 Research
 Open Access
 Published:
Analytical solutions for systems of partial differential–algebraic equations
SpringerPlus volume 3, Article number: 137 (2014)
Abstract
This work presents the application of the power series method (PSM) to find solutions of partial differentialalgebraic equations (PDAEs). Two systems of indexone and indexthree are solved to show that PSM can provide analytical solutions of PDAEs in convergent series form. What is more, we present the posttreatment of the power series solutions with the LaplacePadé (LP) resummation method as a useful strategy to find exact solutions. The main advantage of the proposed methodology is that the procedure is based on a few straightforward steps and it does not generate secular terms or depends of a perturbation parameter.
Introduction
As widely known, the importance of research on partial differentialalgebraic 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 (Bartel and Pulch 2006), electrical networks (Ali et al. 20052003; Günther 2000) and mechanical systems (Simeon 1996), 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 RungeKutta method is investigated in (Strehmel and Debrabant 2005). The numerical solution of linear PDAEs with constant coefficients and the study of indices are given in (Lucht et al. 1997a1997b; Lucht and Strehmel 1998; Lucht et al. 1999). 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 (Martinson and Barton 2000).
Higherindex PDAEs (differentiation index greater than one) are known to be difficult to treat even numerically. Often such problems are first transformed to indexone systems before applying numerical integration methods. This procedure called indexreduction, can be very expensive and may change the properties of the solution. Since applications problems in science and engineering often lead to higherindex PDAEs, new techniques are required to solve these problems efficiently.
Modern methods like homotopy perturbation method (HPM) (He 199920001998; VazquezLeal et al. 2012), homotopy analysis method (HAM) (Guerrero et al. 2013), variational iteration method (VIM) (Khan Y, et al. 2012), generalized homotopy method (VazquezLeal 2013), among others, are powerful tools to approximate nonlinear and linear problems. The HPM has been successfully applied to solve various kinds of nonlinear problems in science and engineering, including Volterra’s integrodifferential equation (ElShahed 2005), nonlinear differential equations (He 1998), nonlinear oscillators (He 2004), partial differential equations (PDEs) (He 2005a), bifurcation of nonlinear problems (He 2005b) and boundaryvalue problems (He 2006). Recently, the modifications of the HPM have been used to solve DAEs (Aminikhah and Hemmatnezhad 2011; Asadi et al. 2012; Salehi et al. 2012; Soltanian et al. 2010). Nevertheless, the power series method (PSM) (Forsyth 1906; Ince 1956) is a wellknown classic straightforward procedure from literature that can be applied successfully to solve differential equations of different kind: linear ordinary differential equations (ODEs) (Coddington 1989; Forsyth 1906; Ince 1956; Kreyszig 1999), nonlinear ODEs (Biazar et al. 2005; Fairen et al. 1988; Filipich and Rosales 2002; Filipich et al. 2004; Guzel and Bayram 2005; Kreyszig 1999) and linear PDEs (Kurulay and Bayram 2009), among others. This method establishes that the solution of a differential equation can be expressed as a power series of the independent variable.
In this paper we present the application of a hybrid technique combining PSM, Laplace Transform (LT) and Padé Approximant (PA) (Barker 1975) to find analytical solutions for PDAEs (Ebaid 2011; Gőkdoğan et al. 2012; Merdan et al. 2011; Momani and Ertűrk 2008; Momani et al. 2009; Sweilam and Khader 2009; Tsai and Chen 2010; Yamamoto et al. 2002). Solutions to PDAEs are first obtained in convergent series form using the PSM. To improve the solution obtained from PSM’s truncated series, we apply LT to it, then convert the transformed series into a meromorphic function by forming its PA. Finally, we take the inverse LT of the PA to obtain the analytical solution. This hybrid method (LPPSM), which combines PSM with LaplacePadé posttreatment greatly improves PSM’s truncated series solutions in convergence rate. In fact, the LaplacePadé 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 LPPSM can obtain exact solutions without requiring the indexreduction of the PDAEs. The proposed method does not produce noise terms also known as secular terms as the homotopy perturbation based techniques (Soltanian et al. 2010). This greatly reduces the volume of computation and improves the efficiency of the method in comparison to the perturbation based methods. What is more, LPPSM does not require a perturbation parameter as the perturbation based techniques including HPM. Finally, LPPSM 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 illustrate the basic concept of the PSM. The main idea behind the Padé approximant is given in section “Padé approximant”. In section “LaplacePadé resummation method”, we give the basic concept of the LaplacePadé resummation method. The application of PSM to solve PDAE systems is depicted in section “Application of PSM to solve PDAE systems”. In section “Test problems”, we apply LPPSM to solve two PDAEs problems of indexone and indexthree. In section “Discussion”, we give a brief discussion. Finally, a conclusion is drawn in the last section.
