Research Article | Open Access
Fadi Awawdeh, Ahmad Adawi, Safwan Al-Shara', "Analytic Solution of Multipantograph Equation", Advances in Decision Sciences, vol. 2008, Article ID 605064, 10 pages, 2008. https://doi.org/10.1155/2008/605064
Analytic Solution of Multipantograph Equation
We apply the homotopy analysis method (HAM) for solving the multipantograph equation. The analytical results have been obtained in terms of convergent series with easily computable components. Several examples are given to illustrate the efficiency and implementation of the homotopy analysis method. Comparisons are made to confirm the reliability of the homotopy analysis method.
The delay differential equation where has been studied by numerous authors (e.g., [1–8]). Second-order versions of this equation have also been studied (e.g., [9, 10]). The enduring interest in this equation is due partially to the number of applications it has found such as a current collection system for an electric locomotive, cell growth models, biology, economy, control, and electrodynamics (e.g., [10–13]). The focus of most of the studies made in the complex plane (e.g., [12, 14]) was on solutions on the real line for either the retarded case or the advanced case
In 1999, Qiu et al.  have studied the delay equationwhere and by transforming the proportional delay into the constant delay. They got the sufficient condition of asymptotic stability for the analytic solution, that is,
In [17–19] the Dirichlet series solution of (1.2) is constructed, and the sufficient condition of the asymptotic stability for the analytic solution is obtained. It is proved that the -methods with a variable stepsize are asymptotically stable if
It is well known that for the multipantograph equationwhere the collocation solution associated with the th degree collocation polynomial possesses the optimal superconvergence order at the first step , provided that the collocation parameters are properly chosen in (e.g.,  for , and  for ).
Ishiwata and Muroya  proposed a piecewise -rational approximation with “quasiuniform meshes” which corresponds to the th collocation method, and established the global error analysis of on successive mesh points. This method is more useful than the known collocation method when solving (1.5) in case that a long time integration is needed, that is, if is large, then the number of steps in the method is less than that of the collocation method. Collocation method is useful for computation, but in these mesh divisions, there are problems. For example, if the end point is larger, then the mesh size near the first mesh point becomes too small, compared with the mesh size near the end point. This implies that the total computational cost is higher (see also [22–25].)
In this paper, and in order to overcome such problems, we propose an analytic solution of (1.5) by the HAM addressed in [26–36]. The HAM is based on the homotopy, a basic concept in topology. The auxiliary parameter is introduced to construct the so-called zero-order deformation equation. Thus, unlike all previous analytic techniques, the HAM provides us with a family of solution expressions in auxiliary parameter . As a result, the convergence region and rate of solution series are dependent upon the auxiliary parameter and thus can be greatly enlarged by means of choosing a proper value of . This provides us with a convenient way to adjust and control convergence region and rate of solution series given by the HAM.
2. Description of the Method
In order to obtain an analytic solution of the delay differential equation (1.5), the HAM is employed. Consider the operator where is unknown function and the independent variable. Let denote an initial guess of the exact solution that satisfies , an auxiliary parameter, an auxiliary function, and an auxiliary linear operator with the property when . Then using as an embedding parameter, we construct such a homotopy:
It should be emphasized that we have great freedom to choose the initial guess , the auxiliary linear operator , the nonzero auxiliary parameter , and the auxiliary function .
Enforcing the homotopy (2.2) to be zero, that is,we have the so-called zero-order deformation equation
Thus, according to (2.5) and (2.6), as the embedding parameter increases from to , varies continuously from the initial approximation to the exact solution . Such a kind of continuous variation is called deformation in homotopy.
By Taylor's theorem, can be expanded in a power series of as follows:where
If the initial guess the auxiliary linear parameter , the nonzero auxiliary parameter , and the auxiliary function are properly chosen, so that the power series (2.7) of converges at Then, we have under these assumptions the solution series
For brevity, define the vector
According to the definition (2.8), the governing equation of can be derived from the zero-order deformation equation (2.4). Differentiating the zero-order deformation equation (2.4) times with respect to and then dividing by and finally setting , we have the so-called th-order deformation equationwhere
Proof. If the
series (2.9) converges, we can writeand it holds that
We can verify thatwhich gives us, according to (3.2),
Furthermore, using (3.3) and the definition of the linear operator we have
According to (2.11), we can obtain thatwhich gives, since and
By the definition (2.12) of , it holds that
From (3.7) and (3.8), we haveand, moreover, with the help of (2.11), it holds that
In view of (3.9) and (3.10), must be the exact solution of (1.5).
