8_xxx_celik.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 4, No. 1, 2011, 67-75 ISSN 1307-5543 – www.ejpam.com The Numerical Solution Of Partial Differential-Algebraic Equa- tions (PDAEs) By Multivariate Pade Approximation Muhammed Yiğider 2, Ercan Celik1,∗ 1 Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, Turkey 2 Department of Mathematics, Faculty of Art and Science, Erzincan University, Erzincan, Turkey Abstract. In this paper, Numerical solution of Partial Diferential-Algebraic Equa- tions(PDAEs) is con- sidered by Multivariate Padè Approximations. We applied these method to one example. First Partial Diferential-Algebraic Equation(PDAE) has been converted to power series by two-dimensional difer- ential transformation,Then the numerical solution of equation was put into Multivariate Padè series form. Thus we obtained numerical solution of Partial Diferential-Algebraic Equa- tion(PDAE). 2000 Mathematics Subject Classifications: 35 Key Words and Phrases: Partial Differential-Algebraic Equation (PDAs), two-dimensional differential transformation, Multivariate Padè Approximation 1. Introduction In this study, we consider Linear Partial Differential-Algebraic Equations(PDAEs) of the form Aut(t, x) + Bux x (t, x) + Cu(t, x) = f (t, x) (1) Where t ∈ (0, te) and x ∈ (−l, l) ⊂ R, A, B, C ∈ Rnxn are constant matrices, u, f : � 0, te � x [−l, l] → Rn. We are interested in cases where at least one of the matrices A and Bis singular. The two special cases A= 0 or B = 0 lead to ordinary differential equations or DAEs which are not considered here. Therefore in this paper we assume that none of the matrices A or B is the zero matrix [6, 7]. Many important mathematical models can be expressed in terms of Partial Differential Alge- braic Equations(PDAEs). Such models arise in many areas of mathematics, engineering, the physical sciences and population growth. In resent years, much research has been focused on the numerical solution of Partial Differential-Algebraic Equations(PDAEs). Some numerical methods have been developed, using Runge-Kutta methods [8]. ∗Corresponding author. Email addresses: er elik�atauni.edu.tr (E. Celik), myigider�erzin an.edu.tr (M. Yigider) http://www.ejpam.com 67 c© 2010 EJPAM All rights reserved. M. Yiğider, E. Çelik / Eur. J. Pure Appl. Math, 4 (2011), 67-75 68 The purpose of this paper is to consider the numerical solution of Partial Differential- Algebraic Equations(PDAEs) by using Multivariate Padé Approximations. 2. Two-Dimensional Differential Transformation The basic definition of the two-dimensional differential transform is defined as follows [9, 2, 3, 4, 1]: W (k,h) = 1 k!h! � ∂ k+hw(x , y) ∂ x k∂ yh � 0,0 (2) where w(x , y) is the original function and W (k,h) is the transformed function. The transfor- mation is called T - function and lower case and upper case letters represent the original and transformed functions respectively. The differential inverse transform of W (k,h) is defined as w(x , y) = ∞ ∑ k=0 ∞ ∑ h=0 W (k,h)x k yh (3) and from Eqs.