Basic concept of power series method
It can be considered that a nonlinear differential equation can be expressed as
having as boundary condition
where A is a general differential operator, f(t) is a known analytic function, B is a boundary operator, and Γ is the boundary of domain Ω.
PSM (Forsyth 1906; Ince 1956) establishes that the solution of a differential equation can be written as
where u_{0},u_{1},… are unknowns to be determined by series method.
The basic process of series method can be described as:

1.
Equation (3) is substituted into (1), then we regroup the equation in terms of powers of t.

2.
We equate each coefficient of the resulting polynomial to zero.

3.
The boundary conditions of (1) are substituted into (3) to generate an algebraic equation for each boundary condition.

4.
Aforementioned steps generate an algebraic linear system for the unknowns of (3).

5.
Finally, we solve the algebraic linear system to obtain the coefficients u _{0},u _{1},…
Padé approximant
Given an analytical function u(t) with Maclaurin’s expansion
The Padé approximant to u(t) of order [ L,M] which we denote by [ L/M]_{ u }(t) is defined by (Barker 1975)
where we considered q_{0}=1, and the numerator and denominator have no common factors.
The numerator and the denominator in (5) are constructed so that u(t) and [ L/M]_{ u }(t) and their derivatives agree at t=0 up to L+M. That is
From (6), we have
From (7), we get the following algebraic linear systems
and
From (8), we calculate first all the coefficients q_{ n }, 1≤n≤M. Then, we determine the coefficients p_{ n }, 0≤n≤L from (9).
Note that for a fixed value of L+M+1, the error (6) is smallest when the numerator and denominator of (5) have the same degree or when the numerator has degree one higher than the denominator.
LaplacePadé resummation method
Several approximate methods provide power series solutions (polynomial). Nevertheless, sometimes, this type of solutions lacks of large domains of convergence. Therefore, LaplacePadé (Ebaid 2011; Gőkdoğan et al. 2012; Merdan et al. 2011; Momani and Ertűrk 2008; Momani et al. 2009; Sweilam and Khader 2009; Tsai and Chen 2010; Yamamoto et al. 2002) resummation method is used in literature to enlarge the domain of convergence of solutions or inclusive to find exact solutions.
The LaplacePadé method can be explained as follows:

1.
First, Laplace transformation is applied to power series (3).

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 Padé approximant of order [ L/M]. L and M are arbitrarily chosen, but they should be of smaller value than the order of the power series. In this step, the Padé 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.
Application of PSM to solve PDAE systems
Since many application problems in science and engineering are often modelled by semiexplicit PDAEs, we consider therefore the following class of PDAEs
where u_{ k }: $[\phantom{\rule{0.3em}{0ex}}0,T]\times [a,b]\to {\mathbb{R}}^{{m}_{k}}$, k=1,2 and b>a.
System (10)(11) is subject to the initial condition
and some suitable boundary conditions
where g(x) is a given function.
We assume that the solution to initial boundaryvalue problem (10)(13) exists, is unique and sufficiently smooth.
To simplify the exposition of the PSM, we integrate first equation (10) with respect to t and use the initial condition (12) to obtain
It is important to note that the time integration of equation (10) is not relevant to the solution procedure presented here, so one can apply the PSM directly to (10).
In view of PSM, we assume the solution components u_{ k }(t,x),k=1,2 to have the form
where α_{k,n}(x), k=1,2; n=0,1,2,… are unknown functions to be determined later on by the PSM.
Then substitute (15) into system (11)(14) and equate the coefficients of powers of t in the resulting polynomial equations to zero to get an algebraic linear system for these coefficients. Finally, we use equation (15) to obtain the exact solution components u_{ k }, k=1,2 as series. The solutions series obtained from PSM may have limited regions of convergence, even if we take a large number of terms. Therefore, we apply the LaplacePadé resummation method to PSM truncated series to enlarge the convergence region as depicted in the next section.
