A Reliable Algorithm for Solving System of Multi-Pantograph Equations
RANIA SAADEH
Department of Mathematics,
Zarqa University,
Zarqa 13110,
JORDAN
Abstract. In this article, a new series solution of a system of pantograph equations is established using the residual
power series method (RPSM). The proposed method produces the solution in terms of a convergent infinite series,
requiring no linearization, perturbation or discretization, in some cases it reproduces the exact solutions. We apply
the RPSM to solve the multi-pantograph equations, and we show that the outcomes are very accurate. Some
examples are given to demonstrate the simplicity and efficiency of the proposed method. Comparisons to the
Laplace decomposition approach are made to verify the efficiency and applicability of the presented method in
solving similar problems.
Key-words: Residual power series method; Pantograph equations; System of initial value problems.
Received: September 24, 2022. Revised: October 28, 2022. Accepted: November 9, 2022. Published: December 1, 2022.
1 Introduction
The pantograph equation, which is one of the most
important kinds of delay differential equations, [1],
[2], [3], [4], [5] and [6], has been studied extensively
owing to the numerous applications in which these
equations arise. The name pantograph originated
from the work of the researchers, [1], on the
collection of current by the pantograph head of an
electric locomotive, this equation appeared in
modeling various problems in engineering and
sciences such as biology, economy, control,
population studies and electrodynamics, [7], [8], [9],
[10].
In the last years, extensive work dealt with the
pantograph equation. Several methods have been
used to solve different types of the pantograph
equation, such as Adomian's decomposition method,
[5], [6], the homotopy perturbation method [7],
Variational iteration method, [8], [9], Runge–Kutta
methods, [10], the reproducing kernel space method,
[11], Taylor polynomials approach, [12], [13], one-
leg -methods [14], Spectral methods, [15],
differential transformation method, [16],
Discontinuous Galerkin methods, [17], Bessel matrix
and collocation methods, [18], [19], Chebyshev
polynomials method, [20], Laplace decomposition
algorithm (LDA) [21], [22], and so on [23], [24],
[25], [26], [27], [28], [29], [30].
The purpose of this paper is to extend the
application of the residual power series (PSR)
method [31], [32] to provide a symbolic approximate
solution for a system of multi-pantograph equations:
(1.1)
Subject to the initial conditions
, (1.2)
(1.2)
Where are constants, are analytical
functions, and .
The RPS was developed in [31] as an efficient
method for determining the coefficients of the power
series solution of the first and second-order fuzzy
differential equation. It has been successfully applied
in the numerical solution of the generalized Lane-
Emden equation, which is a highly nonlinear
problem, [32]. The RPS method is effective and easy
to construct a power-series solution for strongly
linear and nonlinear equations without linearization,
perturbation, or discretization. In contrast to the
classical power series (CPS) methods, the RPS
method does not need to compare the coefficients of
the corresponding terms, and a recursion relation is
not required. This method computes the coefficients
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of the power series by a chain of linear equations of
one variable. The RPS method is an alternative
procedure for obtaining an analytic Taylor series
solution of the system of multi-pantograph equations.
By using residual error concept, we get a series
solution, in practice a truncated series solution. For
linear problems, the exact solution can be obtained
by a few terms of the RPS method solution. As we
shall see later, the exact solution is available when
the solution is polynomial. Moreover, the solutions
and all their derivatives are applicable for each
arbitrary point in the given interval. It does not
require any converting while switching from the first
order to the higher order; as a result, the method can
be applied directly to the given problem by choosing
an appropriate value for the initial guess
approximation.
This paper is organized as follows: in the next
section, we state some definitions and theorems that
help us to construct the proposed method. In section
3, we present the basic idea of the power series
method. In section 4 we extend the PSR method to
provide a symbolic approximate series solution for a
system of multi-pantograph equations. In section 5,
numerical examples are given to illustrate the
capability of the proposed method. Section 6 is the
brief conclusion of this paper. Finally, some
references are listed at the end.
2 Preliminaries
In this section, we introduce some definitions and
theorems related to Taylor's series and analytic
functions.
