Hermite Wavelet Method for Fractional Nondelay Differential Equation

We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method.


Introduction
The future state of a physical system depends not only on the present state but also on its past history.Functional differential equations provide a mathematical model for such physical systems in which the rate of change of the system may depend on the influence of its hereditary effects.Delay differential equations have numerous applications in mathematical modeling [1], for example, physiological and pharmaceutical kinetics, chemical kinetics, the navigational control of ships and aircrafts, population dynamics, and infectious diseases.
Delay differential equation is a generalization of the ordinary differential equation, which is suitable for physical system that also depends on the past data.During the last decade, several papers have been devoted to the study of the numerical solution of delay differential equations.Therefore different numerical methods [2][3][4][5][6][7] have been developed and applied for providing approximate solutions.
Method of steps is easy to understand and implement.In the method of steps [8], we convert the delay differential equation to a nondelay differential equation.The method of steps is utilized in [9] for solving integer order delay differential equations.
Hermite wavelet method [10] is implemented for finding the numerical solution of the boundary value problems and compares the obtained solutions with exact solution.In [11], authors utilized the physicists Hermite wavelet method for solving linear singular differential equations.According to our information, Hermite wavelet method has not been implemented for delay differential equations.
In the present work, we established a technique by combining both the method of steps and the Hermite wavelets method for solving the fractional delay differential equation.We also implemented the Hermite wavelet method for solving fractional delay differential equation, as described in Example 6, which was not implemented before.Shifted Chebyshev nodes are used as the collocation points.Comparison of solutions by these two methods, proposed method and Hermite wavelet method, with each other and with exact solution are also presented.
Since m = 2 −1  and method converges if m → ∞; that is, when we use higher order Hermite polynomials  − 1 or use higher level of resolution  or use both higher  and , we get more accurate results.

Procedure for Implementation of Proposed Scheme
The method of steps [8]  Proposed method consists of two methods, method of steps and Hermite wavelet method.We first implement the method of steps to the fractional delay differential equation (11) and get the fractional nondelay differential equation by utilizing initial function, (), and then we utilize the Hermite wavelet method for solving the obtained fractional nondelay differential equation.It is a fractional nondelay differential equation because  0 ( − ) and   0 ( − ) are known.

Hermite Wavelet Method for Fractional Nondelay Differential Equation.
We solve the obtained fractional nondelay differential equation ( 12) on [, ] by using the Hermite wavelet method.The procedure for implementation of Hermite wavelet method for fractional differential equation is as follows.

Numerical Solutions
In this section, we utilize the proposed scheme for finding the numerical solution of linear and nonlinear fractional delay differential equations.The notations  pro ,  exact , and  abs represent the solution by proposed method, exact solution, and their absolute error, respectively.We use the results up to 100 decimal places.Through this work we use Caputo derivatives.For the details of fractional derivatives and integrals we refer the readers to [13].
The results obtained by the proposed method, by taking  = 0.5,  = 1,  = 3, and  = 0.01 − , are shown in Figure 1 along with the exact solution.Table 1 indicates that results obtained from the proposed method are closer to exact solution and better than the method [2]. abs and error represent the absolute error by proposed method and method [2], respectively.
Example 2. Consider the following fractional delay differential equation: subject to the initial conditions (0) = 1,   (0) = −1, and   (0) = 1.The exact solution, when  = 1, is () =  − .2, proposed method provides more accurate results as compared to Adomian decomposition method [3].These results are obtained by fixing  = 25 and  = 1, at  = 3. Solutions by the proposed method at different values of  are shown in Figure 2, which shows that solutions by proposed method at different  converge to the exact solution at  = 3, when  approaches to 3.
Solution by proposed method  By fixing  = 1 and  = 4, we plot the solutions by proposed method at different values of  and exact solution at  = 1, as shown in Figure 3.It shows that proposed solution approaches to the exact solution while  approaches to 1.
Solution by proposed method x-axis y(x) y(x) at  = 0.5 y(x) at  = 0.6 y(x) at  = 0.7 y(x) at  = 0.8 y(x) at  = 0.9 y(x) at  = 1 Exact solution at  = 1  Solution by proposed method at different values of  is plotted in Figure 5, which shows that proposed solution converges to the exact solution when  approaches to 2. According to Table 5, absolute error reduces while increasing .

Solution by proposed method
x-axis y(x) y(x) at  = 0.7 y(x) at  = 0.8 y(x) at  = 0.9

Comparison of Proposed Method and Hermite Wavelet Method
Example 6.Consider the fractional nonlinear neutral delay differential equation subject to the initial conditions, (0) = 1.The exact solution [15], when  = 1, is () =   .

Hermite Wavelet Method for Fractional Delay Differential
Equation.We can approximate the solution of (23) by the Hermite wavelet method as In the delay equations, we also have to approximate the delay unknown function (/2) in terms of the Hermite wavelet series at delay time as We call this series as the delay Hermite wavelet series.Substituting ( 24) and ( 25) in ( 23), we get the residual as × ( Set the residual (26) to be equal to zero at the set of Chebyshev nodes,   = (1/2) cos((2 + 1)/2  ) + (1/2),  = 0, 1, 2, . . .,  − 1, on interval [0, 1], we get We get 2 −1  − 1 equations from (27) by using Chebyshev nodes   .One more equation is obtained from the condition of (23); that is, We obtained 2 −1  nonlinear equations along with 2 −1  unknown coefficients   , which is solved by Newton iterative method to get   's and used in (24) to get the approximate solution by Hermite wavelet method.We fix  = 9,  = 1, and implement the Hermite wavelet and method to (23).The results are shown in Table 6 along with the absolute errors.According to Table 6, Hermite wavelet and proposed methods give good results; that is, for some points proposed method is more accurate as compared to Hermite wavelet method and vice versa.
For the problem (23), run time of proposed method and Hermite wavelet method is 1.21 and 26.21 seconds, respectively.Proposed method is more efficient than the Hermite wavelet method. HWM and  HWM represent the solution by Hermite wavelet method and their absolute error, respectively.
For this purpose, we use Maple 13 in system with Core Duo CPU 2.00 GHz and RAM 2.50 GB.

Conclusion
It is shown that proposed method gives excellent results when applied to different fractional linear and nonlinear delay differential equations.The results obtained from the proposed method are more accurate and better than the results obtained from other methods, as shown in Tables 1-4.The solution of the fractional delay differential equation converges to the solution of integer delay differential equation, as shown in Figures 2-5.According to the convergence analysis, error by the proposed method reduces while increasing , as shown in Table 5.Table 6 indicates that both Hermite wavelet method and proposed method give good results.Proposed method is more efficient than the Hermite wavelet method.
Solution by proposed method y(x) at  = 0.5 Exact solution at  = 0.5

Figure 4 :
Figure 4: Solution by the proposed method at different ,  = 5 and exact solution at  = 1.

Figure 5 :
Figure 5: Solution by the proposed method at different ,  = 5 and exact solution at  = 2.

Table 1 :
[2]parison of the solution by proposed method at  = 3, with Exact solution and method[2].

Table 5 :
Absolute errors by using proposed method at different  and  = 2.