(2) and (3) can be concluded w(x , y) = ∞ ∑ k=0 ∞ ∑ h=0 1 k!h! � ∂ k+hw(x , y) ∂ x k∂ yh � 0,0 x k yh. (4) 3. Multivariate Padé Approximations Consider the bivariate function f (x , y) with Taylor series development f (x , y) = ∞ ∑ i, j=0 ci j x i y j (5) around the origin. We know that a solution of univariate Padé approximation problem for f (x) = ∞ ∑ i=0 ci x i (6) is given by p(x) = � � � � � � � � � ∑m i=0 ci x i x ∑m−1 i=0 ci x i · · · xn ∑m−n i=0 ci x i cm+1 cm · · · cm+1−n ... ... . . . ... cm+n cm+n−1 · · · cm � � � � � � � � � (7) and q(x) = � � � � � � � � � 1 x · · · xn cm+1 cm · · · cm+1−n ... ... . . . ... cm+n cm+n−1 · · · cm � � � � � � � � � (8) M. Yiğider, E. Çelik / Eur. J. Pure Appl. Math, 4 (2011), 67-75 69 Let us now multiply jth row in p(x) and q(x) by x j+m−1 ( j = 2, . . . , n+ 1) and afterwards divide jth column in p(x) and q(x) by x j−1( j = 2, . . . , n+ 1). This results in a multiplication of numerator and denominator by xmn. Having done so, we get p(x) q(x) = � � � � � � � � � ∑m i=0 ci x i ∑m−1 i=0 ci x i · · · ∑m−n i=0 ci x i cm+1 xm+1 cm xm · · · cm+1−n xm+1−n ... ... . . . ... cm+n xm+n cm+n−1 xm+n−1 · · · cm xm � � � � � � � � � � � � � � � � � � 1 1 · · · 1 cm+1 xm+1 cm xm · · · cm+1−n xm+1−n ... ... . . . ... cm+n xm+n cm+n−1 xm+n−1 · · · cm xm � � � � � � � � � (9) (D = det Dm,n 6= 0). This quotient of determinants can also immediately be written down for a bivariate function f (x , y). The sum ∑k i=0 ci x i shall be replaced kth partial sum of the Taylor series development of f (x , y) and the expression ck x k by an expression that contains all the terms of degree k in f (x , y). Here a bivariate term ci j x i y j is said to be of degree i + j. If we define p(x , y) = � � � � � � � � � � ∑m i+ j=0 ci j x i y j ∑m−1 i+ j=0 ci j x i y j · · · ∑m−n i+ j=0 ci j x i y j ∑ i+ j=m+1 ci j x i y j ∑ i+ j=m ci j x i y j · · · ∑ i+ j=m+1−n ci j x i y j ... ... . . . ... ∑ i+ j=m+n ci j x i y j ∑m i+ j=m+n−1 ci j x i y j · · · ∑m i+ j=m ci j x i y j � � � � � � � � � � (10) and q(x , y) = � � � � � � � � � 1 1 · · · 1 ∑ i+ j=m+1 ci j x i y j ∑ i+ j=m ci j x i y j · · · ∑ i+ j=m+1−n ci j x i y j ... ... . . . ... ∑ i+ j=m+n ci j x i y j ∑m i+ j=m+n−1 ci j x i y j · · · ∑m i+ j=m ci j x i y j � � � � � � � � � (11) Then it is easy to see that p(x , y) and q(x , y) are of the form p(x , y) = ∑mn+m i+ j=mn ai j x i y j q(x , y) = ∑mn+n i+ j=mn bi j x i y j (12) We know that p(x , y) and q(x , y) are called Padé equations [5]. So the multivariate Padé approximant of order (m, n) for f (x , y) is defined as, rm,n(x , y) = p(x , y) q(x , y) (13) M. Yiğider, E. Çelik / Eur. J. Pure Appl. Math, 4 (2011), 67-75 70 4. Numerical Example: The test problem considers the following Partial Differential-Algebraic Equation(PDAE) [8]:    0 1 0 0 0 1 0 0 0    ut +    0 0 −1 0 −1 −1 0 0 0    ux x +    −1 −1 −1 0 −1 0 0 0 1    u= f (14) x ∈ [−0.5,0.5], t ∈ [0,1]. Where f1 = −x(x − 1)(2 sin t + cos t)− (et + t5)(x2− x + 2) f2 = x(x − 1)(et + 5t4 − cos t)− 2(et + t5 + cos t) f3 = −x(x − 1)(et + t5). The exact solution is u(x , t) =    x(x − 1) sin(t) x(x − 1) cos(t) x(x − 1)(et + t5)    (15) Equivalently, Equation (14) can be written as    0 1 0 0 0 1 0 0 0       u1t u2t u3t    +    0 0 −1 0 −1 −1 0 0 0       u1x x u2x x u3x x    +    −1 −1 −1 0 −1 0 0 0 1       u1 u2 u3    =    f1 f2 f3    (16) u2t − u3x x − u1 − u2 − u3 = f1 u3t − u2x x − u3x x − u2 = f2 (17) −u3 = f3 By using