Test problems
In this section, we will demonstrate the effectiveness and accuracy of the LPPSM presented in the previous section through two PDAE systems of indexone and indexthree.
Nonlinear indexone system
Consider the following nonlinear indexone PDAE which arises as a similarity reduction of NavierStokes equations (Budd et al. 1994)
where 0<x<1 and t>0.
System (16)(17) is subject to the following initial condition
and boundary conditions
The exact solution of problem (16)(19) is
Since one time differentiation of equation (17) determines u_{2t} in terms of u and its space derivatives, then PDAE (16)(17) is indexone. Note that no initial condition is prescribed for the variable u_{2} as this is determined by the PDAE.
In order to simplify the exposition of the PSM presented in section “Application of PSM to solve PDAE systems” to solve (16)(17), we first integrate equation (16) with respect to t and use the initial condition (18) to get
In view of the PSM, we assume the solution components u_{ k },k=1,2 to have the form
where α_{k,n}(x),k=1,2; n=0,1,2,… are unknown functions to be determined later on by the PSM.
Then, we substitute (22) into equations (17) and (21) to get
where (^{′}) denotes the ordinary derivative with respect to x.
Equating the coefficients of powers of t to zero in (24) then solving the resulting equation for α_{2,n}(x) and using the boundary conditions (19), we have
Now equation (23) can be written as a series
where
Equating all coefficients of powers of t to zero in (26), yields α_{1,0}(x)= cosπ x and the recursive formula for α_{1,n}(x)
From recursion (27), we get α_{1,1}(x)=π^{2} cosπ x and α_{1,2}(x)=(π^{4}/2) cosπ x.
From equation (25), we get α_{2,0}(x)=(1/π) sinπ x, α_{2,1}(x)=π sinπ x and α_{2,2}(x)=(π^{3}/2) sinπ x. Using (22) and the coefficients recently obtained, we have
and
Similarly, the coefficients α_{1,n}(x) and α_{2,n}(x) for n≥3 can be found from (27) and (25) respectively.
The solutions series obtained from the PSM may have limited regions of convergence, even if we take a large number of terms. Accuracy can be increased by applying the LaplacePadé posttreatment. First, we apply tLaplace transform to (28) and (29). Then, we substitute s by 1/t and apply tPadé approximant to the transformed series. Finally, we substitute t by 1/s and apply the inverse Laplace stransform to the resulting expressions to get the approximate or exact solutions.
Applying Laplace transforms to u_{1}(t,x) and u_{2}(t,x) yields
and
For the sake of simplicity let s=1/t, then
and
All of the [ L/M]tPadé approximants of (32) and (33) with L ≥1 and M ≥1 and L+M≤3 yield
and
Now since t=1/s, we obtain ${\left[L/M\right]}_{{u}_{1}}$ and ${\left[L/M\right]}_{{u}_{2}}$ in terms of s as follows
Finally, applying the inverse LT to the Padé approximants (36) and (37), we obtain the approximate solution which is in this case the exact solution (20) in closed form.
Linear indexthree system
Consider the following indexthree PDAE system
where 0<x<1 and t>0.
System (38)(40) is subject to the following initial conditions
and the boundary conditions
The exact solution of problem (38)(43) is
Since three time differentiations of equation (40) determine u_{3t} in terms of the solution u and its space derivatives, then PDAE (38)(40) is indexthree. Therefore, this PDAE is difficult to solve numerically. Moreover no initial condition is prescribed for the variable u_{3} as this is determined by the PDAE.
In order to simplify the exposition of the LPPSM presented in section “Application of PSM to solve PDAE systems” to solve (38)(43), we first integrate equations (38) and (39) twice with respect to t and use the initial conditions (41)(42) to get
In view of the PSM, we assume the solution components u_{ k }(t,x),k=1,2,3 to have the form
where α_{k,n}(x),k=1,2,3; n=0,1,2,… are unknown functions to be determined later on by the PSM.
Substituting (47) into equations (40), (45) and (46) we get the system
and
where (^{′}) denotes the ordinary derivative with respect to x.
