Analytical solutions for the fractional Klein-Gordon equation

Transcription

1 Computational Methods for Differential Equations Vol. 2, No. 2, 214, pp Analytical solutions for the fractional Klein-Gordon equation Hosseni Kheiri Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Samane Shahi Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Aida Mojaver Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran aida Abstract In this paper, we solve a inhomogeneous fractional Klein-Gordon equation by the method of separating variables. We apply the method for three boundary conditions, contain Dirichlet, Neumann, and Robin boundary conditions, and solve some examples to illustrate the effectiveness of the method. Keywords. Fractional Klein-Gordon equation, Mittag-effler, Method of separating variables, Caputo derivative. 21 Mathematics Subject Classification. 35R11; A Introduction Partial fractional differential equation frequently appear in variety of applications [9, 1], so lots of scientists have been focused on solving them, but unfortunately most of these equations do not have exact solutions, so for solving them numerical and approximative methods must be used. For instance variational method [11], supper symmetric method [15, 3, 2], finite difference method (FDM), and functional variable method [7], are some of these methods. In [14], a novel predictor-corrector method, called Jacobian-predictor-corrector approach, for the numerical solutions of fractional ordinary differential equations, which are based on the polynomial interpolation, was presented. The authors of [13], proposed the Cantor-type cylindricalcoordinate method in order to investigate a family of local fractional differential operators on Cantor sets. In [1], the authors studied an initial value problem for a fractional differential equation using the Riemann-iouville fractional derivative. The Klein-Gordon equation has important applications in mathematical physics such as solid-state physics, nonlinear optics, and quantum mechanics [4, 5, 11, 12]. In this paper we consider the fractional Klein-Gordon equation as follows D α t u(x,t) u xx (x,t)+u(x,t) = f(x,t). (1.1) In (1.1) by α = 2 we can get the PDE Klein-Gordon equation. We try to solve (1.1) by the method of separating of variables. For this purpose in section 2 we give 99

2 1 H. KHEIRI, S. SHAHI, AND A. MOJAVER some preliminaries, section 3 is contained the exact solution of inhomogeneous fractional Klein-Gordon equation with three boundary conditions, Drichlet, Neumann, and Robin boundary condition, then in section 4 we propose some examples related to section Preliminaries In this section, we introduce some necessary definitions from fractional calculus. You can find some extra details to fractional derivatives [6, 9, 1]. Definition 1.([6]) A function f : R R + is said to be in the space C ν, with ν R, if it can be written as f(x) = x p f 1 (x) with p > ν, f 1 (x) C[, ) and it is said to be in the space Cν m if f (m) C ν for m N {}. Definition 2.([8]) The Riemann-iouville fractional integral of f C ν with order α > and ν 1 is defined as: J α f(t) = 1 Γ(α) t (t τ)α 1 f(τ)dτ, α >, t >, J f(t) = f(t). (2.1) Definition 3.([6]) The Riemann-iouville fractional derivative of f C 1 m with order α > and m N {}, is defined as: D α t f(t) = dm dt m J m α f(t), m 1 < α m, m N. (2.2) Definition 4.([6]) The Caputo fractional derivative of f C m 1 with order α > and m N {}, is defined as: C D α t f(t) = { J m α f (m) (t), m 1 < α m, m N d m f(t) dt m, α = m. (2.3) Definition 5.([6]) A two-parameter Mittag-effler function is defined by the following series t k E α,β (t) = Γ(αk +β). (2.4) Definition 6.([8]) A multivariate Mittag-effler function is defined as E (a1,a 2,,a n),b(z 1,z 2,,z n ) = l 1+l 2+ +l n=k k! l 1! l 2! l n! Γ(b+ n i=1 z li i, (2.5) n a i l i ) where b >, l 1,l 2,, l n, z i <,i = 1,2,,n. Definition 7. et us define the aplace-transform (T) operator ϕ on a function u(x,t), (t ) by ϕ{u(x,t);t s} = i=1 e st u(x,t)dt, (2.6)

