Abstract
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.
1. 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 [2–7] 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.
2. Preliminaries
2.1. Hermite Wavelets
The Hermite polynomials , of order , are defined on the interval and given by the following recurrence formulae: The polynomials are orthogonal with respect to the weight function ; that is, The discrete wavelets transform is defined as The set forms an orthogonal basis of . That is, The Hermite wavelets are defined on interval by where , is the level of resolution, , , is the translation parameter, is the order of the Hermite polynomials, .
2.2. Function Approximations
We can expand any function into truncated Hermite wavelet series as where and are () matrices, given by
3. Convergence Analysis
Let be a Hilbert space for which form an orthonormal sequence in . Let ; we have where is an inner product of and . Equation (8) can be written as For simplicity, let ; we can write (9) as where , , , and ; .
By following the procedure in [12], we obtained the convergence of all the orthogonal wavelet methods for all levels of resolution ; that is, converges to , as .
Since and method converges if ; that is, when we use higher order Hermite polynomials or use higher level of resolution or use both higher and , we get more accurate results.
4. Procedure for Implementation of Proposed Scheme
The method of steps [8] is used to convert the discrete delay differential equations to nondelay differential equations on a given interval. Consider the following fractional delay differential equation with discrete delay: where is a source function and is a continuous linear or nonlinear function. Also is constant, is delay, and is called delay argument. The delay is called constant delay, time dependent delay, and state dependent delay if the delay is constant, function of time , and function of time and , respectively.
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.
4.1. Method of Steps
In the fractional delay differential equation the solution is known on , say , and call this solution ; that is, , which is known. Now the fractional delay differential equation on takes the form subject to the initial conditions , .
It is a fractional nondelay differential equation because and are known.
4.2. 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.
Step 1. Approximate the unknown function of (12) by the Hermite wavelet method as
Step 2. Substitute (13) in (12) to get the residual
Step 3. Set the residual (14) to be equal to zero at the set of Chebyshev nodes, , , on interval . Consider
We obtain equations, where is the number of conditions of the delay equation. According to (11), two conditions are given, , so we get equations from (15) by using Chebyshev nodes . Two more equations are obtained from the conditions of (11); that is,
We obtained equations either linear or nonlinear along with unknown coefficients , which is solved by Newton iterative method to get ’s and use it in (13) to get the approximate solution. Denote the obtained solution as , which is defined on .
Continue the procedure for the subsequent interval; delay differential equation on becomes
subject to the initial conditions , , which is again a fractional nondelay differential equation and solve it by the Hermite wavelet method to get on . This procedure may be continued for subsequent intervals.
5. 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 , , and represent the solution by proposed method, exact solution, and their absolute error, respectively. We use the results up to decimal places. Through this work we use Caputo derivatives. For the details of fractional derivatives and integrals we refer the readers to [13].
5.1. Linear Delayed Fractional Differential Equations
Example 1. Consider the fractional delay differential equation The exact solution [2], when , is .
The results obtained by the proposed method, by taking , , , and , 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]. 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 , , and . The exact solution, when , is .
According to Table 2, proposed method provides more accurate results as compared to Adomian decomposition method [3]. These results are obtained by fixing and , at . 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 , when approaches to .
Example 3. Consider the fractional pantograph equation
The exact solution, when , is .
By fixing and , we plot the solutions by proposed method at different values of and exact solution at , as shown in Figure 3. It shows that proposed solution approaches to the exact solution while approaches to .
Comparison of proposed solution at , , and , with Adomian decomposition method [3] and the spline function technique [4], is shown in Table 3. The notations , , and represent the absolute error by proposed method, Adomian decomposition method, and the spline function technique, respectively. We can get more accurate results while increasing .
Example 4. Consider the following fractional neutral functional differential equation with proportional delay:
which has the exact solution, when , which is .
We implement the proposed method by fixing , , at different values of , the results are shown in Figure 4 along with the exact solution at . Table 4 shows that proposed method gives more accurate results as compared to the spectral shifted Legendre-Gauss collocation (SLC) method [5], the reproducing kernel Hilbert space method (RKHSM) [6], and a Runge-Kutta-type (RKT) method [7].
5.2. Nonlinear Delayed Fractional Differential Equations
Example 5. Consider the fractional nonlinear delay differential equation
subject to the initial conditions, and . The exact solution [14], when , is .
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 . According to Table 5, absolute error reduces while increasing .
5.3. Comparison of Proposed Method and Hermite Wavelet Method
Example 6. Consider the fractional nonlinear neutral delay differential equation subject to the initial conditions, . The exact solution [15], when , is .
5.4. 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 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, , , on interval , we get We get equations from (27) by using Chebyshev nodes . One more equation is obtained from the condition of (23); that is, We obtained nonlinear equations along with 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 , , and implement the Hermite wavelet and proposed 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 and seconds, respectively. Proposed method is more efficient than the Hermite wavelet method. and 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.
6. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors are grateful to the anonymous reviewers for their valuable comments which led to the improvement of the paper.