On numerical solution of fractional order boundary value problem with shooting method

In this study the shooting method is used for calculation of the second order boundary value problem with fractional order. This method is found to be useful for the application. The accuracy of the shooting method is tested. Some examples are given to illustrate the efficiency of the method with respect to different value of fractional orders.


Introduction
The history of fractional calculus and fractional differential equations date back to 1695 (refer to [6] for a brief history). Studies on fractional calculus goes back to studies on calculus [11]. Numerous publications relevant with fractional differential equations have been published in time. Some of them are [1]- [4], [7], [8], [10]. Also some important books on fractional field can remark as [3] and [7]. Related with our subject, Mohamed and Mahmoud have investigated fractional Euler's method and modified trapezoid rule in ( [7]). Additionally Tong, Feng and LV have also investigated the error analysis of the fractional Euler's method in ([9]).
In this study a second order boundary value problem (9) is considered. First of all this BVP has been solved analytically and evaluated the value of this exact solution at the certain h points. Then the shooting method has been used to evaluate the numerical solution of this boundary value problem by using the formula (7)-(8) which is given in ( [7]). During the application of the shooting method, for predictor the fractional Euler method and for corrector the modified trapezoid rule for systems has been applied to solve the system of IVP (10)- (11). This solution has implemented by taking 1 , 1 m n   and we have seen that the solution values of the shooting method are compatible with exact solution values at the same h point. Then the calculations have been repeated with different , m n values which are fractional orders. Consequently the results was have been submitted in a table.

Shooting method
For the use of the shooting method, firstly need the changing of the boundary value problem to system of initial value problems and then need to assume a guess value at the lower bound of the interval. After these, the solution goes on like the solution of system of differential equations.
Let us consider a second order boundary value problem boundary conditions. At first boundary value problem, which is given, is converted to two first order differential equation: After deciding the h step size we use one of the known methods like Euler, Runge-Kutta, etc. to solve these equations. Then we gain the first numerical result of the BVP that we consider subject to the first guess value  . The solution is valid if the solution is enough closer to the upper bound  , else the procedure will repeat with another guess.
After two guesses, instead of making the third guess arbitrary, we can use the interpolation to obtain the third and if necessary the next guesses [5].

For predictor the Fractional Euler method and for corrector the Modified Trapezoid Rule formula for systems
In this section the basic formula for the numerical solution of the system of fractional differential equations will be given. Let us consider the system of fractional differential equations. To solve this system numerically the basic formula that we use is:

Numerical example
Example: Let us consider the    

Conclusion
In this study a boundary value problem, which has exact solution is considered. The shooting method is used for the numerical solution of this boundary value problem. To solve the two initial value problems which arise when applying shooting method, the formula for predictor the fractional Euler method and for corrector the modified trapezoid rule for systems is used. Because of the incorrect solution values of the boundary value problem subject to arbitrary guesses the third guess is made by using the interpolation formula. The solution was repeated with this guess value. The results that obtained were compared with exact solution values and approximated the target value (the other boundary condition of boundary value problem) with high sensivity. Additionally, for different , m n values, numerical solution values at certain h points are in line with the exact solution values of the boundary value problem. Finally, the shooting method can be used for the numerical solution of the fractional order boundary value problem as a result of this consistence of the values in the Table 1.