Definition 2.1. A function is called analytic at
, where I is an open interval, if it can be represented
in a form of a power series as
. (2.1)
Taking we get the Maclaurin series
Theorem 2.1 [22] There are only three possibilities
for the convergence conditions of the power series
(2.1):
(i) The series converges only when and the
radius of convergence is zero..
(ii) The series converges for all, and the radius
of convergence is .
(iii) There is a positive number such that the
series converges if and diverges
if.
Here is called the radius of convergence.
Theorem 2.2. [22] If has a power series
representation as follows:
Then its coefficients are given by the formula:
Theorem 2.3 (Convergence Analysis) [22]
If we have and
for all and then the
series of the numerical solutions converges to the
exact solution.
3 Adapting RPSM to Solve Multi-
Pantograph Equations
In this section, we introduce the procedure of using
RPSM in solving multi pantograph systems (1.1) and
(1.2).
We present a simple algorithm that explains the
method and illustrates the steps of the RPSM in
handling the proposed problem.
To apply the RPSM, we firstly assume that the
solutions of system (1.1) and (1.2) have the forms:
(3.1)
(3.1)
Where
Since satisfies the initial conditions (1.2),
are the zeroth RPS solutions of the
IVP (1.1) and (1.2).Thus, the solutions have the
form:
(3.2)
(3.2)
And the th-approximate solutions will be:
(3.3)
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Following that, we define the th-residual functions
of system (1.1) as:
(3.4)
(3.4)
And the following residual functions:
(3.5)
It is obvious that: Res for each
where is the radius of convergence of the power
series (3.1). This shows that these residual functions
are infinitely many times differentiable at . On
the other hand,
(3.6)
In fact, these relations are fundamental rules in
RPSM, for the proof and more details, see [31], [32].
Moreover, a special case of (3.6) is:
(3.7)
In order to obtain the th-approximate solutions of
system (1.1) and (1.2), we substitute the th-
truncated series (3.3) into Eq. (3.4) to get:
(3.8)
To obtain the first approximate solution, we
substitute and into Eq. (3.8), and using
(3.7):
, , we get:
.
Thus, the first approximation for the system of multi-
pantograph equations (1.1) and (1.2) can be
expressed as:
.
Similarly, to find the second approximation, we
differentiate both sides of (3.8) with respect to and
substitute and to get:
According to (3.7), we have the values of as
follows:
Thus, the second approximation for the system of multi-
pantograph equations (1.1) and (1.2) will be:
Completing in the same manner, we can obtain the
rest of the coefficients recursively. Thus the series
solution of
of the multi-pantograph equations (1.2) and (1.2) are
obtained. Moreover, higher accuracy can be achieved
by evaluating more components of the solution.
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4 Numerical Example and Discussion
In this section, we consider four interesting examples
of the multi pantograph equations, we apply the
RPSM to solve them and analyze the results. The
results demonstrate the efficiency and accuracy of the
presented technique. We mention that all numerical
computations are performed using Mathematica
software package.
Example 4.1. Consider the two-dimensional
pantograph equations:
(4.1)
Subject to the initial conditions:
. (4.2)
The exact solution of system (4.1) and (4.2) is
, .
According to the residual functions (3.5), we obtain:
Res
Re
(4.3)
According to the initial conditions (4.2), we can
determine the first coefficients of the power series as:
and
.
Hence, the power series solution of system (4.1) can
be expressed as:
It is clear that the first approximations of the series
solution for system (4.1) and (4.2) is of the form:
(4.4)
To find the values of the coefficients and , we
substitute the equations in system (4.4) into (4.3) to
get the following st-residual functions of Eqs. (4.1):
(4.5)
Setting in (4.5) and use the fact (3.6), then we
obtain and
Thus, the first approximations of the series solution
of (4.1) and (4.2) are:
The second approximations of the series solution of
(4.1) and (4.2) have the forms: (4.6)
In order to find the values of the coefficients, and
, we substitute (4.6) into (4.3) to get the form of
the nd-residual functions of (4.1) which is:
. (4.7)
Differentiate the both sides of Eqs. (4.7) with respect
to as follows:
Res
Re
Substituting in (4.8) and using the fact in (3.6)
leads to
, and
.