The HAM provides an analytical solution in terms of an infinite power series. However, there is a practical need to evaluate this solution, and to obtain numerical values from the infinite power series. The consequent series truncation, as well as the practical procedure conducted to accomplish this task, transforms the otherwise analytical results into an exact solution, which is evaluated to a finite degree of accuracy. In order to investigate the accuracy of the HAM solution with a finite number of terms, three examples were solved. The HAM results were compared with the exact solutions. The impact of the term numbers in the series solution and truncation process was assessed by evaluating the HAM results for different terms in the series. By increasing the number of the HAM terms, the percentage of error decreases. It is also observed that the HAM results with 10 terms have acceptable accuracy compared to the exact solutions. Therefore, it may be concluded that the use of 10 terms in the series yields accurate results with HAM solution sufficiently. MATLAB 7 is used to carry out the computations.
Defining that with the property where is the integral constant and using , the th-order deformation equations (2.11) for becomes
Example 4.1. We consider the following pantograph differential equation: The exact solution is Note that we still have freedom to choose the auxiliary parameter . To investigate the influence of on the solution series (2.9), we can consider the convergence of some related series such as and so on. However, is dependent of . Let denote a set of all possible values of by means of which the corresponding series of converges. According to Theorem 3.1, for each , the corresponding series of converges to the same result. The curve versus contains a horizontal line segment above the the valid region We call such a kind of curve the -curve , which clearly indicates the the valid region of a solution series. The so-called -curve of is as shown in Figure 1. From Figure 1 it is clear that the series of is convergent when Using we have from (2.9) and (4.1) that the ten terms approximate solution obtained by HAM are We see that HAM solution is very close to the exact solution. It may be concluded that the use of 10 terms in the homotopy series yields accurate results.
Example 4.2. Next, we consider the nonhomogeneous delay equation By means of the -curve, it is reasonable to choose . We have for the ten terms approximate solution obtained by HAM as follows: In view of (4.5), we can conclude that the exact solution is Ishiwata and Muroya  proposed a piecewise -rational approximation with “quasiuniform meshes” which corresponds to the th collocation method. For , and the errors at the first mesh point are shown in the third column of Table 1. In Table 1, The accuracy of the HAM is examined by comparing (4.5) with the available exact and the -rational approximation method.
Example 4.3. In the last example, we consider the pantograph equation The exact solution is . By means of the -curve, it is reasonable to choose . We have for The first nine terms of the series (4.7) are coinciding with the first nine terms of the Taylor series of . Figure 2 shows plots of ten and twenty terms approximation of
5. Discussion and Conclusion
In this paper, the HAM was employed to solve the multipantograph differential equation. Unlike the traditional methods, the solutions here are given in series form. The approximate solution to the equation was computed with no need for special transformations, linearization, or discretization. It was shown that the HAM solutions are very close to the exact solutions. It may be concluded that the use of a few terms in the series yields accurate results with HAM solution sufficiently. HAM is a powerful tool for solving analytically nonlinear equations.
- P. O. Fredrickson, “Dirichlet series solution for certain functional differential equations,” in Japan-United States Seminar on Ordinary Differential and Functional Equations, M. Uradbe, Ed., vol. 243 of Lecture Notes in Mathematics, pp. 247–254, Springer, Berlin, Germany, 1971.
- L. Fox, D. F. Mayers, J. R. Ockendon, and A. B. Tayler, “On a functional differential equation,” Journal of the Institute of Mathematics and Its Applications, vol. 8, pp. 271–307, 1971.
- F. Gross, “On a remark of Utz,” The American Mathematical Monthly, vol. 74, pp. 1107–1108, 1967.
- A. Iserles and Y. Liu, “On neutral functional-differential equations with proportional delays,” Journal of Mathematical Analysis and Applications, vol. 207, no. 1, pp. 73–95, 1997.
- E. Ishiwata, “On the attainable order of collocation methods for the neutral functional-differential equations with proportional delays,” Computing, vol. 64, no. 3, pp. 207–222, 2000.
- R. J. Oberg, “Local theory of complex functional differential equations,” Transactions of the American Mathematical Society, vol. 161, pp. 269–281, 1971.
- W. R. Utz, “The equation ,” Bulletin of the American Mathematical Society, vol. 71, no. 1, p. 138, 1965.
- B. van Brunt, J. C. Marshall, and G. C. Wake, “Holomorphic solutions to pantograph type equations with neutral fixed points,” Journal of Mathematical Analysis and Applications, vol. 295, no. 2, pp. 557–569, 2004.
