A COLLOCATION METHOD FOR SOLVING FRACTIONAL ORDER LINEAR SYSTEM

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1 J Indones Math Soc Vol 23, No (27), pp A COLLOCATION METHOD FOR SOLVING FRACTIONAL ORDER LINEAR SYSTEM M Mashoof, AH Refahi Sheikhani 2, and H Saberi Najafi 3 Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran mashoofmohammad@yahoocom 2 Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran *Corresponding author: ah refahi@liauacir, 3 Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran hnajafi@guilanacir Abstract In the current paper we propose a collocation method to achieve an algorithm for numerically solving of fractional order linear systems where a fractional derivative is defined in the Caputo form We have used the Taylor collocation method for solving fractional differential equations; a collocation method which is based on taking the truncated Taylor expansions of the vector function s solution in the fractional order linear system and substituting their matrix forms into the system Through using collocation points we have obtained a system of linear algebraic equation The method has been tested by some numerical examples Key words and Phrases: Fractional system, collocation method, numerical simulation Abstrak In the current paper we propose a collocation method to achieve an algorithm for numerically solving of fractional order linear systems where a fractional derivative is defined in the Caputo form We have used the Taylor collocation method for solving fractional differential equations; a collocation method which is based on taking the truncated Taylor expansions of the vector function s solution in the fractional order linear system and substituting their matrix forms into the system Through using collocation points we have obtained a system of linear algebraic equation The method has been tested by some numerical examples Kata kunci: Sistem fraksional, metode kolokasi, simulasi numerik 2 Mathematics Subject Classification: 26A33, 74H5 Received: , revised: 3--27, accepted:

2 28 M Mashoof etal Introduction Fractional differential equations have been generalized from integer order derivatives through replacing the integer order derivatives by fractional ones The field of fractional differential equations has received attention and interest only in the past 2 years or so [2, 4, 3] In recent years, studies concerning the application of the fractional differential equations in science has attracted more interest among scholars [9, 5]; readers can refer to [7, 8] for the theory and applications of fractional calculus in this regard For instance, [, 5] formulated the motion of a rigid plate immersing in a Newtonian fluid They show that the use of fractional derivatives for the mathematical modelling of viscoelastic materials is quite natural [9] It should be mentioned that the main reason for the theoretical development here is the wide use of polymers in various fields of engineering [9] Moreover, in 99, S Westerlund in a paper on electrochemically polarizable media proposed the use of fractional derivatives for the description of propagation of plane electromagnetic waves in an isotropic and homogeneous lossy dielectric [9] Caputo suggested the fractional order version of the relationship between the electric field and electric flux density [9] Recently, fractional derivatives have been used to new applications in neural networks and control system [5, 6, 2] Several methods such as Haar-wavelet operational matrix method have exited to solve the fractional linear system [6] In [6] the authors introduced Haar-wavelet operational matrix method for fractional control system and translated the control system with initial condition into a Sylvester equation A typical n-term linear non-homogeneous fractional order differential equation in domain can be described as the following form a n (D αn t y) + + a (Dt α y) + a (Dt α y) = u () A fractional order system described by n-term fractional differential () can be rewritten to the state-space representation in the form [2, 24]: ad β t x = Ax + Bu y = Cx (2) For this reason, the behavior of output in system (2) is useful Recently, the collocation method has become a very useful technique for solving differential equations [8, 3, ] In the current study, we present a numerical solution of the fractional order system through the use of Taylor collocation method for a system of the form ad α t x = Ax + Bu y = Cx + Du, t η, x() = ( λ, λ 2,, λ n ) T, (3) with < α, where A, B, C and D are n n, n m, p n and p m matrices, and u is an m-vector function, and a D α t x is the αth-order (always fractional) derivative of x in the Caputo form By using this method, we can translate (3) into an algebraic linear equation that can be solved through the use of existing methods [2, 22]

