Modified Reduced Differential Transform Method for Partial Differential-Algebraic Equations

Abstract

This work presents the application of the reduced differential transform method (RDTM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-two and index-three are solved to show that RDTM can provide analytical solutions for PDAEs in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Pade resummation method as a useful technique to find exact solutions. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend on a perturbation parameter.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Brahim Benhammouda 1 and Hector Vazquez-Leal 2 and Arturo Sarmiento-Reyes 3

Academic Editor:Dogan Kaya

1, Abu Dhabi Men's College, Higher Colleges of Technology, P.O. Box 25035, Abu Dhabi, UAE
2, Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltran S/N, 91000 Xalapa, VER, Mexico
3, National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro No. 1, Santa Maria Tonantzintla, 72840, PUE, Mexico

Received 9 May 2014; Accepted 9 September 2014; 6 November 2014

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.

1. Introduction

As widely known, the importance of research on partial differential-algebraic equations (PDAEs) is that many phenomena, practical or theoretical, can be easily modelled by such equations. Those kinds of equations arise in fields like nanoelectronics [1], electrical networks [2-4], and mechanical systems [5], among others.

In recent years, PDAEs have received much attention. Nevertheless, the theory in this field is still young. For linear PDAEs, the convergence of Runge-Kutta method is investigated in [6]. The numerical solution to linear PDAEs with constant coefficients and the study of indices are given in [7-10]. Linear and nonlinear PDAEs are characterized by means of indices which play an important role in the treatment of these equations. The differentiation index is defined as the minimum number of times that all or part of the PDAE must be differentiated with respect to time, in order to obtain the time derivative of the solution, as a continuous function of the solution and its space derivatives [11].

Higher-index PDAEs (differentiation index greater than one) are known to be difficult to treat even numerically. Often such problems are first transformed to index-one systems before applying numerical integration methods. This procedure called index-reduction can be very expensive and may change the properties of the solution. Since application problems in science and engineering often lead to higher-index PDAEs, new techniques are required to solve these problems efficiently.

Modern methods like differential transform method (DTM) [12, 13], reduced differential transform method (RDTM) [14-16], homotopy perturbation method (HPM) [17, 18], homotopy analysis method (HAM) [19], variational iteration method (VIM) [20], and generalized homotopy method [21], among others, are powerful tools to approximate linear and nonlinear problems. Recently, the modifications of the HPM have been used to solve DAEs [22-25]. Besides, the multivariate Pade series [26] was applied to solve PDAEs. Analytical solutions aid researchers in studying the effect of different variables or parameters of functions under consideration easily [27].

Among the abovementioned methods, the DTM is highlighted by its simplicity and versatility to solve nonlinear differential equations. This method does not rely on a perturbation parameter or a trial function like other popular approximation methods. In [12], the DTM was introduced to the engineering field as a tool to find approximate solutions of electrical circuits. DTM produces approximations based on an iterative procedure derived from the Taylor series expansion. This method is very effective and powerful for solving various kinds of differential equations as nonlinear biochemical reaction model [13], two-point boundary-value problems [28], differential-algebraic equations [29], the KdV and mKdV equations [30], the Schrodinger equations [31], fractional differential equations [32], and the Riccati differential equation [33], among others. Later, the RDTM [14-16] was proposed in order to provide a simplified (but not less powerful) version of DTM. The computation resources required by RDTM are much less than those for DTM. Moreover, using RDTM, the solution to initial valued problems can be expressed as an infinite power series. Later, taking advantage of the resummation methods capabilities [34-38], the domain of convergence of such power series can be extended leading in some cases to the exact solution. As well, the RDTM has been applied successfully to problems such as fractional differential equations [39], generalized KdV equations [16], generalized Hirota-Satsuma coupled KdV equation [40], Fornberg-Whitham equations [41], Newell-Whitehead-Segel equation [42], time-fractional telegraphic equation [43], radial diffusivity equation [44], and nonlinear evolutions equations [45].

