- Open Access
Analytical solutions for systems of partial differential–algebraic equations
© Benhammouda and Vazquez-Leal; licensee Springer. 2014
Received: 4 December 2013
Accepted: 5 March 2014
Published: 10 March 2014
This work presents the application of the power series method (PSM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-one and index-three are solved to show that PSM can provide analytical solutions of PDAEs in convergent series form. What is more, we present the post-treatment of the power series solutions with the Laplace-Padé (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.
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 (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 Runge-Kutta 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).
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 applications problems in science and engineering often lead to higher-index PDAEs, new techniques are required to solve these problems efficiently.
Modern methods like homotopy perturbation method (HPM) (He 199920001998; Vazquez-Leal et al. 2012), homotopy analysis method (HAM) (Guerrero et al. 2013), variational iteration method (VIM) (Khan Y, et al. 2012), generalized homotopy method (Vazquez-Leal 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 integro-differential equation (El-Shahed 2005), nonlinear differential equations (He 1998), nonlinear oscillators (He 2004), partial differential equations (PDEs) (He 2005a), bifurcation of nonlinear problems (He 2005b) and boundary-value 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 well-known 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 Laplace-Padé post-treatment greatly improves PSM’s truncated series solutions in convergence rate. In fact, the Laplace-Padé 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 index-reduction 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 “Laplace-Padé resummation method”, we give the basic concept of the Laplace-Padé 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 index-one and index-three. In section “Discussion”, we give a brief discussion. Finally, a conclusion is drawn in the last section.
Basic concept of power series method
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.
where u0,u1,… are unknowns to be determined by series method.
Equation (3) is substituted into (1), then we regroup the equation in terms of powers of t.
We equate each coefficient of the resulting polynomial to zero.
The boundary conditions of (1) are substituted into (3) to generate an algebraic equation for each boundary condition.
Aforementioned steps generate an algebraic linear system for the unknowns of (3).
Finally, we solve the algebraic linear system to obtain the coefficients u 0,u 1,…
where we considered q0=1, and the numerator and denominator have no common factors.
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.
Laplace-Padé resummation method
Several approximate methods provide power series solutions (polynomial). Nevertheless, sometimes, this type of solutions lacks of large domains of convergence. Therefore, Laplace-Padé (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.
First, Laplace transformation is applied to power series (3).
Next, s is substituted by 1/t in the resulting equation.
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.
Then, t is substituted by 1/s.
Finally, by using the inverse Laplace s transformation, we obtain the exact or approximate solution.
Application of PSM to solve PDAE systems
where u k : , k=1,2 and b>a.
where g(x) is a given function.
We assume that the solution to initial boundary-value problem (10)-(13) exists, is unique and sufficiently smooth.
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).
where αk,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 Laplace-Padé resummation method to PSM truncated series to enlarge the convergence region as depicted in the next section.
In this section, we will demonstrate the effectiveness and accuracy of the LPPSM presented in the previous section through two PDAE systems of index-one and index-three.
Nonlinear index-one system
where 0<x<1 and t>0.
Since one time differentiation of equation (17) determines u2t in terms of u and its space derivatives, then PDAE (16)-(17) is index-one. Note that no initial condition is prescribed for the variable u2 as this is determined by the PDAE.
where αk,n(x),k=1,2; n=0,1,2,… are unknown functions to be determined later on by the PSM.
where (′) denotes the ordinary derivative with respect to x.
From recursion (27), we get α1,1(x)=-π2 cosπ x and α1,2(x)=(π4/2) cosπ x.
Similarly, the coefficients α1,n(x) and α2,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 Laplace-Padé post-treatment. First, we apply t-Laplace transform to (28) and (29). Then, we substitute s by 1/t and apply t-Padé approximant to the transformed series. Finally, we substitute t by 1/s and apply the inverse Laplace s-transform to the resulting expressions to get the approximate or exact solutions.
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 index-three system
where 0<x<1 and t>0.
Since three time differentiations of equation (40) determine u3t in terms of the solution u and its space derivatives, then PDAE (38)-(40) is index-three. Therefore, this PDAE is difficult to solve numerically. Moreover no initial condition is prescribed for the variable u3 as this is determined by the PDAE.
where αk,n(x),k=1,2,3; n=0,1,2,… are unknown functions to be determined later on by the PSM.
where (′) denotes the ordinary derivative with respect to x.
Similarly, the coefficients α1,n(x), α2,n(x) and α3,n-2(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 t-Padé 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 t-Padé 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.
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.