3 CMDE Vol. 2, No. 2, 214, pp and denote it by ϕ{u(x,t);t s} = (u(x,t)), where s is the T parameter. For our purpose here, we shall take s to be real and positive. Consequently, the T of Mittag-effler function has the following form (E α,β (t)) = e st E α,β (t)dt = 1 s k+1 Γ(αk +β). (2.7) emma 2.1([8]). et µ > µ 1 > µ 2 >... > µ n, m i 1 < µ i m i, m i N = N {}, d i R,i = 1,2,...,n. Consider the initial value problem n (D µ y)(x) λ i (D µi y)(x) = g(x), i=1 y (k) () = c k R, k =,1,...,m 1, m 1 < µ m, where the function g(x) is assumed to lie in C 1, if µ N, in C 1, 1 if µ / N and the unknown function y(x) is to be determined in the space C 1 m. This has solution m 1 y(x) = y g (x)+ c k u k (x), x, where and y g (x) = x u k (x) = xk k! + n t µ 1 E (.),µ (t)g(x t)dt i=l k +1 d i x k+µ µi E (.),k+1+µ µi (x), k =,1,...,m 1, fulfills the initial conditions u (l) k () = δ kl, k,l =,1,...,m 1. The function E (.),σ (x) = E (µ µ1,...,µ µ n),σ(d 1 x µ µ1,...,d n x µ µn ), is a particular case of the multivariate Mittag-effler function (see [8]) and the natural numbers l k,k =,1,...,m 1, are determined from the condition { mlk k +1, m lk +1 k. In the case m i k,i = 1,2,...,n, we set l k :=, and if m i k +1,i = 1,2,...,n, then l k := n. 3. Nonhomogeneous fractional Klein-Gordon equation 3.1. Dirichlet boundary condition. In this section, we determine the solution of the fractional Klein-Gordon equation (1.1), with the initial condition u(x,) = φ(x), u t (x,) = ψ(x), x, (3.1) and the inhomogeneous boundary conditions u(,t) = µ 1 (t), u(,t) = µ 2 (t), t,

4 12 H. KHEIRI, S. SHAHI, AND A. MOJAVER where µ 1 (t) and µ 2 (t) are nonzero smooth functions with order-one continuous derivative and φ(x) and ψ(x) are a continuous functions satisfying φ() = µ 1 () and φ() = µ 2 (). Now, for applying the method of separating variables with nonhomogeneous boundary conditions, we first transform it into homogeneous ones by taking u(x,t) = W 1 (x,t)+v 1 (x,t), where W 1 (x,t) is an unknown function and V 1 (x,t) = µ 2(t) µ 1 (t) x+µ 1 (t), (3.2) which satisfies the following boundary conditions V 1 (,t) = µ 1 (t), V 1 (,t) = µ 2 (t). (3.3) Furthermore W 1 (x,t) satisfies the problem with homogeneous boundary conditions as follows: Dt α W 1(x,t) 2 W 1 (x,t) x 2 +W 1 (x,t) = f(x,t), W 1 (x,) = g 1 (x), (W 1 ) t (x,) = g 2 (x), x, (3.4) W 1 (,t) =, W 1 (,t) =, t, where f(x,t) = f(x,t) D α t V 1(x,t)+ 2 V 1 (x,t) x 2 V 1 (x,t), (3.5) g 1 (x) = φ(x) µ 2() µ 1 () g 2 (x) = ψ(x) µ 2 () µ 1 () µ 1 (), µ 1 (). For solving the corresponding homogeneous equation in (1.1), with proposed method we set W 1 (x,t) = X(x)T(t) in (3.4), we get the following linear ODE for X(x): X (x) λx(x) =, X() =, X() = (3.6) and a fractional linear ode with the Caputo derivative for T(t) D α t T(t)+(1 λ)t(t) =, (3.7) in which λ is a negative constant. Hence the Sturm-iouville problem, which is given by (3.6), has eigenvalues λ n and corresponding eigenfunctions X n (t) as follows: λ n = n2 π 2 2, X n(t) = sin nπ x, n = 1,2,. (3.8) Now, we want to find a solution of the inhomogeneous problem in (3.4) of the form W 1 (x,t) = B n (t)sin( nπ x). (3.9)