Therefore, in this paper we present the application of a hybrid technique combining RDTM, Laplace transform, and Pade approximant [46] to find analytical solutions for PDAEs [34-38]. Solutions to PDAEs are first obtained in convergent series form using the RDTM. To improve the solution obtained from RDTM's truncated series, we apply Laplace transform to it and then convert the transformed series into a meromorphic function by forming its Pade approximant. Finally, we take the inverse Laplace transform of the Pade approximant to obtain the analytical solution. This hybrid method (LPRDTM) which combines RDTM with Laplace-Pade posttreatment greatly improves RDTM's truncated series solutions in convergence rate. In fact, the Laplace-Pade resummation method enlarges the domain of convergence of the truncated power series and often leads to the exact solution.

It is important to remark that LPRDTM can obtain exact solutions without requiring an index-reduction for the PDAEs. The proposed method does not produce noise terms also known as secular terms as the homotopy perturbation based techniques [22]. This property of RDTM greatly reduces the volume of computation and improves the efficiency of the method in comparison to the perturbation based methods. What is more, LPRDTM does not require a perturbation parameter like the perturbation based techniques (including HPM). Finally, LPRDTM is straightforward and can be programmed using computer algebra packages like Maple or Mathematica.

The rest of this paper is organized as follows. In the next section, we describe how the RDTM can be applied to solve PDAEs. The main idea behind the Pade approximant is given in Section 3. In Section 4, we give the basic concept of the Laplace-Pade resummation method. In Section 5, we apply LPRDTM to solve two PDAEs problems of index-two and index-three. In Section 6, we give a brief discussion. Finally, a conclusion is drawn in the last section.

2. Reduced Differential Transform Method (RDTM)

In this section we will describe the reduced differential transform method to solve PDAEs.

Definition 1.

If a function u ( t , x ) is analytical and continuously differentiable with respect to time t and space x in the domain of interest Ω , then [figure omitted; refer to PDF] is the transformed function of u ( t , x ) .

Definition 2.

The differential inverse transform of ^{_{U k} ( x ) k = 0 n} is defined by [figure omitted; refer to PDF] Substituting (1) into (2), we deduce that [figure omitted; refer to PDF] From the above definitions, it is easy to see that the concept of the RDTM is obtained from the power series expansion. To illustrate the application of the proposed RDTM to solve PDAEs, we consider the following nonlinear PDAE system: [figure omitted; refer to PDF] where A and B are n × n square matrices with A singular, N u ( t , x ) is a nonlinear differential operator, and g is a known analytical function.

PDAE (4) is supplied with some consistent initial conditions: [figure omitted; refer to PDF] In contrast to parabolic or hyperbolic initial value problems, initial conditions for PDAEs cannot be prescribed for all components of the solution vector arbitrarily as initial conditions have to fulfill certain consistency conditions.

RDTM establishes that the solution to a differential equation can be written as [figure omitted; refer to PDF] where _{U 0} x , _{U 1} x , ... are unknown functions to be determined by RDTM.

Applying the RDTM to initial condition (5) and PDAE (4), respectively, we obtain the transformed initial condition [figure omitted; refer to PDF] and the recursion system [figure omitted; refer to PDF] where ( ^{d 2} / d ^{x 2} ) _{U k} ( x ) , N _{U 0} ( x ) , ... , _{U k} ( x ) , and G ( x ) are the reduced differential transforms of ( ^{∂ 2} / ∂ ^{x 2} ) u ( t , x ) , N ( u ( t , x ) ) , and g ( t , x ) , respectively.

Substituting (7) into (8) and solving the resulting system, we determine the unknown functions _{U k} ( x ) , k = 0,1 , 2 , ... . Then, the differential inverse transformation of the set of functions ^{_{U k} ( x ) k = 0 n} gives the approximate solution [figure omitted; refer to PDF] where n is the approximation order of the solution. The exact solution to problem (4)-(5) is then given by [figure omitted; refer to PDF] If _{U k} ( x ) and _{V k} ( x ) are the reduced differential transforms of u ( t , x ) and v ( t , x ) , respectively, then the main operations of RDTM are shown in Table 1.

Table 1: Main operations of RDTM.