3 A Collocation Method 29 This paper is organized as follows We review some basic definitions of fractional derivative operators The function approximations and two applicable algorithms are presented in section 3 Section 4 contains four examples, and finally, conclusions are presented in section 5 2 Basic Definition and Theorem Here and in this part of the study, we deal with fractional calculus, definitions and theorems; see [23, 4, 7] for more details in this regard Definition 2 A real function f(x), x is said to be in space C µ, µ R if there exists a real number p(> µ), such that f(x) = x p f (x) where f (x) [, ), and it is said to be in the space C m µ iff f m C µ, m N Definition 22 The Riemann-Liouville fractional derivative of order α with respect to the variable x and with the starting point at x = a is adt α f(x) = Γ( α+m+) dm+ dx x m+ a (x τ)m α f(τ)dτ ; m α < m +, d m+ dx m+ ; α = m + N Definition 23 The Riemann-Liouville fractional integral of order α is ad α t f(x) = Γ(α) x a (4) (x τ) α f(τ)dτ ; α > (5) Definition 24 The fractional derivative of f(x) in the Caputo form is defined as D α t f(x) = Γ(n α) x (x τ) n α f (n) (τ)dτ; n < α n, n N, x >, f C n For the Caputo s derivative we have Dt α C =, and C is a constant Moreover, we have { ; n N, n < α, Dt α x t = (7) ; n N, n < α Γ(n+) Γ(n+ α) xn Definition 25 (Fractional Derivative of a Vector) If X(x) = (X (x) X n (x)) T is a vector function, we define Dx α X (x) Dx α Dx α X 2 (x) X(x) = (8) Dx α X n (x) (6)

4 3 M Mashoof etal Now consider the fractional differential equations as D α t x = Ax + q, (9) where < α <, A is an N N matrix, and q : [, h] C N Two following theorems show the form of the general solution of (9) where E α is the Mittag- Leffler function Definition 26 The Mittag-Leffler function with parameter α is given by z k E α (z) =, R(α) >, z C Γ(αk + ) k= It is obvious that if α =, then E α (z) = e z Theorem 27 If λ,, λ N, be the eigenvalues of A and u (),, u (N) will be the corresponding eigenvectors; then the general solution of the homogeneous differential equation D α t x = Ax will be x = N c l u (l) E α (λ l x α ), () l= with certain constants c l C The unique solution of this differential equation subject to the initial condition x() = x is characterized by the linear system Proof See [] x = (u (),, u (N) )(c,, c N ) T () For the inhomogeneous boundary value problem, we have the following theorem Theorem 28 The general solution of the boundary value problem (9) has the form x = x hom +x inhom where x hom is the general solution of the associated homogeneous problem and x inhom is a particular solution of the inhomogeneous problem Proof See [] Following we recall the generalization of Taylor formula which forms the basis of our numerical method Theorem 29 (generalized Taylor formula) Suppose that Dx kα f(x) C(a, b] for k =,,, n + where < α ; then one has n f(x) = i= (x a) iα Γ(iα + ) Diα x f(a) + Dx n+ f(ζ) Γ((n + )α + ) (x a)(n+)α, (2) with a ζ x, for all x (a, b], where D nα t = D α t D α t D α t (n s) Proof See [7]

5 A Collocation Method 3 We can use the generalized Taylor formula in the matrix form as follows where f(x) = X(x)M v + DN+ t f(ζ) Γ((n + )α + ) (x a)(n+)α, (3) X(x) = ( (x a) α (x a) 2α (x a) Nα ) (N+), (4) and and Γ() Γ(α+) M = Γ(2α+), (5) Γ(Nα+) v = ( D α x f(a) D α x f(a) D 2α x f(a) D Nα x f(a) ) ; (6) hence we can approximate f(x) as follows Moreover, we have and f(x) X(x)M v (7) D α x f(x) D α x (X(x)M v) = (D α x X(x))M v, (8) Dx α (X(x)) = ( Dx α Dx α (x a) α Dx α (x a) 2α Dx α (x a) ) Nα (N+) ( Γ(α+) Γ(Nα+) = Γ() Γ((N )α+) ) (N+) (t a)nα ; (9) thus where hence D α x (X(x)) = X(x)M, Γ(α+) Γ() Γ(2α+) Γ(α+) M = Γ(Nα+) Γ((N )α+) ; (2) D α x f(x) X(x)M M v (2)

