# Analytic Solution of Multipantograph Equation

Authors: Fadi  Awawdeh; Ahmad Adawi; Safwan Al-Shara'
Journal: Journal of Applied Mathematics and Decision Sciences
 (2008)

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## Abstract

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.


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## Body

## 1. Introduction

The  delay differential equation(1.1)y′(t)=λy(t)+∑i=1kμiy(fi(t)),t>0,y(0)=y0, where λ,μ1,μ2,…,μk,y0∈ℂ, 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 0<q<1 or the advanced case q>1.In 1999, Qiu et al. [15] have studied the delay
equation(1.2)y′(t)=λy(t)+∑i=1kμiy(qit),y(0)=y0,where 0<qk<qk−1<⋯<q1<1 and λ,μ1,μ2,…,μk,y0∈ℂ, by transforming the proportional delay into
the constant delay. They got the sufficient condition of asymptotic stability
for the analytic solution, that is,(1.3)Reλ<0,∑i=1k|μi|<−Reλ.Liu and Li in [16, 17] proved the existence and
uniqueness of analytic solution of (1.2) for any λ,μ1,μ2,…,μk,y0∈ℂ,
and the analytic solution is asymptotically stable if(1.4)Reλ<0,∑i=1k|μi|<|λ|.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 θ-methods with a variable stepsize are
asymptotically stable if  1/2<θ≤1.It is well known that for the multipantograph equation(1.5)y′(t)=λy(t)+∑i=1kμiy(qit)+f(t),0<t<T,y(0)=α,where 0<qk<qk−1<⋯<q1<1, the collocation solution associated with
the mth degree collocation
polynomial possesses the optimal superconvergence order 2m+1 at the first step t=h,
provided that the collocation m parameters are properly chosen in (0,1) (e.g., [5] for f(t)=0,
and [20] for f(t)≠0).Ishiwata and Muroya [21] proposed a piecewise (2m,m)-rational approximation with “quasiuniform meshes” which corresponds to the mth collocation method, and established the
global error analysis of O(h2m) 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 T 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 t=T 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 h 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 h.
As a result, the convergence region and rate of solution series are dependent
upon the auxiliary parameter h and thus can be greatly enlarged by means of
choosing a proper value of h.
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 N,(2.1)N[y(t)]=∂y(t)∂t−λy(t)−∑i=1kμiy(qit)−f(t)=0,where y(t) is unknown function and t the independent variable. Let y0(t) denote an initial guess of the exact solution y(t) that satisfies y0(0)=α, h≠0 an auxiliary parameter, H(t)≠0 an auxiliary function, and L an auxiliary linear operator with the property L[y(t)]=0 when y(t)=0.
Then using q∈[0,1] as an embedding parameter, we construct such a
homotopy:(2.2)(1−q)L[ϕ(t;q)−y0(t)]−qhH(t)N[ϕ(t;q)]=H^[ϕ(t;q);y0(t),H(t),h,q].It should be emphasized that we have great freedom to
choose the initial guessy0(t),
the auxiliary linear operator L,
the nonzero auxiliary parameter h,
and the auxiliary function H(t).Enforcing the homotopy (2.2) to be zero, that is,(2.3)H^[ϕ(t;q);y0(t),H(t),h,q]=0,we have the so-called zero-order
deformation equation(2.4)(1−q)L[ϕ(t;q)−y0(t)]=qhH(t)N[ϕ(t;q)].Whenq=0,
the zero-order deformation equation (2.4) becomes(2.5)ϕ(t;0)=y0(t)and when q=1,
since h≠0 and H(t)≠0,
the zero-order deformation equation (2.4) is equivalent to(2.6)ϕ(t;1)=y(t).Thus, according to (2.5) and (2.6), as the embedding
parameter q increases from 0 to 1, ϕ(t;q) varies continuously from the initial
approximation y0(t) to the exact solution y(t).
Such a kind of continuous variation is called deformation in homotopy.By Taylor's theorem,ϕ(t;q) can be expanded in a power series of q as follows:(2.7)ϕ(t;q)=y0(t)+∑m=1∞ym(t)qm,where(2.8)ym(t)=1m!∂mϕ(t;q)∂qm|q=0.If the initial guessy0(t), the auxiliary linear parameter L,
the nonzero auxiliary parameter h,
and the auxiliary function H(t) are properly chosen, so that the power series
(2.7) of ϕ(t;q) converges at q=1. Then, we have under these assumptions the solution
series(2.9)y(t)=ϕ(t;1)=∑m=0∞ym(t).For brevity, define the vector(2.10)y→n(t)={y0(t),y1(t),y2(t),…,yn(t)}.According to the definition (2.8), the governing
equation of ym(t) can be derived from the zero-order deformation
equation (2.4). Differentiating the zero-order deformation equation (2.4) m times with respect to q and then dividing by m! and finally setting q=0,
we have the so-called mth-order deformation
equation(2.11)L[ym(t)−χmym−1(t)]=hH(t)ℜm(y→m−1(t)),ym(0)=0,where(2.12)ℜm(y→m−1(t))=1(m−1)!∂m−1N[ϕ(t;q)]∂qm−1|q=0=ym−1′(t)−λym−1(t)−∑i=1kμiym−1(qit)−(1−χm)f(t),χm={0,m≤11,m>1.