Function	Reduced differential transform
α u ( t , x ) ± β v ( t , x )	α U k ( x ) ± β V k ( x )
u ( t , x ) v ( t , x )	∑ r = 0 k U r ( x ) V k - r ( x )
∂ ∂ t L L L L [ u ( t , x ) ]	( k + 1 ) U k + 1 ( x )
∂ ∂ x [ u ( t , x ) ]	d d x L L L L L U k ( x )
x m t n	{ x m , k = n 0 , k ...0; n
x m t n u ( t , x )	x m U k - n ( x )
e λ t	λ k k !
sin ... ... ( ω t + α )	ω k k ! L L L L L L L sin ... ... ( π k 2 + α )
cos ... ( ω t + α )	ω k k ! L L L L L L L cos ... ( π k 2 + α )

The process of RDTM can be described as follows.

(1) Apply the reduced differential transform to the initial conditions.

(2) Apply the reduced differential transform to the PDAE to obtain a recursion system for the unknown functions _{U 0} x , _{U 1} x , ... .

(3) Use the transformed initial conditions and solve the recursion system for the unknown functions _{U 0} x , _{U 1} x , ... .

(4) Use differential inverse transform formula (9) to obtain an approximate or exact solution for the PDAE.

The solutions series obtained from RDTM may have limited regions of convergence, even if we take a large number of terms. Therefore, we propose to apply the Laplace-Pade resummation method to RDTM truncated series to enlarge the convergence region as depicted in the next section.

3. Pade Approximant

Let u ( t ) be an analytical function with Maclaurin's expansion: [figure omitted; refer to PDF] Then the Pade approximant to u t of order [ L , M ] which we denote by [ L / M _{] u} t is defined by [46] [figure omitted; refer to PDF] where we considered _{q 0} = 1 , and the numerator and denominator have no common factors.

The numerator and the denominator in (12) are constructed so that u t and [ L / M _{] u} t and their derivatives agree at t = 0 up to L + M . That is, [figure omitted; refer to PDF] From (13), we have [figure omitted; refer to PDF] From (14), we get the following algebraic linear systems: [figure omitted; refer to PDF] From (15), we calculate first all the coefficients _{q n} , 1 ...4; n ...4; M . Then, we determine the coefficients _{p n} , 0 ...4; n ...4; L , from (16).

Note that, for a fixed value of L + M + 1 , error (13) is the smallest when the numerator and denominator of (12) have the same degree or when the numerator has one degree higher than the denominator.

4. Laplace-Pade Resummation Method

Several approximate methods provide power series solutions (polynomial). Nevertheless, sometimes, this type of solutions lacks large domains of convergence. Therefore, Laplace-Pade [34-38] resummation method is used in literature to enlarge the domain of convergence of solutions or inclusive to find exact solutions.

The Laplace-Pade method can be explained as follows.

(1) First, Laplace transformation is applied to power series (9).

(2) Next, s is substituted by 1 / t in the resulting equation.

(3) After that, we convert the transformed series into a meromorphic function by forming its Pade approximant of order [ N / M ] . N and M are arbitrarily chosen, but they should be of smaller values than the order of the power series. In this step, the Pade approximant extends the domain of the truncated series solution to obtain better accuracy and convergence.

(4) Then, t is substituted by 1 / s .

(5) Finally, by using the inverse Laplace s transformation, we obtain the exact or approximate solution.

5. Test Problems

In this section, we will demonstrate the effectiveness and accuracy of the LPRDTM described in the previous sections through two PDAE systems of index-two and index-three.

5.1. Nonlinear First Order Index-Two PDAE

Consider the following nonlinear index-two PDAE: [figure omitted; refer to PDF] a coupled system of two parabolic equations and one algebraic equation, with - ∞ < x < + ∞ and t ...5; 0 .

This PDAE is subject to the following initial conditions: [figure omitted; refer to PDF] Note here that no initial condition is prescribed for the variable _{u 3} as this is determined by the PDAEs (17)-(19) and (20). Moreover, since differentiating (19) twice with respect to time and using (17)-(19) determine ∂ _{u 3} / ∂ t in terms of _{u 1} , _{u 2} and their space derivatives, then the index of PDAEs (17)-(19) is two. Therefore, this PDAE is difficult to solve numerically.