Thus, the second approximations of the series
solution of (4.1) and (4.2) can be written as:
(4.9)
Continuing with similar fashion, the series solutions
of and will be:
(4.10)
The closed form of above solutions, when are
, which are the exact
solutions.
Example 4.2. Consider the system of multi-
pantograph equations [21]:
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(4.11)
Subject to the initial conditions: (4.12)
According to residual functions in (3.5), we obtain:
Res
Re
(4.13)
The first approximations of the series solution of
(4.11) and (4.12) have the form:
(4.14)
To find the values of the coefficients and ,
substitute Eqs. (5.14) into Eqs. (5.13) to obtain the
st-residual function which of the form:
(4.15)
If we set in Eq. (5.15) and use the fact
, then we obtain , and
. Thus, the first approximations of the series
solution for Eqs. (5.11) and (5.12) are:
(4.16)
By continuing with the similar arguments of Example
(4.1), we get the series solutions of and
as follows:
(4.17)
Which are the expansions of the exact solutions:
, .
Example 4.3. Consider the following system of
multi-pantograph equations:
(4.18)
Subject to the initial conditions: (4.19)
Which have the exact solution:
, .
As in the previous examples, the initial guesses
approximation as:
And
,
Then the power series expansions of the solution take
the form:
(4.20)
Consequently, the first approximations of the series
solution of (4.18) and (4.19) are:
(4.21)
and thest-residual functions of Eqs. (4.19) are:
(4.22)
Setting in (4.21) and using the fact in (3.7), one
can get and
Thus, the second approximations of the series
solutions of (4.18) and (4.19) are:
(4.23)
and the nd –residual functions of (4.18) are:
(4.24)
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Using the fact in (3.7) for reduces a system of
two linear equations with two variables and .
The solution of this system gives and
.
It is easy to discover that each of the coefficients
and for in the expansions (4.20)
vanished. In other words, we have:
(4.25)
Thus, the analytic approximate solution of system
(4.18) and (4.19) coincide with the exact solution,
which is a powerful merit in RPSM, that is it gives
the exact solution if it is a polynomial.
Example 4.4. Consider the three-dimensional
pantograph equations :
(4.26)
Subject to the initial conditions:
(4.27)
Which has the exact solution,
and .
Repeating the same steps in the previous examples,
we can find the numerical solution of system (4.26)
and (4.27) as:
(4.28)
For the third example which are the exact solutions
, and , .
To show the accuracy of the presented method, we
report two types of errors. The first one is the
residual error, and defined as:
(4.29)
While the exact error, is defined, by:
(4.30)
Where,
is the th-order approximation of
obtained by the RPS method, and is the exact
value of , . We introduce Table 1,
Table 2 and Table 3, below to show the related errors of
Without loss of generality, we will test the accuracy of
the presented method for the fourth example.
In Table 1,2 and 3, the residual errors, exact errors and
the exact errors obtained by the Laplace decomposition
algorithm (LDA), [21], have been calculated for various
values of in to compare the th-order
approximate RPS method solution with LDA. From the
tables, it can be seen that the RPS method provides us
with the accurate approximate solution of system (4.26)
and (4.27). Moreover, we can control the error also by
evaluating more components of the solution.
Table 1. Exact and residual error of of Example (4.4)
Exact Error(LDA)
Exact Error(RPS)
Residual Error(RPS)
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Table 2. Exact and residual error of of Example (4.4)
Exact Error(LDA)
Exact Error(RPS)
Residual Error(RPS)
Table 3. Exact and residual error of of Example (4.4)
Exact Error(LDA)
Exact Error(RPS)
Residual Error(RPS)
5 Conclusion
The aim of this work is to propose an efficient
algorithm of the solution of the system of pantograph
equations. We extended the RPS method to solve this
class of systems of IVPs. We conclude that the RPS
method is a powerful and efficient technique in
constructing approximate series solutions of linear
and nonlinear IVPs of different types. The proposed
algorithm produced a rapidly convergent series
without requiring perturbations, discretization, or
other restrictive assumptions which may change the
structure of the problem being solved. We believe
that the efficiency of the RPS method gives it a much
wider applicability. In the future, we will expand the
applications of the presented method to solve more
physical and engineering problems.
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