## 3. Convergence

Theorem 3.1.
As long as the series (2.9) converges, it must be the exact solution of
the multipantograph equation (1.5).Proof.
If the
series (2.9) converges, we can write(3.1)S(t)=∑m=0∞ym(t)and it holds that(3.2)limm→∞ym(t)=0.
We can verify that(3.3)∑m=1n[ym(t)−χmym−1(t)]=y1+(y2−y1)+⋯+(yn−yn−1)=yn(t),which gives us, according to
(3.2),(3.4)∑m=1∞[ym(t)−χmym−1(t)]=limn→∞yn(t)=0.
Furthermore, using (3.3) and the definition of the
linear operator L, we have(3.5)∑m=1∞L[ym(t)−χmym−1(t)]=L[∑m=1∞[ym(t)−χmym−1(t)]]=0.
According to (2.11), we can obtain that(3.6)∑m=1∞L[ym(t)−χmym−1(t)]=hH(t)∑m=1∞ℜm(y→m−1(t))=0,which gives, since h≠0 and H(t)≠0,(3.7)∑m=1∞ℜm(y→m−1(t))=0.
By the definition (2.12) of ℜm(y→m−1(t)),
it holds that(3.8)∑m=1∞ℜm(y→m−1(t))=∑m=1∞[ym−1′(t)−λym−1(t)−∑i=1kμiym−1(qit)−(1−χm)f(t)]=∑m=0∞ym′(t)−λ∑m=0∞ym(t)−∑m=0∞∑i=1kμiyn(qit)−f(t)=S'(t)−λS(t)−∑i=1kμiS(qit)−f(t).
From (3.7) and (3.8), we have(3.9)S′(t)=λS(t)+∑i=1kμiS(qit)+f(t)and, moreover, with the help of
(2.11), it holds that(3.10)S(0)=∑m=0∞ym(0)=y0(0)+∑m=1∞ym(0)=y0(0)=α.
In view of (3.9) and (3.10), S(t) must be the exact solution of (1.5).


## 4. Examples

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 thatL[ϕ(t;q)]=∂ϕ(t;q)/∂t, with the
property L[C]=0, where C is the integral constant and using H(t)=1,
the mth-order deformation equations (2.11) for m≥1 becomes(4.1)ym(t)=χmym−1(t)+h∫0t[ym−1′(τ)−λym−1(τ)−∑i=1kμiym−1(qiτ)−(1−χm)f(τ)]dτ.Example 4.1.
We
consider the following pantograph differential equation:(4.2)y′(t)=−y(t)+14y(12t)−14e−0.5t,y(0)=1.  The exact solution is  y(t)=e−t.
Note that we still have freedom to choose the
auxiliary parameter  h.
To investigate the influence of  h
on the solution series (2.9), we can consider
the convergence of some related series such as  y′(0),y′′(0),y′′′(0),  and so on. However,  y′′(0)  is dependent of  h.
Let  Rh  denote a set of all possible values of  h  by means of which the corresponding series of  y′′(0)  converges. According to Theorem 3.1, for each  h∈Rh,
the corresponding series of  y′′(0)  converges to the same result. The curve  y′′(0)  versus  h
contains a horizontal line segment above the
the valid region  Rh.  We call such a kind of curve the  h-curve [33], which clearly indicates the the valid region  Rh  of a solution series. The so-called  h-curve of  y′′(0)  is as shown in Figure 1. From Figure 1 it is clear
that the series of  y′′(0)  is convergent when  −2≤h≤0.  Using  h=−1,
we have from (2.9) and (4.1) that the ten terms
approximate solution obtained by HAM are  (4.3)∑m=010ym(t)=1−t+12t2−16t3+124t4−1120t5+1720t6−15040t7+140320t8−1362880t9+6.3×10−8t10≃∑k=010(−1)ktkk!.  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.Figure 1
Theh-curve of y′′(0). Solid line: 10th-order approximation of y′′(0).Example 4.2.
Next, we consider the nonhomogeneous delay equation(4.4)y′(t)=−y(t)+12y(12t)+cost+sint−12sin12t,0≤t≤2π,y(0)=0.  By means of the  h-curve, it is reasonable to choose  h=−1.5.
We have for  t>0
the ten terms approximate solution obtained by
HAM as follows:  (4.5)∑m=010ym(t)=t−16t3+1120t5−15040t7+1362880t9+1.6×10−7t10≃∑m=09(−1)k(2k+1)!t2k+1.
In view of (4.5), we can conclude that  the
exact solution is  y(t)=sint.  Ishiwata and Muroya [21] proposed a piecewise  (2m,m)-rational approximation  Q2m,m(t)
with   “quasiuniform meshes” which
corresponds to the mth collocation method. For  m=2,
and  h=2(6+n),n=0,1,…,4,  the errors  e(h)=|Q4,2(h)−y(h)|  at the first mesh point  t1=h
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  (2m,m)-rational approximation method.Table 1
Comparison of the results of the HAM and the(2m,m)-rational approximation.
nHAM(2m,m)-rational approximation003.8391471⋯E−0716.93⋯E−182.613675⋯E−0823.46⋯E−181.70118⋯E−0931.23⋯E−311.0844⋯E−1044.04⋯E−366.83⋯E−12Example 4.3.
In the last example, we consider the
pantograph equation(4.6)y′(t)=−y(t)−e−0.5tsin(0.5t)y(0.5t)−2e−0.75tcos(0.5t)sin(0.25t)y(0.25t),y(0)=1.  The exact solution is  y(t)=e−tcost.
By means of the  h-curve, it is reasonable to choose  h=−1. We
have for  t>0,(4.7)∑m=010ym(t)=1−t+13t3−16t4+130t5−1630t7+12520t8−122680t9−13628800t10.The first nine terms of the
series (4.7) are coinciding with the first nine terms of the Taylor series of e−tcost. Figure 2 shows plots of ten and twenty terms
approximation of y(t).Figure 2
Plots of ten “**” and twenty “oo” terms approximations for y(t) “−” versus t.


## 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.



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