A new perspective on the numerical solution for fractional

In the present manuscript, a new numerical scheme is presented for solving the time fractional nonlinear Klein-Gordon equation. The approximate solutions of the fractional equation are based on cubic B-spline collocation finite element method and L2 algorithm. The fractional derivative in the given equation is handled in terms of Caputo sense. Using the methods, fractional differential equation is converted into algebraic equation system that are appropriate for computer coding. Then, two model problems are considered and their error norms are calculated to demonstrate the reliability and efficiency of the proposed method. The newly calculated error norms show that numerical results are in a good agreement with the exact solutions.


INTRODUCTION
Fractional differential equations own a deep history and also rich theory.Its past is as long as classical calculus and up to date since 1695 .Over the years, many mathematician and physicist have been attracted by fractional calculus because of its wide application areas, longterm memory and chaotic behaviour such as physics, biology, finance, fluid dynamics, engineering etc.The development and obtaining numerical and exact solutions of the equations, containing fractional derivative and integral, have gained great and significant importance.So, various methods have been investigated for this purpose.Among others,some of them are [1,2,3,4,5,6,7].In this study, we are going to concern with obtaining numerical solutions of time fractional Klein Gordon equation in terms of Caputo sense derivative which is one of the fundamental equations seen in fractional calculus.The mathematical expression of the equation is given as *Sorumlu Yazar (Corresponding Author) e-posta : bkaraagac@adiyaman.edu.tr D u x t au x t bu x t cu x t u x t f x t subject to the following initial and boundary conditions where (.) t D  symbolizes th  order fractional derivative according to time variable and the range of  is (1, 2] . x t is a known forced term and in addition to these terms ,, abc and  are real constants and also c can seen as a variable coefficients in some examples.For 2   , we get the classical Klein Gordon equation which appears in classical relativistic and quantum mechanics and analysing of wave propagation in linear dispersive media.Additionally, The fractional Klein-Gordon equation has many application in nonlocal quantum field theory and stochastic quantization of nonlocal fields [8] The equation has been solved by several authors using different methods and techniques.Among others, Nagy [7] has solved the problem using a method consisting of expanding the required approximate solution as the elements of Sinc functions along the space direction and shifted Chebyshev polynomials of the second kind for the time variable.Kheri et.al. [9] have solved inhomogenous fractional Klein-Gordon equation by the method of separating variables and applied the method for three boundary conditions.Mohebbi et al. [10] have applied a high-order difference scheme for the solution of some time fractional partial differential equations including linear time fractional Klein-Gordon and dissipative Klein-Gordon equations.Lyu and Vong [11] have considered difference schemes for nonlinear timefractional Klein-Gordon type equations.Khader et al. [12] have implemented the Chebyshev spectral method for solving the non-linear fractional Klein-Gordon equation and considered the fractional derivative in the Caputo sense.Alqahtani [13] has implemented the spectral collocation method with the help of the Legendre polynomials for solving the nonlinear Fractional (Caputo sense) Klein-Gordon Equation.
For all that, recent developments in computational methods are lead to improving new numerical methods for solving fractional or ordinary order partial differential equations.One can receive more information about newly research in Refs [ 14,15,16,17,] where In order to define all spline basis in same typical element   .
Moreover the approximate solution can be written in terms of the basis given in (5) as The nodal values of First all of , we will start to progress with a simple linearization choosing = N zm u .Then substituting (2) into Eq.(1) and using (7), we have For going on to obtain numerical scheme, time dependent element shape functions   t  s' are discretized using 2 L algorithm given in (1.3), forward difference and crank-Nicolson formula as At the end, after some calculation and simplification we we get a algebraic equation system consisting   parameters and ,   Now we get a system consisting of   1 N  equations and   3 N  unknown variables.Eq. ( 10) is valid for only interior nodal points so to obtain unique solution one must apply boundary conditions given in (2) to numerical scheme.For this purpose, we employ from the system we get

   
11 NN    system of equations.At the last , we have a iterative system.Now, we need an initial vector for begining iteration, so one can obtain   0 n  parameters easily by using initial conditions as   (11) as seen from the Eq. ( 11), there exist   system can now be solved with any algorithm and iteration can be started.

Numerical Tests for Time Fractional Klein Gordon Equation
In the third section of the manuscript, we are going to demonstrate efficiency and applicability of numerical method using two test problems.For where u and N U represent exact and numerical solutions, respectively.And the order of convergence is calculated with the following formula; and with non-zero right boundary condition exact solution of the example is given as First of all, the results are calculated for various space step    when partition number is chosen as = 10

N
. Also applied method has more converges results for all  values when partition number is chosen as > 10

CONCLUSION
In conclusion, this study has introduced to obtain numerical approximations using the finite element method.We have started with the application of the method to the time fractional Klein Gordon equation.
Then, the problem has converted into ordinary differential equation system with the help of cubic Bspline basis and element shape parameters.Using mathematical coding, the newly obtained numerical scheme is solved with an iterative method uses an initial vector to purpose of generate approximate solution for two test problem.Additionally, the newly numerical results with the calculation have shown that has a reasonable agreement with exact ones.After all the numerical experiments examined in this paper, we can conclude that finite element collocation method and L2 algorithm can be a useful tool for obtaining numerical solutions of wide variety problems on fractional order partial differential equations.

3 . 1 . Example 1 :
are able to compare exact ones with numerical ones.In the following first numerical experiment, we have taken time fractional Klein-Gordon equation with the values of the coefficients as = 1So we can rewrite the equation given in (1) with the forced term, as follows with the boundary conditions and the exact solution is

Figure 1 . 3 
Figure 1.The numerical solutions of Time Fractional Klein Gordon equation for = 1.3 

Figure 2 . 8 
Figure 2. The numerical solutions of Time Fractional Klein Gordon equation for = 1.8 

Figure 3 . 3 
Figure 3.The numerical solutions of Time Fractional Klein Gordon equation given in example 2 for = 1.3 

Figure 4 .
Figure 4.The numerical solutions of Time Fractional Klein Gordon equation given in example 2 for = 1.8 

Application of collocation cubic B-spline FEM method to the time fractional Klein Gordon equation In
this part of paper, we are going to obtain numerical solution for the fractional Klein Gordon equation with the help of finite element formulation and cubic spline basis.
and therein.The manuscript consists of four parts.The first part presents an introduction to the model problem and some research papers on it.The second one covers application of cubic B-spline collocation method to the problem and obtaining numerical formulation.Two different examples of time fractional Klein Gordon equation and their numerical results are considered in the third part for different values of constants and forced term.The last one is conclusion 2. j t as the grid points for time and t  are grid size.So time discretization for two examples, since the exact solutions of the examples are known, we are going to calculate error norms

Table 1 .
Tables 1, 2 and 3, when the number of time step size are the same, to increase number of collocation points lead to a decrease in the error norms.Additionally, for the collocation finite element method, time step sizes as important as collocation points.So one can see, decreasing of time step sizes results decreasing in the error norms.A representation of the h and time step   2 L and L  are presented in Tables1 and 2, respectively.In addition, error norms and orders are

Table 3 .
The error norms and orders for various

Table 7 .
A

Table 8 .
A

Table 10 .
Numerical results have same behaviour as first example i.e decreasing in time and space step size ends up with decreasing in the error norms.At the end of this part, numerical simulations of example 2 are depicted for the various choosing of  parameters in Figures1 and 2. A representation of the

Table 11 .
A representation of the

Table 12 .
A representation of the