5 CMDE Vol. 2, No. 2, 214, pp We suppose that the series can be differentiated term by term. For determining B n (t), weexpand f(x,t) asafourierseriesbytheeigenfunctions{sin( nπ x)}asfollows in which f(x,t) = f n (t)sin( nπ x), (3.1) f n (t) = 2 f(x,t)sin( nπ x)dx. (3.11) Now, if we substitute (3.9) and (3.1) into (3.4), we have Dt α B n (t)sin( nπ ( n 2 x)+ π 2 ) 2 +1 B n (t)sin( nπ x) = f n (t)sin( nπ (3.12) x). By orthogonality properties of sin( nπ x), we obtain Dt α B n(t)+ (1+ n2 π 2 ) B n (t) = f n (t). (3.13) Since W 1 (x,t) satisfies the initial condition in (3.4), we must have which gives B n () = 2 2 B n ()sin( nπ x) = g 1(x), (3.14) B n t () = 2 B n t ()sin(nπx ) = g 2(x), (3.15) g 1 (x)sin( nπ x)dx, (3.16) g 2 (x)sin( nπx )dx, n = 1,2, For each value of n, (3.14) and (3.16) give a fractional initial value problem. emma 2.1 implies that the fractional initial value problem has the following solution B n (t) = t τ α 1 E (α,α) ( ( nπ )2 1)τ α fn (t τ)dτ +B n ()u (t)+b n ()u 1().

6 14 H. KHEIRI, S. SHAHI, AND A. MOJAVER Therefore we obtain the solution of the (3.4) with the form W 1 (x,t) = B n (t)sin( nπx ) = sin( nπx [ t ) τ α 1 E (α,α) ( (( nπ )2 1)τ α fn (t τ)dτ Hence the solution of (1.1) is the form as u(x,t) = B n (t)sin( nπx )+µ 1(t)+ µ 2(t) µ 1 (t) x = sin( nπx [ t ) τ α 1 E (α,α) ( (( nπ ] )2 1)τ α fn (t τ)dτ +µ 1 (t)+ µ2(t) µ1(t) x. ]. (3.17) (3.18) 3.2. Neumann boundary condition. In this subsection, we obtain the solution of the fractional Klein-Gordon equation (1.1) with the initial and Neumann boundary conditions u(x,) = φ(x), u t (x,) = ψ(x), x, (3.19) u x (,t) = µ 1 (t), u x (,t) = µ 2 (t), t, in which φ(x), ψ(x), µ 1 (t), µ 2 (t) are as defined in subsection 3.1. For solving the problem with inhomogeneous boundary conditions, in a similar way, transform it into a homogeneous boundary condition. Thus we suppose that u(x,t) = W 2 (x,t)+v 2 (x,t), where W 2 (x,t) is an unknown function and V 2 (x,t) = µ 2(t) µ 1 (t) x 2 +µ 1 (t)x, (3.2) 2 which satisfies the following boundary conditions: (V 2 ) x (,t) = µ 1 (t), (V 2 ) x (,t) = µ 2 (t) (3.21) and the function W 2 (x,t) satisfies in problem with homogeneous boundary conditions as follows: Dt α W 2(x,t) 2 W 2 (x,t) x 2 +W 2 (x,t) = f(x,t), W W 2 (x,) = g 1 (x), 2(x,) (3.22) t = g 2 (x), x, (W 2 ) x (,t) =, (W 2 ) x (,t) =, t, in which f(x,t) = f(x,t) D α t V 2 (x,t)+ µ 2(t) µ 1 (t) g 1 (x) = φ(x) x2 2 [µ 2() µ 1 ()] µ 1 ()x, µ 2(t) µ 1 (t) x 2 µ 1 (t)x, 2 g 2 (x) = ψ(x) x2 2 [µ 2 () µ 1 ()] µ 1 ()x. (3.23)