Applying the reduced differential transform to initial conditions (20) and PDAEs (17)-(19), respectively, we get [figure omitted; refer to PDF] and the recursion system [figure omitted; refer to PDF] for k = 0,1 , 2 ... .

System (22) can be written as [figure omitted; refer to PDF] for k = 1,2 , 3 ... .

Using (21) and solving the 3 × 3 algebraic linear system (23) for _{U 1 , k} x , _{U 2 , k} x , and _{U 3 , k - 1} x for k = 1,2 , 3 ... , we have [figure omitted; refer to PDF] Then, using (9) and (24), we get the fourth order approximation solution: [figure omitted; refer to PDF] The solutions series obtained from the RDTM may have limited regions of convergence, even if we take a large number of terms. Therefore, we propose to apply the t -Pade approximation technique to these series to increase the convergence region. First t -Laplace transform is applied to (25). Then, s is substituted by 1 / t and the t -Pade approximant is applied to the transformed series. Finally, t is substituted by 1 / s and the inverse Laplace s -transform is applied to the resulting expression to obtain the approximate or exact solution.

Applying Laplace transforms to _{u 1} t , x , _{u 2} t , x , and _{u 3} t , x yields [figure omitted; refer to PDF] For simplicity let s = 1 / t ; then [figure omitted; refer to PDF] All of the [ L / M ] t -Pade approximants of (27) with L ...5; 1 and M ...5; 1 and L + M ...4; 4 yield [figure omitted; refer to PDF] Now since t = 1 / s , we obtain _{L / M u 1} , _{L / M u 2} , and _{L / M u 3} in terms of s as follows: [figure omitted; refer to PDF] Finally, applying the inverse Laplace transform to Pade approximants (29) yields an approximate solution which in this case is the exact solution: [figure omitted; refer to PDF]

5.2. Linear Second Order Index-Three PDAE

Consider the following index-three PDAE [47]: [figure omitted; refer to PDF] a coupled system of two hyperbolic equations and one algebraic equation, with - ∞ < x < + ∞ and t ...5; 0 .

System (31)-(33) is subject to the following initial conditions: [figure omitted; refer to PDF] Note here that no initial condition is prescribed for the variable _{u 3} as this is determined by the PDAEs (31)-(33) and (34). Moreover, since differentiating (33) three times with respect to time and using (31)-(33) determine ∂ _{u 3} / ∂ t in terms of _{u 1} , _{u 2} and their space derivatives, then the index of PDAEs (31)-(33) is three. Therefore, this PDAE is difficult to solve numerically.

Applying the reduced differential transform to initial conditions (34) and PDAEs (31)-(33), respectively, we obtain [figure omitted; refer to PDF] and the recursion system [figure omitted; refer to PDF] System (36) can be written as [figure omitted; refer to PDF] Using (35) and solving the 3 × 3 algebraic linear system (37) for _{U 1 , k} x , _{U 2 , k} x , and _{U 3 , k - 2} x for k = 2,3 , 4 ... , we have [figure omitted; refer to PDF] Using (9) and (38), we get the approximate solution [figure omitted; refer to PDF] Similarly, the coefficients _{U 1 , k} x , _{U 2 , k} x , and _{U 3 , k - 2} x for k ...5; 5 can be found from (37). The solutions series obtained from the RDTM may have limited regions of convergence, even if we take a large number of terms. Therefore, we propose to apply the t -Pade approximation technique to these series to increase the convergence region. First t -Laplace transform is applied to (39). Then, s is substituted by 1 / t and the t -Pade approximant is applied to the transformed series. Finally, t is substituted by 1 / s and the inverse Laplace s -transform is applied to the resulting expressions to get the approximate or exact solutions.