System (48)(50) can be rewritten as series
Equating the coefficient of powers of t to zero in (51) then solving the resulting system we find the coefficients α_{k,n}(x), for k=1,2,3 and n=0,1,2,…
and the nonsingular algebraic linear system for the unknown functions α_{1,n}, α_{2,n} and α_{3,n2}
Solving system (52) exactly, we obtain the recursions
where ${\delta}_{n}\left(x\right)={\alpha}_{1,n2}^{\mathrm{\prime \prime}}\left(x\right)cos\mathrm{\pi x}\underset{2,n2}{\overset{\mathrm{\prime \prime}}{\alpha}}\left(x\right)sin\mathrm{\pi x}$.
For n=2,3,4, we have δ_{ n }(x)=0 and hence
and
Using (47) and the coefficients recently obtained, we get
and
Similarly, the coefficients α_{1,n}(x), α_{2,n}(x) and α_{3,n2}(x) for n≥5 can be found from (53). The solutions series obtained from the PSM may have limited regions of convergence, even if we take a large number of terms. Therefore, we apply the tPadé approximation technique to these series to increase the convergence region. First t Laplace transform is applied to (54), (55) and (56). Then, s is substituted by 1/t and the tPadé 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
and
For the sake of simplicity let s=1/t, then
and
All of the [ L/M] tPadé approximants of (60), (61) and (62) with L≥1 and M≥1 and L+M≤3 yield
and
Now since t=1/s, we obtain ${\left[L/M\right]}_{{u}_{1}}$, ${\left[L/M\right]}_{{u}_{2}}$ and ${\left[L/M\right]}_{{u}_{3}}$ in terms of s as follows
and
Finally, applying the inverse Laplace transform to the Padé approximants (66), (67) and (68), we obtain the approximate solution which is in this case the exact solution (44) in closed form.
Discussion
In this paper we presented the power series method (PSM) as a useful analytical tool to solve partial differentialalgebraic equations (PDAEs). Two PDAE problems of indexone and indexthree were solved by this method leading to the exact solutions. The method has successfully handled the indexthree PDAE without the need for a preprocessing step of indexreduction. For each of the two problems solved here, the PSM transformed the PDAE into an easily solvable linear algebraic system for the coefficient functions of the power series solution. To improve the PSM solution, a LaplacePadé (LP) posttreatement is applied to the PSM’s truncated series leading to the exact solution. Additionally, the solution procedure does not involve unnecessary computation like that related to noise terms (Soltanian et al. 2010). This greatly reduces the volume of computation and improves the efficiency of the method. It should be noticed that the high complexity of these problems was effectively handled by LPPSM method due to the malleability of PSM and resummation capability of LaplacePadé. What is more, there is not any standard analytical or numerical methods to solve higherindex PDAEs, converting the LPPSM method into an attractive tool to solve such problems.
On one hand, semianalytic methods like HPM, HAM, VIM among others, require an initial approximation for the sought solutions and the computation of one or several adjustment parameters. If the initial approximation is properly chosen the results can be highly accurate, nonetheless, no general methods are available to choose such initial approximation. This issue motivates the use of adjustment parameters obtained by minimizing the leastsquares error with respect to the numerical solution.
On the other hand, PSM or LPPSM methods do not require any trial equation as requisite for the starting the method. What is more, PSM obtains its coefficients using an easy computable straightforward procedure that can be implemented into programs like Maple or Mathematica. Finally, if the solution of the PDAE is not expressible in terms of known functions then the LP posttreatement will provide a larger domain of convergence.
Conclusion
This work presented LPPSM method as a combination of the classic PSM and a resummation method based on the Laplace transforms and Padé approximant. Firstly, the solutions of PDAEs are obtained in convergent series forms using PSM. Next, in order to enlarge the domain of convergence of the truncated power series, a posttreatment combining Laplace transform and Padé approximant is applied. This technique that we call LPPSM greatly improves PSM’s truncated series solutions in convergence rate, and often leads to the exact solution. Additionally, PSM is an attractive tool, because it does not require of 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 LPPSM as a handy tool with high potential to solve linear/nonlinear higherindex PDAEs. Additionally, the LPPSM does not require an indexreduction to solve higherindex PDAEs. Furthermore, we obtained successfully the exact solutions of such two problems highlighting the efficiency of LPPSM. What is more, the proposed method is based on a straightforward procedure, suitable for engineers. Finally, further research should be performed to solve other higherindex nonlinear PDAE systems.