7 CMDE Vol. 2, No. 2, 214, pp Now, for solving the corresponding homogeneous equation in (3.22) by the method of separating variables (assuming that f(x,t) = ), we suppose that W 2 (x,t) = X(x)T(t) and substituting it in (3.22), we obtain a linear ode for X(x) and a linear FDE for T(t) as X (x)+λx(x) =, X() = X() =, D α t T(t)+(1 λ)t(t) =. (3.24) The Sturm-iouville problem, which is given by (3.24), has eigenvalues and corresponding eigenfunctions as follows: λ n = n2 π 2 2, X n(x) = cos( nπx ), n = 1,2,...,. (3.25) Now we are going to find a solution of the inhomogeneous problem in (3.22) which takes the form W 2 (x,t) = B n (t)cos( nπ x). (3.26) For determining B n (t), we expand f(x,t), as a Fourier series by the eigenfunctions {cos( nπ x)}, as follows: f(x,t) = f n (t)cos( nπ x), (3.27) in which the Fourier coefficients are f n (t) = 2 f(x,t)cos( nπ x)dx. (3.28) Then substituting (3.26), (3.27) into (3.22) implies Dt α B n (t)+ (1+ n2 π 2 ) B n (t) = f n (t), (3.29) 2 Since W 2 (x,t) fulfills the initial conditions in (3.22), we have B n ()cos( nπ x) = g 1(x), (3.3) which yields B n () = 2 B n t ()cos(nπx ) = g 2(x), (3.31) B n t () = 2 g 1 (x)cos( nπ x)dx, (3.32) g 2 (x)cos( nπx )dx.

8 16 H. KHEIRI, S. SHAHI, AND A. MOJAVER Hence lemma 2.1 implies that the fractional initial value problem has the solution as u(x,t) = B n (t)cos( nπx ) = µ 1(t)x+ µ 2(t) µ 1 (t) x cos( nπ [ t ] (3.33) x) τ α 1 E (α,α) ( (nπ) 2 1)τ α fn (t τ)dτ Robin boundary condition. In this subsection, we want to find the solution of the (1.1) with the Robin boundary conditions u(x,) = φ(x), u t (x,) = ψ(x), x, (3.34) u(,t)+α 1 u x (,t) = µ 1 (t), u(,t)+β 1 u x (,t) = µ 2 (t), t. where α 1 and β 1 are nonzero constants. for solving this equation, we should translate the nonhomogeneous boundary conditions to the homogenous ones. So u(x,t) = W 3 (x,t)+v 3 (x,t), where W 3 (x,t) is a new unknown function and V 3 (x,t) = µ 1(t) µ 2 (t) α 1 β 1 x (+β 1)µ 1 (t) α 1 µ 2 (t), (3.35) α 1 β 1 so we will have { V3 (,t)+α 1 (V 3 ) x (,t) = µ 1 (t), V 3 (,t)+β 1 (V 3 ) x (,t) = µ 2 (t). The function W 3 (x,t) is the solution of problem with homogeneous boundary conditions: where and D α t W 3(x,t)+(W 3 ) xx (x,t)+w 3 (x,t) = f(x,t), W 3 (x,) = g 1 (x), (W 3 ) t (x,) = g 2 (x), x, W 3 (,t)+α 1 (W 3 ) x (,t) =, W 3 (,t)+β 1 (W 3 ) x (,t) =. (3.36) f(x,t) = f(x,t) D α t V 3(x,t)+(V 3 ) xx (x,t) V 3 (x,t), (3.37) g 1 (x) = φ(x) V 3 (x,t), g 2 (x) = ψ(x) µ 1() µ 2() α 1 β 1 x+ (+β 1)µ 1() α 1 µ 2(). (3.38) α 1 β 1 ike two previous subsections first we solve the homogeneous equation ( f(x,t) = ). By assuming W 3 (x,t) = X(x)T(t), and substituting it in (3.36), an ordinary linear differential equations obtained for X(x) as follows: { X (x)+λ 2 X(x) =, X()+α 1 X () =, X()+β 1 X () =.