6 32 M Mashoof etal 3 Numerical method In this section, we present two algorithms for numerical solution of system (3) based on Taylor collocation method For this, we use equations (4) and (9) for each array of vector function x, and it s fractional derivative, namely as x j x N j = XM v j, (22) and D α t x j D α t x N j = XM M v j (23) If we use (22) and (23) to approximate x j and D α t x j for j =, n; then we have a system of linear equations Theorem 3 If we approximate x j and D α t x j for j =, n, by (22) and (23) respectively; then we have a system of linear equations as X F ṽ = b where ṽ = (v, v,, v) T Proof By substituting equations (22) and (23) in equation (3) we have X( n i = i j m a ij M v i + (a jj M M M )v j ) = b ji u i ; j =,, n, (24) i= or X(A I M )M v = β, (25) X X where X =, X I N+ I N+ I, I N+ M M M =, M M M M =, M

7 v = v v 2 v n, A Collocation Method 33 β = Bu, (26) and a I N+ a n I N+ A I = (27) a n I N+ a nn I N+ Dispersing equation (24) by the collocation points t i, i =,,, N, we can obtain where X(t i )F v = β(t i ), F = (A I M )M (28) Now we find the matrix representation of the initial condition x(a), X(a)M v j = λ j for j =, 2,, n, (29) or X(a)M v = λ λ = ( λ λ n ) T (3) To obtain the numerical solution of (3), by replacing X(t N )F v = β(t N ) with equation (29), we have the new algebraic equation, X(t i )F v = β(t i ) for i =,, N (3) X(a)M v = λ or in blocked matrix form X F ṽ = b, (32) where X = X(t ) X(t ) X(t N ) X(a) (N+) (N+),

8 34 M Mashoof etal F F F = F M v β(t ) ṽ =, b = v β(t N ) v λ (N+) (N+) (N+), (N+) (33) From above discussions we found that v is (N +) and ṽ is n(n +) 2 ; hence if in (3) we set { ( ) T ṽ = y y 2 y n(n+) y n(n+)2 n y n(n+) 2, (34) y i = y jn(n+)+i for i =, 2,, n(n + ) and j =, 2,, N; hence we obtain a system of n(n + ) 2 linear algebraic equations with n(n + ) 2 unknown Taylor coefficients for x i, i =, 2,, n From above discussions we find that v i = ( y (i )(N+)+ y (i )(N+)+2 y (i )(N+)+N y i(n+) ), (35) which is the vector of Taylor coefficients for x i, i =, 2,, n The algorithm is Proposition 32 Input A, B, x(a), u, N, a, b Output x = (x,, x n ) T Compute M, M 2Set M = diag(m,, M ), M = diag(m,, M ), I = diag(i N+,, I N+ ) 3Compute F = (A I M ) 4for i =,, N do Set t i = a + i b a N compute X(t i) Construct X(t i ) = diag(x(t i ) X(t i )), β(t i ) = Bu(t i ) end do 5Set b = (β(t ),, β(t N ), x(a)) T, X = diag(x(t ),, X(t N ), X(a)), F = diag(f,, F, M ) 6Solve X F ṽ = b such that ṽ i = ṽ jn(n+)+i, i =, 2,, n(n + ) and j =, 2,, N 7for i =, 2,, n do x i XM (ṽ (i )N+i,, ṽ i(n+) ) end do

