M = 1, N = Hence the required solution is, = E ( t cos (3 t + 4sin (3 t y = E ( t cos (3 t 5sin (3 t Numerical simulation in Figure 3 shows that for = 06, 08 and 10 after the initiation ( t = 0 of the system t ( and yt ( both oscillate, Period of oscillation changes with decrease of Case-III In this case roots of the equations (9 being equal, that is λ1 = λ = λ Then one solution will be of the form 1 = AE( λt and y1 = BE( λt and the other solution will be = ( At 1 + A E( λt and y = ( Bt 1 + B E( λt Hence the general solution is, = c1ae( λt + c( At 1 + A E( λt y = c1be( λt + c( Bt 1 + B E( λt where AA, 1, A, BB, 1, B, c1, c are arbitry constants Eample 4: D = 4 y,0 < 1 with (0 =, y(0 = 1(1 D y = + y Let = AE( λt and y = BE( λt

9 American ournal of Mathematical Analysis 80 be the solutions of the above differential equation, then putting in the above equation we get A(4 λ B = 0 A+ ( λ B = 0 (13 Figure 3 Numerical simulation of the solutions of fractional differential equation in Eample-3 for t ( and yt ( for different Values of = 06, 08 and 10 Figure (e & (f and Figure (g & (h represents the same figure only length of -ais is changed here to represent the prominent initial values Eliminating A and B as in previous eamples, we get, 4 λ 1 = 0 giving λ = 3, 3 1 λ For λ = 3 from equation (4 we get A= B = 1 Thus 1 = E(3 t and y1 = E(3 t is one solution of the equation The second solution is as in classical integer order differential equation [[1], pp307] = ( At 1 + A E (3 t and y = ( Bt 1 + B E (3 t -th order differentiating for above and y we obtain the following D 3( At 1 A E (3 t (1 A 1 E (3 t = + +Γ + and D y 3( Bt 1 B E (3 t (1 B 1 E (3 t = + +Γ + Putting the above obtained result in the given equation (1 we get, 3( At 1 + A E(3 t +Γ (1 + A1E(3 t = E(3 t (4 A1 B1 t + E(3 t (4 A B 3( Bt 1 + B E(3 t +Γ (1 + B1E(3 t = E(3 t ( A1+ B1 t + E(3 t ( A + B 3( At 1 + A +Γ (1 + A1 = (4 A1 B1 t + (4 A B 3( Bt 1 + B +Γ (1 + B1 = ( A1+ B1 t + ( A + B

10 81 American ournal of Mathematical Analysis Comparing the coefficients and simplifying we get A1 = B1, A B = A1Γ (1 + = B1Γ (1 + for simple non-ero values we take A1 = B1 = 1, A B =Γ (1 +, for simplicity take B = 0 then A =Γ (1 + Thus the other solution is = t +Γ (1 + E (3 t and y= t E (3 t Figure 4 Numerical simulation of the solutions of fractional differential equation in Eample-3 for t ( and yt ( for different Values of = 0, 04, 06, 08 and 10 Figure (i & (k and Figure (j & (l represents the same figure only length of -ais 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 [[1], pp307] = c1e(3 t + c t +Γ (1 + E (3 t and y = c1e(3 t + ct E(3 t

11 American ournal of Mathematical Analysis 8 Putting the initial condition and solving we get Hence the solution is, (0 = and y(0 = 1 1 c1 = 1 and c = Γ (1 + 1 = E(3 t + t +Γ (1 + E (3 t Γ (1 + and y E (3 t 1 = + t E (3 t Γ (1 + Numerical simulation in Figure 4 shows that t ( and yt ( start from and 1 respectively; and both of them start to grow This time interval to grow decreases as increases Moreover growth of t ( and yt ( is higher for is for lower values As increases growth rate of the solution decreases This implies for lower values of, t ( and yt ( grow initially slowly Once they start to grow, their growth rate is very high whereas for higher values t ( and yt ( start to grow sooner but their growth rate is low From the above discussion of the three cases one can state a theorem in the following form Theorem 4: The solutions of the system of differential equations are Case (i Case (ii a b D X = AX, Where A =, X = c d y 1 1 λ1 λ X = c AE ( t + c BE ( t, Where ABc,, 1and care the arbitry constants, λ1+ λ = a+ d, λλ 1 = ad bc, λ λ X = c1ae( λt + c( Bt 1 + B E( λt, Where AB, 1, B, c1and care the arbitry constants, λ = a+ d, λ Case (iii = ad bc A1cos ( qt Asin ( qt + ( Acos ( qt + A1sin ( qt X = E ( pt B1cos ( qt Bsin ( qt + ( Bcos ( qt + B1sin ( qt Where A1, A, B1, Bare the arbitry constants, p = a+ d, p + q = ad bc 3 Application of the Above Formulation in Real Life Problem: Consider the following fractional damped oscillator, formulated by umarie fractional derivative ( [ ] [ ] D + a D + b = 0 (31 Let d D [ ] = y then the given equation reduce dt to the following system of equation d y D [ y] = = ay b dt (3 d D [ ] = = y dt The above system of equation can be written in the form Let d D dt 0 1 = = D y d y b a y dt = AE( λt and y = BE( λt be solutions of the differential equations Then A(0 λ + B = 0 ba + ( a λ B = 0 (33 For the above system of equation the auiliary equation is, 0 λ 1 = 0 giving λ + aλ+ b = 0 b a λ Here the discriminant is 4( a b We consider the case when 1 a b< 0 then the eigen-values are λ, λ = p± iq where p = a, q = b a Then from (33 putting 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, = E ( p + iq t = E ( pt cos ( qt + sin ( qt 1 = E ( pt cos ( qt and = E ( pt sin ( qt Thus the general solution can be written in the form as in classical integer order differential equation [[1], pp305]