9 CMDE Vol. 2, No. 2, 214, pp Note that the eigenvalues are countable and can be listed in a sequence λ 1 < λ 2 <... < λ n <..., with lim λ n =, (3.39) n and corresponding eigenfunctions are X n (x) = α 1 λ n cos(λ n x)+sin(λ n x), n = 1,2,... (3.4) we can show that where Xn 2 (x)dx = α 1 λ n cos(λ n x)+sin(λ n x) 2 dx = α2 1λ n 1 2 tanλ n. 1+tan 2 λ n α 1tan 2 λ n 1+tan 2 λ n + (α2 1λ ), (3.41) 2 tanλ n = (α 1 β 1 )λ n 1+α 1 β 1 λ 2. n The soloution for (3.36) is of the form W 3 (x,t) = B n (t)x n (t), (3.42) with following initial conditions: W 3 (x,) = B n ()( α 1 λ n cosλ n x+sinλ n x) = g 1 (x), (W 3 ) t (x,) = B n ()( α 1λ n cosλ n x+sinλ n x) = g 2 (x). (3.43) We assume that the series can be differentiated term by term. In order to determine B n (t), we expand f(x,t) as follows f(x,t) = f n (t)( α 1 µ n cosµ n x+sinµ n x), (3.44) where then f n (t) = b 1 n f(ξ,t)(α 1 µ n cosµ n ξ +sinµ n ξ)dξ, (3.45) D α B n (t)x n +λ 2 n B n (t)x n + B n (t)x n = f n (t)x n. (3.46) By orthogonality properties of X n (x), we get D α t B n(t)+(1+λ 2 n )B n(t) = f n (t). (3.47)

10 18 H. KHEIRI, S. SHAHI, AND A. MOJAVER According to lemma 2.1 where so B n (t) = t τ α 1 E α,α ( (λ 2 n +1)τα ) f n (t τ)dτ +u (t)b n ()+u 1 (t)b n(), (3.48) u (t) = 1 (1+λ 2 n)e α,α+1 ( (λ 2 n +1)t α ), u 1 (t) = t (1+λ 2 n )E α,α+2( (λ 2 n +1)tα ), u(x,t) = W 3 (x,t)+v 3 (x,t) = B n (t)x n + µ 1(t) µ 2 (t) α 1 β 1 x (+β 1)µ 1 (t) α 1 µ 2 (t). (3.49) α 1 β 1 4. Examples In this section, we consider three examples with different initial and boundary conditions and source terms. We show that the solution obtained above agree with those established in these examples. Example 1. Consider the following inhomogeneous fractional Klein-Gordon equation D α t u(x,t) 2 u(x,t) x 2 +u(x,t) = f(x,t), (4.1) with the initial condition and Dirichlet boundary conditions as follows: in which u(x,) =, u t (x,) =, x 1, (4.2) u(,t) = t 3, u(1,t) = t 3 +t 2, t, Γ(3) f(x,t) = sin(3πx) Γ(3 α) t2 α +x Γ(3) Γ(3 α) t2 α + + Γ(4) Γ(4 α) t3 α +(9π 2 +1)t 2 sin(3πx)+t 2 x+t 3. For solving the problem, we first transform it into a homogeneous boundary condition. Therefore suppose that u(x,t) = W 1 (x,t)+v 1 (x,t) = W 1 (x,t)+t 2 x+t 3, where W 1 (x,t) is an unknown function and satisfies in problem with homogeneous boundary conditions as: Dt α W 1(x,t) 2 W 1 (x,t) x 2 +W 1 (x,t) = f(x,t), W 1 (x,) =, (W 1 ) t (x,) =, x 1, (4.3) W 1 (,t) =, W 1 (1,t) =,