References
Ali G, Bartel A, Günther M: Parabolic differentialalgebraic models in electrical network design. SIAM J Mult Model Sim 2005, 4(3):813838. 10.1137/040610696
Ali G, Bartel A, Günther M, Tischendorf C: Elliptic partial differentialalgebraic multiphysics models in electrical network design. Math Model Meth Appl Sci 2003, 13(9):12611278. 10.1142/S0218202503002908
Aminikhah H, Hemmatnezhad M: An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations. Appl Math Lett 2011, 24: 15021508. 10.1016/j.aml.2011.03.032
Asadi MA, Salehi F, Hosseini MM: Modification of the homotopy perturbation method for nonlinear system of secondorder BVPs. J Comput Sci & Comput Math 2012, 2(5):2328.
Barker GA: Essentials of Padé Approximants. London: Academic Press; 1975.
Bartel A, Pulch R: A concept for classification of partial differentialalgebraic equations in nanoelectronics. 2006.http://wwwnum.math.uniwuppertal.de/fileadmin/mathe/wwwnum/preprints/amna_06_07.pdf Preprint BUWAMNA 06/07, .
Biazar J, Ilie M, Khoshkenar A: A new approach to the solution of the prey and predator problem and comparison of the results with the Adomian method. Appl Math Comput 2005, 171: 486491. 10.1016/j.amc.2005.01.040
Budd CJ, Dold JW, Stuart AM: Blowup in a system of partial differential equations with conserved first integral, part 2: problems with convection. SIAM J Appl Math 1994, 54(3):610640. 10.1137/S0036139992232131
Coddington EA: An introduction to ordinary differential equations. New York: Dover Publications; 1989.
Ebaid AE: Commun Nonlinear Sci Numerical Simul. 2011, 16(1):528536. 10.1016/j.cnsns.2010.03.012
ElShahed M: Application of He’s homotopy perturbation method to Volterra’s integrodifferential equation. Int J Nonlinear Sci Numer Simul 2005, 6(2):163168.
Fairen V, Lopez V, Conde L: Power series approximation to solutions of nonlinear systems of differential equations. Am J Phys 1988, 56: 5761. 10.1119/1.15432
Filipich CP, Rosales MB: A Straightforward approach to solve ordinary nonlinear differential systems. Mecanica Computacional 2002, 21: 15491568.
Filipich CP, Rosales MB, Buezas F: Some nonlinear mechanical problems solved with analytical solutions. J Latin Am Appl Res 2004, 34: 101109.
Forsyth A: Theory of differential equations. Cambridge, University Press; 1906. pp 78–90
Gőkdoğan A, Merdan M, Yildirim A: The modified algorithm for the differential transform method to solution of Genesio systems. Commun Nonlinear Sci Numerical Simul 2012, 17(1):4551. 10.1016/j.cnsns.2011.03.039
Guerrero F, Santonja FJ, Villanueva RJ: Solving a model for the evolution of smoking habit in Spain with homotopy analysis method. Nonlinear Anal: Real World Appl 2013, 14(1):549558. 10.1016/j.nonrwa.2012.07.015
Günther M: A joint DAE/PDE model for interconnected electrical networks. Math Comput Model Dyn Syst 2000, 6(2):114128. 10.1076/13873954(200006)6:2;1M;FT114
Guzel N, Bayram M: Power series solution of nonlinear first order differential equation systems. Trakya Univ J Sci 2005, 6(1):107111.
He JH: Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999, 178(3–4):257262.
He JH: A coupling method of homotopy technique and a perturbation technique for non linear problems. Int J Nonlinear Mech 2000, 35(1):3743. 10.1016/S00207462(98)000857
He JH: Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput Methods Appl Mech Eng 1998, 167(1–2):6973.
He JH: The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004, 151(1):287292. 10.1016/S00963003(03)003412
He JH: Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals 2005a, 26(3):695700. 10.1016/j.chaos.2005.03.006
He JH: Homotopy perturbation method for bifurcation of nonlinear problems. Int J Non Sci Numer Simul 2005b, 6(2):207208.
