A Four-Stage Fifth-Order Trigonometrically Fitted Semi-Implicit Hybrid Method for Solving Second-Order Delay Differential Equations.

1. Introduction

Research reveals that things depend on not only the current state of a system but also the past states, resulting in differential equations with a time delay. This kind of equations is called delay differential equations (DDEs) where the derivative at any time depends on the solution at prior times and is best known as model that incorporates past history. It is a more realistic model which includes some of the past history of the system to determine the future behavior. In this paper, we deal with numerical method for solving second-order delay differential equations (DDEs) with constant delay which can be written in the form of

[mathematical expression not reproducible], (1)

where [tau] is the delay term and the first derivative does not appear explicitly. There are many applications which are well known related to DDEs such as population dynamics, epidemiology, and reforestation.

Equation (1) can be solved using methods such as direct multistep method, Runge-Kutta Nystrom (RKN) method, and hybrid method. There has been a growing interest in the field of DDEs; Kuang [1] in his book discussed delay differential equations with applications in population dynamics. Bt Ismail and Suleiman [2] studied the P-Stability and Q-Stability of singly diagonally implicit Runge-Kutta method for delay differential equations. Taiwo and Odetunde [3] studied delay differential equations using a decomposition method. Ismail et al. [4] used Runge-Kutta method and Hermite interpolation to solve delay differential equations. Hoo et al. [5] constructed a direct Adams-Moulton Method for solving second-order delay differential equations. Some other authors also derived block multistep method to solve delay differential equations; such work can be seen in San et al. [6], Radzi et al. [7], and Ishak et al. [8].

In our previous work Ahmad et al. [9] have derived semi-implicit hybrid method of four stages and fifth order denoted as SIHM4(5), where we incorporated the phase-lag and amplification equations, in the derivation, so that we obtained method which has higher order of dissipation and dispersion. But in this paper, we are going to construct a new semi-implicit hybrid method (NSIHM) of four stages and fifth order using the algebraic order conditions given in Coleman [10]; the derivation also incorporates the simplifying conditions as well as the technique of minimization of the error constant. The method is then trigonometrically fitted using similar approach as in [11, 12] so that it has a higher order of dissipation and dispersion; this approach is simpler than incorporating the phase-lag and amplification equations into the derivation. In all the previous work regarding phase-fitted methods, the methods are used to solve oscillatory second-order ordinary differential equations (ODEs). Here the trigonometrically fitted method is used for solving retarded second-order delay differential equations which are oscillatory in nature. The efficiency of the new method will be compared with several other existing explicit and implicit methods of RKN type and hybrid methods.

2. Derivation of Four-Stage Fifth-Order Semi-Implicit Hybrid Methods

An s-stage semi-implicit hybrid method for the numerical integration of the IVPs is given as

[mathematical expression not reproducible], (2)

where i = 1, ..., s and i [greater than or equal to]. The equations of form (2) are defined as

[mathematical expression not reproducible], (3)

where the first two nodes are [c.sub.1] = -1 and [c.sub.2] = 0 and i = 3, ..., s, while functions [f.sub.n-1] = f([t.sub.n-1], [y.sub.n-1]) and [f.sub.n] = f([t.sub.n], [y.sub.n]). The coefficients of [b.sub.i], [c.sub.i], and [a.sub.ij] can be represented in Butcher tableau as follows:

[mathematical expression not reproducible] (4)

The coefficients of the diagonal element ([gamma]) are always equal for this method. Here, we derive the four-stage fifth-order NSIHM based on the order conditions, simplifying conditions, and minimization of the error constant [C.sb.p+1] of the method. The error constant is defined by

[mathematical expression not reproducible], (5)

where k is the number of trees of order p + 2 (p([t.sub.i]) = p + 2) and [e.sub.p+1] ([t.sub.i]) is the local truncation error defined in Coleman [10]. The order conditions defined in Coleman [10] for hybrid method up to order six are listed as follows:

(i) Order 2:

[s.summation over (i=1)][b.sub.i] = 1. (6)

(ii) Order 3:

[s.summation over (i=1)] [b.sub.i][c.sub.i] = 0. (7)

(iii) Order 4:

[mathematical expression not reproducible]. (8)

(iv) Order 5:

[mathematical expression not reproducible]. (9)

(v) Order 6:

[mathematical expression not reproducible], (10)

where value of i [greater than or equal to] j [greater than or equal to] k. For [c.sub.i], the method needs to satisfy the simplifying condition which is