12 83 American ournal of Mathematical Analysis = E ( pt C1cos ( qt + Csin ( qt Where C 1 and C are arbitrary constants Here (0 = and D (0 = 1 and solving we get C1 = 1 and C = (1 p / q Hence the solution is 1 p = E ( pt cos ( qt + sin ( qt q Figure 5 Numerical simulation of the solutions of fractional differential equation in equation (31 for different Values = 0, 04, 06, 08, and 10 for b= and for a=01 & a=0 The numerical simulation of the solutions of the differential equation (31 has shown in Figure 5, for b =, left hand figures for a = 01 and right hand figures for a = 0 Figures 4 (a and (b are drawn for = 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 = 08, it is clear from the figure in both the cases the amplitude of the oscillation decreases with time and ultimately amplitude tends to ero Figures 4 (e & (f and (g & (h and (i & (j are drawn for = 06, 04and 0 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

13 American ournal of Mathematical Analysis 84 modified fractional derivative (with umarie s fractional derivative formulation in terms of Mittag-Leffler function and the generalied 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 umarie fractional derivative gives a conjugation with classical methods of solution of system of linear integer order differential equations, by usage of Mittag-Leffler and generalied 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 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(3/14/46/014- BRNS with BSC BRNS, title Characteriation of unreachable (Holderian functions via Local Fractional Derivative and Deviation Function References [1] K S Miller, B Ross An Introduction to the Fractional Calculus and Fractional Differential Equationsohn Wiley & Sons, New York, NY, USA; 1993 [] S Das Functional Fractional Calculus nd Edition, Springer- Verlag 011 [3] I Podlubny Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA 1999; 198 [4] K Diethelm The analysis of Fractional Differential equations Springer-Verlag, 010 [5] A Kilbas, H M Srivastava, Trujillo Theory and Applications of Fractional Differential Equations North-Holland Mathematics Studies, Elsevier Science, Amsterdam, the Netherlands, [6] M Caputo, Linear models of dissipation whose q is almost frequency independent, Geophysical ournal of the Royal Astronomical Society, 1967 vol 13, no 5, pp [7] G umarie Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions Further results, Computers and Mathematics with Applications, 006 (51, [8] U Ghosh, S Sengupta, S Sarkar and S Das Characteriation of non-differentiable points of a function by Fractional derivative of umarie type European ournal of Academic Essays (: 70-86, 015 [9] K Diethelm, N Ford and A D Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 9, 3-, 00 [10] Y Hua, Y Luoa, Z Lu, Analytical solution of the linear fractional differential equation by Adomian decomposition method ournal of Computational and Applied Mathematics 15 ( [11] S S Ray, RK Bera An approimate solution of a nonlinear fractional differential equation by Adomian decomposition method Applied Mathematics and Computation 167 ( [1] O Abdulai, I Hashim, S Momani, Application of homotopyperturbation method to fractional IVPs, Comput Appl Math 16 ( [13] R Yulita Molliq, MSM Noorani, I Hashim, RR Ahmad Approimate solutions of fractional Zakharov Kunetsov equations by VIM ournal of Computational and Applied Mathematics 33( [14] VS Ertürk, S Momani Solving systems of fractional differential equations using differential transform method ournal of Computational and Applied Mathematics 15 ( [15] U Ghosh, S Sengupta, S Sarkar and S Das Analytic solution of linear fractional differential equation with umarie derivative in term of Mittag-Leffler function American ournal of Mathematical Analysis 3( , 015 [16] G umarie Fourier s Transformation of fractional order via Mittag-Leffler function and modified Riemann-Liouville derivatives Appl Math & informatics [17] Mittag-Leffler, G M Sur la nouvelle fonction E (, C R Acad Sci Paris, (Ser II 137, (1903 [18] Erdelyi, A Asymptotic epansions, Dover (1954 [19] Erdelyi, A (Ed Tables of Integral Transforms vol 1, McGraw- Hill, 1954 [0] Erdelyi, A On some functional transformation Univ Potitec Torino 1950 [1] S L Ross Differential equations (3rd edition ohn Wiley & Sons 1984