Applying Laplace transforms to _{u 1} t , x , _{u 2} t , x , and _{u 3} t , x yields [figure omitted; refer to PDF] For simplicity let s = 1 / t ; then [figure omitted; refer to PDF] All of the [ L / M ] t -Pade approximants of (41) with L ...5; 1 and M ...5; 1 and L + M ...4; 3 yield [figure omitted; refer to PDF] Now since t = 1 / s , we obtain _{L / M u 1} , _{L / M u 2} , and _{L / M u 3} in terms of s as follows: [figure omitted; refer to PDF] Finally, applying the inverse Laplace transform to Pade approximants (43) we obtain an approximate solution which in this case is the exact solution: [figure omitted; refer to PDF]

6. Discussion

In this paper, we presented the reduced differential transform method (RDTM) as a useful analytical tool to solve partial differential-algebraic equations (PDAEs). The coupling of RDTM and Laplace-Pade enabled us to obtain the exact solution to two PDAE problems of index-two and index-three without the need for a preprocessing step of index-reduction. This is a relevant result given the fact that a higher-index PDAE is often difficult to treat numerically without reducing its index to one and that the index-reduction can be very expensive and may not preserve the properties of the solution to the original PDAE. For each of the two problems solved here, the RDTM transformed the PDAE into an easily solvable linear algebraic recursion system for the coefficient functions of the power series solution. As aforementioned, in order to enlarge the domain of convergence of the RDTM power series solution, a Laplace-Pade resummation was applied to the RDTM's truncated series leading to the exact solution.

The RDTM solution procedure does not involve unnecessary computation like that related to noise terms [22], which is a common problem for approximation methods like the HPM or others. This property of RDTM greatly reduces the volume of computation and improves the efficiency of the method. It should be noted that the high complexity of these problems was effectively handled by LPRDTM method due to the malleability of RDTM and resummation capability of Laplace-Pade. What is more, there is not any standard analytical or numerical method to solve higher-index PDAEs, converting the LPRDTM method into an attractive tool to solve such problems.

On the one hand, semianalytic methods like HPM, HAM, and VIM, among others, require an initial approximation for the solutions sought and the computation of one or several adjustment parameters. If the initial approximation is properly chosen, then the results can be highly accurate. Nonetheless, there is no general method to choose such initial approximation. This issue motivates the use of adjustment parameters obtained by minimizing the least-squares error with respect to the numerical solution. On the other hand, RDTM or LPRDTM methods do not require any trial equation or a procedure for least-squares error minimization. As well, RDTM obtains its coefficients using an easily computable straightforward procedure that can be implemented into programmes like Maple or Mathematica.

It is important to remark that even if the Laplace-Pade resummation strategy fails to obtain the exact solution to the PDAE under study, it can still produce a good approximation with an enlarged domain of convergence. The treatment of higher-index PDAEs is still an open issue in science and requires further research.

7. Conclusion

This work presented LPRDTM method as a combination of the RDTM and a resummation method based on Laplace transforms and the Pade approximant. Firstly, the solutions of PDAEs are obtained in convergent series forms using RDTM. Next, in order to enlarge the domain of convergence of the truncated power series, a posttreatment combining Laplace transforms and the Pade approximant is applied. This technique that we call LPRDTM greatly improves RDTM's truncated series solutions in convergence rate and often leads to the exact solution. Additionally, RDTM is an attractive tool, because it does not require a perturbation parameter to work and it does not generate secular terms (noise terms) as other semianalytical methods like HPM, HAM, or VIM.

By solving two problems, we presented the LPRDTM as a handy tool with a great potential to solve linear/nonlinear higher-index PDAEs. Additionally, the LPRDTM does not require an index-reduction to solve higher-index PDAEs. Furthermore, we obtained successfully the exact solutions of such two problems highlighting the efficiency of the LPRDTM. The proposed method is based on a straightforward procedure, producing highly accurate approximations. Therefore, it is suitable for engineers and in particular for those in fields of mechanics, electronics, and electrical engineering where application problems give rise to higher-index PDAEs. Finally, further research should be performed to solve other higher-index nonlinear PDAE systems.

Acknowledgment

Hector Vazquez-Leal gratefully acknowledges the financial support provided by the National Council for Science and Technology of Mexico (CONACyT) through Grant CB-2010-01 no. 157024.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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