9 A Collocation Method 35 If a t b and (b a) are large, we can divide I = [a, b] into I = [a, t ], I 2 = [t, t 2 ],, I N = [t N, b] and approximate x on subinterval I k+ and use the approximated solution at t k+ to initial condition of next subinterval The following Lemma shows that by this procedure error can be decreased Lemma 33 Let x be bounded on [a, b] If we use I = [a, t ], I 2 = [t, t 2 ],, I N = [t N, b] and approximate x on subinterval I k+ and use approximated solution at t k+ to initial condition of next subinterval, absolute error at t = b decreases Proof From (2), for interval I, if Dt N+ x i (ζ) < l i ; then we have e = x i (b) x N i (b) = Dt N+ x i (ζ) (b a)(n+)α Γ((N+)α+) < l i (b a)(n+)α Γ((N+)α+) = L i (Nh) (N+)α, where h = b a N Moreover, for subinterval I j we have thus e j = x i (t j ) x N i (t j ) < l i (t j t j ) (N+)α Γ((N + )α + ) = L ih (N+)α ; e + e e N < L i Nh (N+)α < L i (Nh) (N+)α = e (36) This shows that the following algorithm can be more efficient than algorithm 32 at t = b Proposition 34 Input A, B, x(a), u, N, a, b Output x(b) for i =,,, N do Set I i = [t i, t i + ], a = t i, b = t i+ Utilize algorithm 32 and approximate x(t i+ ) Set x(a) = x(t i+ ) 2 Set x(b) = x(t N ) end do 4 Numerical Examples To illustrate the method and algorithms referred to in the previous section, we consider the following examples In order to show the efficiency of the method for solving fractional order linear systems, we apply it to solve different types of

10 36 M Mashoof etal fractional linear systems whose exact solutions are known We use 2 to compare the exact and numerical solutions Example 4 Let us consider the following system with the initial conditions ( ) ( ) ( ) ( ) Dt α 3 x = x + ; t ; x() =, (37) the exact solution when α = is ( ) x x = x 2 = ( ) 22 e 2t 3e 6t 8 e 2t 22 + e 2t 3e 6t (38) We approximately solve the above system for N = and obtain the approximate solution for α = as x = 7 t Γ(2) t5 Γ(6) t2 t3 t Γ(3) Γ(4) Γ(5) t t7 Γ(7) Γ(8) t8 Γ(9) t 9 Γ() x 2 = t Γ(2) t5 Γ(6) t Γ(), t2 t3 t Γ(3) Γ(4) Γ(5) t t7 Γ(7) Γ(8) 2 5 t8 Γ(9) t t Γ() Γ() In figure we can see the numerical and exact solutions Example 42 Let us consider following fractional equation the exact solution, when α =, is D α t x = x; x() = ; t ; (39) x = e t, (4) and for α = 2 is x = e t (erf t + ) (4) We approximately solve fractional equation (39) for α = with N = 3 and obtained the approximate solution, x = + t t2 t Γ(2) Γ(3) Γ(4), also for α = 5 and N = 9 we have, x = + t 2 Γ(5) t Γ(2) + 63 t 3 2 Γ(25) +

11 A Collocation Method 37 5 x,x exact solution of x exact solution of x 2 numerical solution of x numerical solution of x 2 / 2/ 3/ 4/ 5/ 6/ 7/ 8/ 9/ Figure Numerical and exact solution of example 4 for α = and N = x exact solution of x numerical solution of x /3 2/3 Figure 2 Numerical and exact solution of example 42 for α = and N = t2 Γ(3) t 5 2 Γ(35) t3 Γ(4) t 7 2 t Γ(45) Γ(5) t 9 2 Γ(55) In figure 2 we can see the numerical and exact solutions of example 42 for α = with N = 3; Moreover, figure 3 shows the results for α = 5 with N = 9 in example 42 Example 43 In this example, we consider a fractional system with three equations, 3 Dt α x = 2 9 x; t ; < α ; x() = 5 (42) 3 6