[s.summation over (i)] [a.sub.ij] = ([c.sub.i.sup.2] + [c.sub.i])/2, for i = 3, ... ,s. (11)

First, we derive the four-stage fifth-order NSIHM using the algebraic order conditions up to order five (see (6) to (9)) and simplifying condition in (11). We obtained the solution for the coefficients in terms of [a.sub.41], [a.sub.43], [a.sub.44], [b.sub.4], and [c.sub.4] listed as follows:

[mathematical expression not reproducible], (12)

where

[mathematical expression not reproducible]. (13)

By minimizing the error norm and letting the values of [a.sub.41] = 1506Y7l77W20, [a.sub.43] = 1410971771120, [a.sub.44] = 1/81, [b.sub.4] = 1/ 81, and [c.sub.4] = 1, we obtained the other coefficients of the four-stage fifth-order semi-implicit hybrid method denoted as NSIHM4(5) which can be written in Butcher tableau as follows:

[mathematical expression not reproducible]. (14)

The norm of the principal local truncation error coefficient for [y.sub.n] is given by

[parallel][[tau].sup.(6)][parallel][sub.2] = 1.1472 x [10.sup.-1], (15)

where [parallel][[tau].sup.(6)][parallel][sub.2] is the norm of the error equations for the sixth-order method.

3. Trigonometrically Fitting the Semi-Implicit Hybrid Method

To trigonometrically fit the new method NSIHM4(5), we consider stage three and stage four of the NSIHM4(5) in (14). The new method which will be derived is denoted as four-stage fifth-order trigonometrically fitted semi-implicit hybrid method or TF-NSIHM4(5) which is of fifth algebraic order which is the same as the algebraic order of NSIHM4(5). Note that trigonometrically fitting the method will not change the algebraic order of the method. The method can be written in Butcher tableau as follows:

[mathematical expression not reproducible] (16)

The values of [a.sub.32], [a.sub.42], [b.sub.1], [b.sub.2], and [b.sub.3] are modified using the trigonometrically fitting technique so that it would improve the accuracy of the method and suitable for solving oscillatory problems.

We require the internal stage (stages 3 and 4) and the updating stage to integrate exactly the linear combination of the functions {sin(vt), cos(vt)} for v [member of] R subject to the fifth-order formulae. Hence, we obtain the following equations:

[mathematical expression not reproducible], (17)

[mathematical expression not reproducible], (18)

[mathematical expression not reproducible], (19)

[mathematical expression not reproducible], (20)

[mathematical expression not reproducible], (21)

[mathematical expression not reproducible]. (22)

Solving (17) to (20) with the choice of coefficients [c.sub.3] = 1, [c.sub.4] = 1, [b.sub.33] = 1/81, [a.sub.41] = 150617/771120, [a.sub.43] = 141097/771120, and [a.sub.44] = 1/81 simultaneously, we obtained

[mathematical expression not reproducible], (23)

where = vh, h is step size, and v is the fitted frequency. Next, using (21) and (22) and another two additional order conditions in (6) and (7) for the fifth-order method which are

[b.sub.1] + [b.sub.2] + [b.sub.3] + [b.sub.4] = 1, -[b.sub.1] + [b.sub.3][c.sub.3] + [b.sub.4][c.sub.4] = 0, (24)

with the choice of coefficients [c.sub.3] = 1, [c.sub.4] = 1, and [b.sub.4] = 1/81,we solve the equations simultaneously to get [b.sub.1], [b.sub.2], and [b.sub.3], which are given as follows:

[mathematical expression not reproducible], (25)

where

Q = [H.sup.2] (cos (H) - 1). (26)

The above formulae can be expressed in Taylor series expansions:

[mathematical expression not reproducible]. (27)

The values of [a.sub.32], [a.sub.42], [b.sub.1], [b.sub.2], and [b.sub.3] are constants for constants v and h and the other coefficients remain the same.

4. Problems Tested and Numerical Results

In this section, the new method NSIHM4(5) and the trigonometrically fitted method, TF-NSIHM4(5), are used to solve a set of oscillatory delay differential equations problems. The delay terms are evaluated using Newton divided different interpolation. Numerical results are tabulated and compared with the existing explicit and implicit methods in the scientific literature. The test problems are listed as follows.

Problem 1. It is as follows:

y" (t) = -1/2y(t) + 1/2y(t - [pi]), 0 [less than or equal to] t [less than or equal to]8n[pi][y.sub.o] = 0. (28)

The fitted frequency is v = 1. Exact solution is y(t) = sin(t), source: Schmitt [13].