11 CMDE Vol. 2, No. 2, 214, pp in which f(x,t) = sin(3πx) [ Γ(3) Γ(3 α) t2 α +(1+9π 2 )t 2]. Now, we are going to solve the corresponding homogeneous equation in (4.3) by the method of separating variables. With similar manner in subsection 3.1, we obtain a Sturm-iouville problem and a linear ODE respect to x and t respectively. The eigenvalues and eigenfunctions of Sturm-iouville are as λ n = n 2 π 2, X n (x) = sin(nπx), n = 1,2,... (4.4) Then, we want to find a solution of the inhomogeneous problem in (4.3) which takes the form W 1 (x,t) = B n (t)sin(nπx). (4.5) If we substitute this in (4.3) and use arguments in subsection 3.1 we have Dt α B n (t)+ (1+ n2 π 2 ) B n (t) = f n (t), (4.6) in which with f n (t) = H(t), n = 3 f(x,t)sin(nπx)dx =, n 3, H(t) = Γ(3) Γ(3 α) t2 α +9π 2 t 2 +t 2. Since W 1 (x,t) satisfies the initial conditions in (4.3), we obtain B n ()sin(nπx) = sin(3πx), (4.7) which yields B n () = , n = 3 sin(3πx) sin(nπx)dx =, n 3. Hence lemma 2.1 implies that this problem has the solution as t τ α 1 E (α,α) ( ((nπ) 2 +1))τ α H(t τ)dτ, n = 3 B n (t) =, n 3. Now, if we take the aplace transform from both side of (4.9) we get (4.8) (4.9) [B n (t)] =, n 3, (4.1)

12 11 H. KHEIRI, S. SHAHI, AND A. MOJAVER and [B 3 (t)] = [ t τ α 1 E (α,α) ( ((3π) 2 +1)τ α )H(t τ)dτ] = [E (α,α) ( ((3π) 2 +1)τ α )][H(t)] [ ] = t α 1 ( 1) k [(3π) 2 +1] k t αk [H(t)] Γ(α+kα) ( ) ( 1) k [(3π) 2 +1] k t αk+α 1 = [H(t)] Γ(α+kα) = ( 1) k [(3π) 2 +1] k [H(t)] s αk+α ( ) 1 2 = s α +(3π) 2 +1 s 3 α + 9π2 s s 3 (4.11) = 2 s 3. From (4.1) and (4.11), we obtain { t B n (t) = 2, n = 3,, n 3. Therefore, the solution of (4.3) is W 1 (x,t) = t 2 sin(3πx). Then the exact solution of the fractional Klein-Gordon equation given in example 1 is u(x,t) = t 2 sin(3πx)+t 2 x+t 3. Example 2. Consider the inhomogeneous fractional Klein-Gordon equation 1.1, with the initial condition and Neumann boundary conditions as follows: in which u(x,) =, x 1, (4.12) u x (,t) = t 3, u x (1,t) = 2t 4 +t 3, t, f(x,t) = Γ(4) Γ(4 α) t3 α (cos(2πx)+x)+ Γ(5) Γ(5 α) t4 α x π 2 t 3 cos(2πx) 2t 4 +t 3 cos(2πx)+x 2 t 4 +xt 3.