In this paper we presented the power series method (PSM) as a useful analytical tool to solve partial differential-algebraic equations (PDAEs). Two PDAE problems of index-one and index-three were solved by this method leading to the exact solutions. The method has successfully handled the index-three PDAE without the need for a preprocessing step of index-reduction. 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 Laplace-Padé (LP) post-treatement 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 Laplace-Padé. What is more, there is not any standard analytical or numerical methods to solve higher-index PDAEs, converting the LPPSM method into an attractive tool to solve such problems.
On one hand, semi-analytic 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 least-squares 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 post-treatement will provide a larger domain of convergence.
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 post-treatment 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 semi-analytical 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 higher-index PDAEs. Additionally, the LPPSM 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 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 higher-index nonlinear PDAE systems.
H. Vazquez-Leal gratefully acknowledge the financial support of the National Council for Science and Technology of Mexico CONACyT through Grant CB-2010-01 #157024.
- Ali G, Bartel A, Günther M: Parabolic differential-algebraic models in electrical network design. SIAM J Mult Model Sim 2005, 4(3):813-838. 10.1137/040610696View ArticleGoogle Scholar
- Ali G, Bartel A, Günther M, Tischendorf C: Elliptic partial differential-algebraic multiphysics models in electrical network design. Math Model Meth Appl Sci 2003, 13(9):1261-1278. 10.1142/S0218202503002908View ArticleGoogle Scholar
- Aminikhah H, Hemmatnezhad M: An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations. Appl Math Lett 2011, 24: 1502-1508. 10.1016/j.aml.2011.03.032View ArticleGoogle Scholar
- Asadi MA, Salehi F, Hosseini MM: Modification of the homotopy perturbation method for nonlinear system of second-order BVPs. J Comput Sci & Comput Math 2012, 2(5):23-28.Google Scholar
- Barker GA: Essentials of Padé Approximants. London: Academic Press; 1975.Google Scholar
- Bartel A, Pulch R: A concept for classification of partial differential-algebraic equations in nanoelectronics. 2006.http://www-num.math.uni-wuppertal.de/fileadmin/mathe/www-num/preprints/amna_06_07.pdf Preprint BUW-AMNA 06/07, .Google Scholar
- 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: 486-491. 10.1016/j.amc.2005.01.040View ArticleGoogle Scholar
- Budd CJ, Dold JW, Stuart AM: Blow-up in a system of partial differential equations with conserved first integral, part 2: problems with convection. SIAM J Appl Math 1994, 54(3):610-640. 10.1137/S0036139992232131View ArticleGoogle Scholar
- Coddington EA: An introduction to ordinary differential equations. New York: Dover Publications; 1989.Google Scholar
- Ebaid AE: Commun Nonlinear Sci Numerical Simul. 2011, 16(1):528-536. 10.1016/j.cnsns.2010.03.012View ArticleGoogle Scholar
- El-Shahed M: Application of He’s homotopy perturbation method to Volterra’s integro-differential equation. Int J Nonlinear Sci Numer Simul 2005, 6(2):163-168.View ArticleGoogle Scholar
- Fairen V, Lopez V, Conde L: Power series approximation to solutions of nonlinear systems of differential equations. Am J Phys 1988, 56: 57-61. 10.1119/1.15432View ArticleGoogle Scholar
- Filipich CP, Rosales MB: A Straightforward approach to solve ordinary nonlinear differential systems. Mecanica Computacional 2002, 21: 1549-1568.Google Scholar
- Filipich CP, Rosales MB, Buezas F: Some nonlinear mechanical problems solved with analytical solutions. J Latin Am Appl Res 2004, 34: 101-109.Google Scholar
- Forsyth A: Theory of differential equations. Cambridge, University Press; 1906. pp 78–90Google Scholar
- 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):45-51. 10.1016/j.cnsns.2011.03.039View ArticleGoogle Scholar
- 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):549-558. 10.1016/j.nonrwa.2012.07.015View ArticleGoogle Scholar
- Günther M: A joint DAE/PDE model for interconnected electrical networks. Math Comput Model Dyn Syst 2000, 6(2):114-128. 10.1076/1387-3954(200006)6:2;1-M;FT114View ArticleGoogle Scholar
- Guzel N, Bayram M: Power series solution of non-linear first order differential equation systems. Trakya Univ J Sci 2005, 6(1):107-111.Google Scholar