12 38 M Mashoof etal exp(z) (erf(z /2 ) + ) exact solution of x numerical solution of x /9 2/9 3/9 4/9 5/9 6/9 7/9 8/9 Figure 3 Numerical and exact solution of example 42 for α = 5 and N = 9 x,x 2,x exact solution of x exact solution of x 2 exact solution of x 3 numerical solution of x numerical solution of x 2 numerical solution of x /5 2/5 3/5 4/5 Figure 4 Numerical and exact solution of example 43 for α = 975 and N = 5 The general solution of this system according to theorems 27 and 28 is given by x = c u E α (λ t α ) + c 2 u 2 E α (λ 2 t α ) + c 3 u 3 E α (λ 3 t α ), (43) where c, c 2, c 3 are constants, λ, λ 2, λ 3 are eigenvalues, and u, u 2, u 3 are the corresponding eigenvectors of A In the figures 4 and 5 the numerical and exact solutions of x for α = 975 are shown for N = 5 and N = 5, respectively From figure 4 we can see that the numerical solution with N = 5 is not a mismatch with the exact solution, but from figure 5 we can see that the numerical solution is in very good agreement with the exact solution for N = 5 Moreover, the errors of numerical solutions for different N are shown in table From table we can see that the numerical solutions are more and more close to the exact solution when the value of N becomes large Example 44 Consider example 43 with t [, 3], N = 6 and divide [, 3] to subintervals [, ], [, 2], [2, 3] Now we solve it with algorithms 3 and 32, and

13 A Collocation Method 39 x,x 2,x exact solution of x exact solution of x 2 exact solution of x 3 numerical solution of x numerical solution of x 2 numerical solution of x /5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 9/5/5/52/53/54/5 Figure 5 Numerical and exact solution of example 43 for α = 975 and N = 5 Table Error of approximated solution in example 43 N error at t = 4 error at t = 2 4 error at t = e e e e-7 6 Table 2 Error of approximated solution in example 44 at t = b by algorithm 32 and algorithm 34 subinterval [, ] [, 2] [2, 3] [, 3] algorithm error at b 9359e e compare the results The numerical results will presented in table 2 From table 2 we see that if we use algorithm 32, the error at t = 3 is which is unacceptable, but with dividing interval [, 3] into three intervals [, ], [, 2], [2, 3] and use algorithm 34, the error at t =, 2 and 3 is 9359e 7, 6742e 5 and 4, respectively This example confirms Lemma 32, and so we see that algorithm 34 is useful In figure 6 we can see the numerical and exact solutions of example 44 with N = 6 by algorithm 32 Figures 7, 8, and 9 show the numerical and exact solutions of example 44 with N = 6 by algorithm 34

14 4 M Mashoof etal x,x 2,x exact solution of x exact solution of x 2 exact solution of x 3 numerical solution of x numerical solution of x 2 numerical solution of x Figure 6 Numerical and exact solution of example 44 on [, 3] by algorithm 32 x,x 2,x exact solution of x exact solution of x 2 exact solution of x 3 numerical solution of x numerical solution of x 2 numerical solution of x Figure 7 Numerical and exact solution of example 44 on [, ] by algorithm 34 5 Conclusions The fractional order linear system plays an important role in physics, chemical mixing, chaos theory, and biological systems In this paper we presented a collocation method to solve fractional order linear system with initial conditions These methods approximate the numeric solution of fractional order linear system through taking the truncated Taylor expansions of the vector function s solution in the fractional order linear system and then substituting their matrix forms into the system This method transforms fractional linear system into an algebraic equation Examples show that the Taylor collocation method has been successfully applied to find the approximate solutions of the fractional linear system Numerical

15 A Collocation Method 4 x,x 2,x exact solution of x exact solution of x 2 exact solution of x 3 numerical solution of x numerical solution of x 2 numerical solution of x Figure 8 Numerical and exact solution of example 44 on [, 2] by algorithm 34 x,x 2,x exact solution of x exact solution of x 2 exact solution of x 3 numerical solution of x numerical solution of x 2 numerical solution of x Figure 9 Numerical and exact solution of example 44 on [2, 3] by algorithm 34 results show that this method is extremely effective and practical for this sort of approximate solutions This method will be applicable in large domains as well References [] Aminikhah, H, Refahi Sheikhani, A and Rezazadeh, H, Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method, Bol Soc Paran Mat, 34 (26), [2] Aminikhah, H, Refahi Sheikhani, A H and Rezazadeh, H, Sub-equation method for the fractional regularized long-wave equations with conformable fractional derivatives, Scientia Iranica, 23 (26), [3] Aminikhah, H, Refahi Sheikhani, A H and Rezazadeh, H, Exact solutions of some nonlinear systems of partial differential equations by using the functional variable method, MATHEMATICA, 56 (24), 3-6

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