the basic definition of the two-dimensional differential transform and taking the transform of Equation (17) can obtain that (k+ 1)U2(k+ 1,h)− (h+ 1)(h+ 2)U3(k,h+ 2)− U1(k,h)− U2(k,h)− U3(k,h) = F1(k,h) (k+ 1)U3(k+ 1,h)− (h+ 1)(h+ 2)U2(k,h+ 2)− (h+ 1)(h+ 2)U3(k,h+ 2)− U2(k,h) = F2(k,h) −U3(k,h) = F3(k,h) Consequently, by substituting the values of Ui . We have obtained u1(x , t) = −x t + x2t + 1 6 x t3 − 1 6 x2t3 − 1 120 x t5 + 1 120 x2t5 + 1 5040 x t7 u2(x , t) = −x + x2+ 1 2 x t2 − 1 2 x2t2 − 1 24 x t4 + 1 24 x2t4 + 1 720 x t6 − 1 720 x2t6 u3(x , t) = −x − x t + x2− 1 2 x t2 + x2t − 1 6 x t3 + 1 2 x2t2 − 1 24 x t4 + 1 6 x2t3 − 121 120 x t5 M. Yiğider, E. Çelik / Eur. J. Pure Appl. Math, 4 (2011), 67-75 71 + 1 24 x2t4 − 1 720 x t6 + 121 120 x2t5 − 1 5040 x t7 + 1 720 x2t6 The Power series u1(x , t),u2(x , t) and u3(x , t) can be transformed into multivariate Padé ap- proximation m= 3, n= 2 p1(x , t) = � � � � � � � −x t + x2t −x t 0 1 6 x t3 x2t −x t 1 6 x2t3 1 6 x t3 x2t � � � � � � � = −0.1666666667x3t5 + 0.1666666667x4t5 − x5t7 + x6t7 q1(x , t) = � � � � � � � 1 1 1 1 6 x t3 x2t −x t 1 6 x2t3 1 6 x t3 x2t � � � � � � � = x4t2 + 0.1666666667x2t4 + 0.02777777778x2t6 + 0.1666666667x4t4 r1(x , t) = p1(x , t) q1(x , t) = −0.1666666667x3t5 + 0.1666666667x4t5 − x5t7 + x6t7 x4t2 + 0.1666666667x2t4 + 0.02777777778x2t6 + 0.1666666667x4t4 p2(x , t) = � � � � � � � −x + x2+ 1 2 x t2 −x + x2 −x −1 2 x2t2 1 2 x t2 x2 − 1 24 x t4 −1 2 x2t2 1 2 x t2 � � � � � � � = −0.2500000000x3t4 − 0.5000000000x5t2 + 0.04166666667x4t4 +0.5000000000x6t2 + 0.1041666667x3t6 + 0.2083333333x5t4 q2(x , t) = � � � � � � � 1 1 1 −1 2 x2t2 1 2 x t2 x2 − 1 24 x t4 −1 2 x2t2 1 2 x t2 � � � � � � � = 0.2500000000x4t4 + 0.5000000000x4t2 + 0.2500000000x4t4 +0.20833333333x3t4 + 0.0208333333x2t6 r2(x , t) = p2(x , t) q2(x , t) M. Yiğider, E. Çelik / Eur. J. Pure Appl. Math, 4 (2011), 67-75 72 = � −0.2500000000x3t4 − 0.5000000000x5t2 + 0.04166666667x4t4 +0.5000000000x6t2 + 0.1041666667x3t6 + 0.2083333333x5t4 � � −0.2500000000x3t4 − 0.5000000000x5t2 + 0.04166666667x4t4 +0.5000000000x6t2 + 0.1041666667x3t6 + 0.2083333333x5t4 � p3(x , t) = � � � � � � � −x − x t + x2− 1 2 x t2 + x2t −x − x t + x2 −x −1 6 x t3 + 1 2 x2t2 −1 2 x t2 + x2t −x t + x2 − 1 24 x t4 + 1 6 x2t3 −1 6 x t3 + 1 2 x2t2 −1 2 x t2 + x2t � � � � � � � = 0.0833333333x2t4 − 0.3333333333x3t3 + 0.50000000000x4t2 + 0.0694444444x2t6 −0.0416666667x3t5 − 0.0416666667x2t5 + 0.2083333333x3t4 + 0.083333333x4t4 −0.3333333333x4t3 r3(x , t) = p3(x , t) q3(x , t) =    0.2083333333x4t4 − 0.0833333333x3t4 − 0.5000000000x5t2 + 0.5000000000x6t2 −0.0694444444x3t6 − 0.1250000000x5t4 + 0.3333333333x4t3 − 0.0416666667x3t5 −0.5000000000x5t3 + 0.0416666667x4t5 + 0.1666666667x6t3       0.0833333333x2t4 − 0.3333333333x3t3 + 0.50000000000x4t2 + 0.0694444444x2t6 −0.0416666667x3t5 − 0.0416666667x2t5 + 0.2083333333x3t4 + 0.083333333x4t4 −0.3333333333x4t3   Table 1: Comparison of the Numeri al Solution of u1(x , t) with Exa t Solutions (t = 0.01) x u1(x , t) r1(x , t) � �u1(x , t)− r1(x , t) � � -0.5 0.007499875000 0.007499874999 1.10-12 -0.4 0.005599906667 0.005599006668 1.10-12 -0.3 0.003899935000 0.003899935000 0 -0.2 0.002399960000 0.002399960000 0 -0.1 0.001099981667 0.001099981668 1.10-12 0.1 -0.008999850001 -0.008999850006 5.10-13 0.2 -0.001599973333 -0.001599973333 0 0.3 -0.002099965000 -0.002099965000 0 0.4 -0.002399960000 -0.002399960000 0 0.5 -0.002499958334 -0.002499958333 1.10-12 M. Yiğider, E. Çelik / Eur. J. Pure Appl. Math, 4 (2011), 67-75 73Table 2: Comparison of the numeri al solution of u2(x , t) with exa t solutions (t = 0.01) x u2(x , t) r2(x , t) � �u2(x , t)− r2(x , t) � � -0.5 0.07499625003 0.07499625011 8.10-10 -0.4 0.5599720002 0.5599720006 4.10-10 -0.3 0. 3899805002 0. 3899805002 2.10-10 -0.2 0.2399880001 0.2399880002 1.10-10 -0.1 0.1099945000 0.1099945001 1.10-10 0.1 -0.08999550004 -0.08999550002 2.10-11 0.2 -0. 1599920001 -0. 1599929999 2.10-10 0.3 -0.2099895001 -0.2099895999 2.10-10 0.4 -0.2399880001 -0.2399889999 2.10-10 0.5 -0.2499875001 -0.2499874995 6.10-10Table 3: Comparison of the numeri al solution of u3(x , t) with exa t solutions (t = 0.01) x u3(x , t) r3(x , t) � �u3(x , t)− r3(x , t) � � -0.5 0.7575376252 0.7575376251 1.10-10 -0.4 0.5656280935 0.5656280938 3.10-10 -0.3 0. 3939195651 0. 3939195651 0 -0.2 0.2424120401 0.2424120401 0 -0.1 0.1111055184 0.1111055184 0 0.1 -0.09090451503 -0.09090451504 1.10-11 0.2 -0. 1616080267 -0. 1616080267 0 0.3 -0.2121105351 -0.2121105350 1.10-10 0.4 -0.2424120401 -0.2424120401 0 0.5 -0.2525125418 -0.2525125415 3.10-10 Figure 1: Values of u1(x , t) and its r3,2(x , t) REFERENCES 74 Figure 2: Values of u2(x , t) and its r3,2(x , t) Figure 3: Values of u3(x , t) and its r3,2(x , t) 5. Conclusions The method has proposed for solving partial differential-algebraic equations(PDAEs). The results of example showed that exactly the same solutions have been obtained with Multivarite Padé approximation. On the other hand the results are quite reliable. Therefore, this method can be applied to many complicated Partial Differential-Algebraic Equations(PDAEs). References [1] G Adomian. Convergent series solution of nonlinear equations, Journal of Computational and applied Mathematits, 11, 225-230,1984. [2] F Ayaz. On the Two-Dimensional Differential Transform Method, Applied Mathematics and Computation, 143:361-374,2003. [3] F Ayaz. Solutions of the System of Differential Equations by Differential Transform Method, Applied Mathematics and Computation, 147:547-567,2004. REFERENCES 75 [4] N Bildik, A Konuralp. Two-Dimensional Differential Transform Method,Adomian’s De- compostion Method and Variational Iteration Method for Partial Differential Equations, International Journal of Computer Mathematics ,Vol.83, 12:973-987,2006. [5] A Cuyt, L Wuytack. Nonlinear Methods in Numerical Analysis, Amsterdam,1987. [6] W Lucht, K Strehmel, C E Liebenow. Linear Partial Differential-Algebraic Equations, Part I, Reports of the Institute of Numerical Mathematics, Report No. 17,1997. [7] W Lucht, K Strehmel, C E Liebenow. Linear Partial Differential-Algebraic Equations, Part II, Reports of the Institute of Numerical Mathematics, Report No. 18, 1997, [8] K Strehmel, K Debrabant. Convergence of Runge-Kutta Methods Applied to Linear Partial Differential-Algebraic Equations, Applied Numerical Mathematics, 53:213-229, 2005, [9] J K Zhou. Differential Transform and its Applications for Electrical Circuits (Wuhan:Huarjung University Press), 1986