13 CMDE Vol. 2, No. 2, 214, pp As before, for solving the problem, we first transform it into a homogeneous boundary condition. Therefore let u(x,t) = W 2 (x,t)+v 2 (x,t) = W 2 (x,t)+t 4 x 2 +t 3 x, where W 2 (x,t) is an unknown function and satisfies in problem with homogeneous boundary conditions as follows: Dt α W 2(x,t) 2 W 2 (x,t) x 2 +W 2 (x,t) = f(x,t), W 2 (x,) =, x 1, (4.13) (W 2 ) x (,t) =, (W 2 ) x (1,t) =, in which ( ) Γ(4) f(x,t) = cos(2πx) Γ(4 α) t3 α +(1+4π 2 )t 3. As before the eigenvalues and eigenfunctions of Sturm-iouville are as follows: λ n = n 2 π 2, X n (x) = cos(nπx), n = 1,2,... (4.14) Then, we obtain a solution of the inhomogeneous problem in (4.13) which takes the form W 2 (x,t) = B n (t)cos(nπx). (4.15) If we substitute this in (4.13) and use arguments in subsection 3.2 we have in which with D α t B n(t)+ ( 1+n 2 π 2) B n (t) = f n (t), (4.16) f n (t) = H(t), n = 2 f(x,t)cos(nπx)dx =, n 2, H(t) = Γ(4) Γ(4 α) t3 α +(1+4π 2 )t 3. Since W 2 (x,t) satisfies the initial conditions in (4.13), we obtain B n ()cos(nπx) = cos(2πx), (4.17) which gives B n () = , n = 2 cos(2πx) cos(nπx)dx =, n 2. Hence lemma 2.1 implies that this problem has the solution as B n (t) = t (4.18) τ α 1 E (α,α) ( ((nπ) 2 +1))τ α fn (t τ)dτ. (4.19)

14 112 H. KHEIRI, S. SHAHI, AND A. MOJAVER By taking the aplace transform from both side of (4.19), we get [B n (t)] =, n 2, (4.2) and [B 2 (t)] = [ t τ α 1 E (α,α) ( ((3π) 2 +1)τ α )H(t τ)dτ] = [E (α,α) ( ((2π) 2 +1)τ α )][H(t)] = [t α 1 ( 1) k [(2π) 2 +1] k t αk [H(t)] Γ(α+kα) ( ) ( 1) k [(2π) 2 +1] k t αk+α 1 = [H(t)] Γ(α+kα) (4.21) = ( 1) k [(2π) 2 +1] k [H(t)] s αk+α ( ) 1 6 = s α +(2π) 2 +1 s 4 α + 24π2 s s 6 = 1 s 4. From (4.2) and (4.21), we obtain B 2 (t) = { t 3, n = 2,, n 2. (4.22) Therefore, the solution of (4.13) is W 2 (x,t) = t 3 cos(2πx). Then the exact solution of the fractional Klein-Gordon equation given in example 2 is u(x,t) = t 3 cos(2πx)+t 4 x 2 +t 3 x.

15 CMDE Vol. 2, No. 2, 214, pp Example 3. Consider the fractional Klein-Gordon problem Dt α u(x,t)+u xx (x,t)+u(x,t) = f(x,t), u(x,) =, u t (x,) =, u(,t) 1 2π u x(,t) = t 4 +t 3 (1 1 2π ), u(1,t) 1 2π u x(1,t) = t 4 +t 3 (2 1 2π ), which so f(x,t) (4.23) = t 4 + Γ(5) Γ(5 α) t4 α +4π 2 t 3 (sin(2nπ)+cos(2nπ)) (4.24) Γ(4) +( Γ(4 α) t3 α +t 3 )(x+1+sin2nπ+cos2nπ), (4.25) V 3 (x,t) = t 3 (x+1)+t 4 (4.26) and from (3.37), we get Γ(4) f(x,t) = (sin2πx+cos2πx)( Γ(4 α) t3 α +t 3 (4π 2 +1)). (4.27) In this example, we have λ n = 2nπ, (4.28) because of orthogonality of X n (x) = sin2πx+cos2πx, fn (t), will be as follows: f n (t) = { Γ(4) Γ(4 α) t3 α +t 3 (4π 2 +1), n = 2,, n 2. (4.29) Now from emma 2.1, B 2 (t) is as B 2 (t) = t τ α 1 E α,α ( (4π 2 +1)τ α Γ(4) ) Γ(4 α) (t τ)3 α (4.3) +(t τ) 3 (4π 2 +1)dτ. (4.31) By applying the aplace transform for both side of the last equation just like what we did in example 1 for (4.11) we can show that (B n (t)) = { 6 s4, n = 2,, n 2. (4.32) Hence B 2 (t) = t 3 (4.33) and as a result, the solution of (4.23) is as follows: u(x,t) = t 3 (x+1+sin2πx+cos2πx). (4.34)

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