- He JH: Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999, 178(3–4):257-262.View ArticleGoogle Scholar
- He JH: A coupling method of homotopy technique and a perturbation technique for non linear problems. Int J Nonlinear Mech 2000, 35(1):37-43. 10.1016/S0020-7462(98)00085-7View ArticleGoogle Scholar
- He JH: Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput Methods Appl Mech Eng 1998, 167(1–2):69-73.View ArticleGoogle Scholar
- He JH: The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004, 151(1):287-292. 10.1016/S0096-3003(03)00341-2View ArticleGoogle Scholar
- He JH: Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals 2005a, 26(3):695-700. 10.1016/j.chaos.2005.03.006View ArticleGoogle Scholar
- He JH: Homotopy perturbation method for bifurcation of nonlinear problems. Int J Non Sci Numer Simul 2005b, 6(2):207-208.Google Scholar
- He JH: Homotopy perturbation method for solving boundary value problems. Phys Lett A 2006, 350(1–2):87-88.View ArticleGoogle Scholar
- Ince E: Ordinary differential equations. Dover, New York: Dover Publications; 1956. pp 189–199Google Scholar
- Khan Y, Vazquez-Leal H, Hernandez-Martinez L, Faraz N: Variational iteration algorithm-II for solving linear and non-linear ODEs. Int J Phys Sci 2012, 7(25):3099-4002.View ArticleGoogle Scholar
- Kreyszig E: Advanced Engineering Mathematics. New York: Wiley & Sons; 1999.Google Scholar
- Kurulay M, Bayram M: A novel power series method for solving second order partial differential equations. Eur J Pure Appl Math 2009, 2(2):268-277.Google Scholar
- Lucht W, Strehmel K, Eichler-Liebenow C: Linear partial differential-algebraic equations part I: Indexes, consistent boundary/initial conditions. Martin-Luther-University, Halle-Wittenberg: Report no. 17, Department of Mathematics and Computer Science; 1997a.Google Scholar
- Lucht W, Strehmel K, Eichler-Liebenow C: Linear partial differential algebraic equations part II. Martin-Luther-University, Halle-Wittenberg: Report no. 18, Department of Mathematics and Computer Science; 1997b.Google Scholar
- Lucht W, Strehmel K: Discretization based indices for semilinear partial differential-algebraic equations. Appl Numer Math 1998, 28(2–4):371-386.View ArticleGoogle Scholar
- Lucht W, Strehmel K, Eichler-Liebenow C: Indexes and special discretization methods for linear partial differential-algebraic equations. BIT 1999, 39(3):484-512. 10.1023/A:1022370703243View ArticleGoogle Scholar
- Martinson WS, Barton PI: A differentiation index for partial differential-algebraic equations. SIAM J Sci Comput 2000, 21(6):2295-2315. 10.1137/S1064827598332229View ArticleGoogle Scholar
- 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):118-123.View ArticleGoogle Scholar
- Momani S, Ertűrk VS: Solutions of non-linear oscillators by the modified differential transform method. Comput & Math Appl 2008, 55(4):833-842.View ArticleGoogle Scholar
- Momani S, Erjaee GH, Alnasr MH: The modified homotopy perturbation method for solving strongly nonlinear oscillators. Comput & Math Appl 2009, 58(11–12):2209-2220.View ArticleGoogle Scholar
- Salehi F, Asadi MA, Hosseini MM: Solving system of DAEs by modified homotopy perturbation method. J Comput Sci & Comp Math 2012, 2(6):1-5.Google Scholar
- Simeon B: Modelling a flexible slider crank mechanism by a mixed system of DAEs and PDEs. Math Model Syst 1996, 2(1):1-18. 10.1080/13873959608837026View ArticleGoogle Scholar
- Soltanian F, Dehghan M, Karbassi SM: Solution of the differential-algebraic equations via homotopy method and their engineering applications. Int J Comp Math 2010, 87(9):1950-1974. 10.1080/00207160802545908View ArticleGoogle Scholar
- Strehmel K, Debrabant K: Convergence of Runge-Kutta methods applied to linear partial differential-algebraic equations. Appl Numer Math 2005, 53(2–4):213-229.Google Scholar
- 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):2134-2141.View ArticleGoogle Scholar
- Tsai PY, Chen CK: An approximate analytic solution of the nonlinear Riccati differential equation. J Franklin Inst 2010, 347(10):1850-1862. 10.1016/j.jfranklin.2010.10.005View ArticleGoogle Scholar
- Vazquez-Leal H: Generalized Homotopy method for solving nonlinear differential equations. Comput Appl Math 2013. doi:10.1007/s40314-013-0060-4Google Scholar
- Vazquez-Leal H, Khan Y, Fernandez-Anaya G, Herrera-May A, Sarmiento-Reyes A, Filobello-Nino U: A general solution for Troesch’s problem. Math Probl Eng 2012. doi:10.1155/2012/208375Google Scholar
- 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):783-798.Google Scholar
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