He JH: Homotopy perturbation method for solving boundary value problems. Phys Lett A 2006, 350(1–2):8788.
Ince E: Ordinary differential equations. Dover, New York: Dover Publications; 1956. pp 189–199
Khan Y, VazquezLeal H, HernandezMartinez L, Faraz N: Variational iteration algorithmII for solving linear and nonlinear ODEs. Int J Phys Sci 2012, 7(25):30994002.
Kreyszig E: Advanced Engineering Mathematics. New York: Wiley & Sons; 1999.
Kurulay M, Bayram M: A novel power series method for solving second order partial differential equations. Eur J Pure Appl Math 2009, 2(2):268277.
Lucht W, Strehmel K, EichlerLiebenow C: Linear partial differentialalgebraic equations part I: Indexes, consistent boundary/initial conditions. MartinLutherUniversity, HalleWittenberg: Report no. 17, Department of Mathematics and Computer Science; 1997a.
Lucht W, Strehmel K, EichlerLiebenow C: Linear partial differential algebraic equations part II. MartinLutherUniversity, HalleWittenberg: Report no. 18, Department of Mathematics and Computer Science; 1997b.
Lucht W, Strehmel K: Discretization based indices for semilinear partial differentialalgebraic equations. Appl Numer Math 1998, 28(2–4):371386.
Lucht W, Strehmel K, EichlerLiebenow C: Indexes and special discretization methods for linear partial differentialalgebraic equations. BIT 1999, 39(3):484512. 10.1023/A:1022370703243
Martinson WS, Barton PI: A differentiation index for partial differentialalgebraic equations. SIAM J Sci Comput 2000, 21(6):22952315. 10.1137/S1064827598332229
Merdan M, Gőkdoğan A, Yildirim A: On the numerical solution of the model for HIV infection of CD4^{+}T cells. Comput & Math Appl 2011, 62(1):118123.
Momani S, Ertűrk VS: Solutions of nonlinear oscillators by the modified differential transform method. Comput & Math Appl 2008, 55(4):833842.
Momani S, Erjaee GH, Alnasr MH: The modified homotopy perturbation method for solving strongly nonlinear oscillators. Comput & Math Appl 2009, 58(11–12):22092220.
Salehi F, Asadi MA, Hosseini MM: Solving system of DAEs by modified homotopy perturbation method. J Comput Sci & Comp Math 2012, 2(6):15.
Simeon B: Modelling a flexible slider crank mechanism by a mixed system of DAEs and PDEs. Math Model Syst 1996, 2(1):118. 10.1080/13873959608837026
Soltanian F, Dehghan M, Karbassi SM: Solution of the differentialalgebraic equations via homotopy method and their engineering applications. Int J Comp Math 2010, 87(9):19501974. 10.1080/00207160802545908
Strehmel K, Debrabant K: Convergence of RungeKutta methods applied to linear partial differentialalgebraic equations. Appl Numer Math 2005, 53(2–4):213229.
Sweilam NH, Khader MM: Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method. Comput & Math Appl 2009, 58(11–12):21342141.
Tsai PY, Chen CK: An approximate analytic solution of the nonlinear Riccati differential equation. J Franklin Inst 2010, 347(10):18501862. 10.1016/j.jfranklin.2010.10.005
VazquezLeal H: Generalized Homotopy method for solving nonlinear differential equations. Comput Appl Math 2013. doi:10.1007/s4031401300604
VazquezLeal H, Khan Y, FernandezAnaya G, HerreraMay A, SarmientoReyes A, FilobelloNino U: A general solution for Troesch’s problem. Math Probl Eng 2012. doi:10.1155/2012/208375
Yamamoto Y, Dang C, Hao Y, Jiao YC: An aftertreatment technique for improving the accuracy of Adomian’s decomposition method. Comput & Math Appl 2002, 43(6–7):783798.
Acknowledgements
H. VazquezLeal gratefully acknowledge the financial support of the National Council for Science and Technology of Mexico CONACyT through Grant CB201001 #157024.
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Brahim Benhammouda and Hector VazquezLeal contributed equally to this work.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Received
Accepted
Published
DOI
Keywords
 Partial differentialalgebraic equations
 Power series method
 Laplace transform
 Padé approximant; Analytical solutions