- B. van Brunt, G. C. Wake, and H. K. Kim, “On a singular Sturm-Liouville problem involving an advanced functional differential equation,” European Journal of Applied Mathematics, vol. 12, no. 6, pp. 625–644, 2001.
- G. C. Wake, S. Cooper, H.-K. Kim, and B. van-Brunt, “Functional differential equations for cell-growth models with dispersion,” Communications in Applied Analysis, vol. 4, no. 4, pp. 561–573, 2000.
- G. Derfel and A. Iserles, “The pantograph equation in the complex plane,” Journal of Mathematical Analysis and Applications, vol. 213, no. 1, pp. 117–132, 1997.
- A. Iserles, “On the generalized pantograph functional-differential equation,” European Journal of Applied Mathematics, vol. 4, no. 1, pp. 1–38, 1993.
- J. R. Ockendon and A. B. Tayler, “The dynamics of a current collection system for an electric locomotive,” Proceedings of the Royal Society of London. Series A, vol. 322, no. 1551, pp. 447–468, 1971.
- M. Buhmann and A. Iserles, “Stability of the discretized pantograph differential equation,” Mathematics of Computation, vol. 60, no. 202, pp. 575–589, 1993.
- L. Qiu, T. Mitsui, and J.-X. Kuang, “The numerical stability of the -method for delay differential equations with many variable delays,” Journal of Computational Mathematics, vol. 17, no. 5, pp. 523–532, 1999.
- D. Li and M. Z. Liu, “The properties of exact solution of multi-pantograph delay differential equation,” Journal of Harbin Institute of Technology, vol. 3, no. 1, pp. 1–3, 2000 (Chinese).
- M. Z. Liu and D. Li, “Properties of analytic solution and numerical solution of multi-pantograph equation,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 853–871, 2004.
- A. Bellen, N. Guglielmi, and L. Torelli, “Asymptotic stability properties of -methods for the pantograph equation,” Applied Numerical Mathematics, vol. 24, no. 2-3, pp. 279–293, 1997.
- A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, UK, 2003.
- Y. Muroya, E. Ishiwata, and H. Brunner, “On the attainable order of collocation methods for pantograph integro-differential equations,” Journal of Computational and Applied Mathematics, vol. 152, no. 1-2, pp. 347–366, 2003.
- E. Ishiwata and Y. Muroya, “Rational approximation method for delay differential equations with proportional delay,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 741–747, 2007.
- H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2004.
- H. Brunner, Q. Hu, and Q. Lin, “Geometric meshes in collocation methods for Volterra integral equations with proportional delays,” IMA Journal of Numerical Analysis, vol. 21, no. 4, pp. 783–798, 2001.
- A. Bellen, “Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay,” IMA Journal of Numerical Analysis, vol. 22, no. 4, pp. 529–536, 2002.
- N. Takama, Y. Muroya, and E. Ishiwata, “On the attainable order of collocation methods for delay differential equations with proportional delay,” BIT Numerical Mathematics, vol. 40, no. 2, pp. 374–394, 2000.
- Z. Abbas and T. Hayat, “Radiation effects on MHD flow in a porous space,” International Journal of Heat and Mass Transfer, vol. 51, no. 5-6, pp. 1024–1033, 2008.
- S. Abbasbandy, “Soliton solutions for the Fitzhugh-Nagumo equation with the homotopy analysis method,” Applied Mathematical Modelling, vol. 32, no. 12, pp. 2706–2714, 2008.
- T. Hayat and M. Sajid, “On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder,” Physics Letters A, vol. 361, no. 4-5, pp. 316–322, 2007.
- M. Inc, “Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity,” Mathematics and Computers in Simulation, vol. 79, no. 2, pp. 189–200, 2008.
- M. Inc, “On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method,” Physics Letters A, vol. 365, no. 5-6, pp. 412–415, 2007.
- M. Inc, “On numerical solution of Burgers' equation by homotopy analysis method,” Physics Letters A, vol. 372, no. 4, pp. 356–360, 2008.
- M. Sajid, T. Hayat, and S. Asghar, “Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt,” Nonlinear Dynamics, vol. 50, no. 1-2, pp. 27–35, 2007.
- S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2004.
- S. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004.
- S. Liao, “An explicit analytic solution to the Thomas-Fermi equation,” Applied Mathematics and Computation, vol. 144, no. 2-3, pp. 495–506, 2003.
- Y. Tan and S. Abbasbandy, “Homotopy analysis method for quadratic Riccati differential equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 3, pp. 539–546, 2008.
Copyright © 2008 Fadi Awawdeh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.