Problem 2. It is as follows:

y" (t) - y(t) + [eta](t) y (t/2) = 0, 0 [less than or equal to] t [less than or equal to] 2[pi]n, (29)

where

[eta](t) = 4 sin(t)/[(2 - 2 cos(t)).sup.1/2], [eta](0) = 4. (30)

Fitted frequency is v = 2. Exact solution is y(t) = sin(t), source: Schmitt [13].

Problem 3. It is as follows:

y" (t) = y(t - [pi]), 0 [less than or equal to] t [less than or equal to] 8[pi], [y.sub.o] = 0. (31)

Fitted frequency is v = 1. Exact solution is y(t) = sin(f), source: Ladas and Stavroulakis [14].

Problem 4. It is as follows:

y" (t) = -sin (t)/2 - sin (t) y(t - [pi]), 0 [less than or equal to] t [less than or equal to] 8[pi], [y.sub.o] = 2. (32)

Fitted frequency is v = 1. Exact solution is y(t) = 2 + sin(f), source: Singh [15].

The following notations are used in Figures 1-4:

(i) SIHM4(5): SIHM derived in Ahmad et al. [9].

(ii) NSIHM4(5): a four-stage fifth-order new semi-implicit hybrid method derived in this paper.

(iii) TF-NSIHM4(5): a four-stage fifth-order new trigonometrically fitted semi-implicit hybrid method derived in this paper.

(iv) DIRKN4(4): a four-stage fourth-order dispersive order six of DIRKN method by Senu et al. [16].

(v) MPAFRKN4(4): modified phase-fitted and amplification-fitted RKN method of four stages and fourth order by Papadopoulos et al. [17].

(vi) EHM4(5): explicit hybrid method of order five derived in Franco [18].

(vii) DIRKN3(4): a three-stage fourth-order dispersive order six of DIRKN method by Senu et al. [19].

(viii) PFRKN4(4): a phase-fitted RKN method of four stages and fourth order by Papadopoulos et al. [20].

A measure of the accuracy is examined using absolute error which is defined by

Absolute error = max {[absolute value of y ([t.sub.n]) - [y.sub.n]]}, (33)

where y([t.sub.n]) is the exact solution and [y.sub.n] is the computed solution. The efficiency curves are presented whereby the logarithms of the maximum global errors are plotted against the CPU time in seconds.

In analyzing the numerical results, methods of the same order or stage are compared. The results are given in Figures 1-4. We observed that all the methods give better accuracy for smaller step size, h. However the new TF-NSIHM4(5) method is the most efficient method in solving oscillatory DDEs compared to other explicit and implicit RKN and hybrid methods.

5. Discussion and Conclusion

In this paper we derived a four-stage fifth-order semi-implicit hybrid method. The method has a minimized local truncation error and it is suitable for directly solving special second-order ODEs. The new method is called NSIHM4(5); it is then trigonometrically fitted so that it is suitable for solving oscillatory problems and denoted as TF-NSIHM4(5).

Both methods are then used for solving oscillatory second-order DDEs; numerical results clearly have shown that the TF-NSIHM4(5) is the most efficient compared to the original NSIHM4(5) method and other existing methods in the scientific literature. Take note also that the existing methods were derived with high order of dispersion and dissipation and purposely derived for solving highly oscillatory problems. Trigonometrically fitting the method improved the efficiency of the NSIHM4(5) hybrid method and it is much easier to derive compared to the approach whereby we have to include the dispersion and dissipation equations in the derivation of the method.

Even though the new TF-NSIHM4(5) is a semi-implicit method and fairly expensive in computation time, it is still more efficient compared to the existing explicit and implicit methods.

http://dx.doi.org/10.1155/2016/2863295

Competing Interests

The authors declare that they have no competing interests.

References

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Sufia Zulfa Ahmad, (1) Fudziah Ismail, (1,2) and Norazak Senu (1,2)

(1) Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

(2) Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to Fudziah Ismail; fudziah_i@yahoo.com.my

Received 23 February 2016; Accepted 26 April 2016

Caption: Figure 1: The efficiency curves for Problem 1 with h = [pi]/4i, for i = 1, ..., 4.

Caption: Figure 2: The efficiency curves for Problem 2 with h = [pi]/16i, for i = 1,..., 4.

Caption: Figure 3: The efficiency curves for Problem 3 with h = [pi]/[2.sup.i], for i = 2, ..., 6.

Caption: Figure 4: The efficiency curves for Problem 4 with h = [pi]/[2.sup.i], for t = 1, ..., 5.