SOLUTION OF SHIFF SYSTEMS BY USING DIFFERENTIAL TRANSFORM METHOD

Abstract

In this paper, we use the differential transform method to solve stiff ordinary differential equations of the first order and an ordinary differential equation of any order by converting it into a system of differential of the order one. Theoretical considerations have been discussed and some examples were presented to show the ability of the method for linear and non-linear systems of differential equations. We use MAPLE computer algebra systems for numerical calculations [13].

Keywords

• Stiff system,the differential transform method,Numerical solution of the ordinary differential equations,MAPLE

SOLUTION OF SHIFF SYSTEMS BY USING DIFFERENTIAL TRANSFORM METHOD

Abstract

Bu çalışmada, stiff adi diferansiyel denklemleri çözmek için diferansiyel dönüşüm metodu kullanıldı ve teorisi tartışıtıldı. Metodun, lineer ve lineer olmayan diferansiyel denklem sistemlerine etkinliğini göstermek için, bazı örnekler verildi. Sayısal hesaplamalarda MAPLE bilgisayar cebiri sistemleri kullanıldı.

Keywords

• stiff sistem,diferansiyel dönüşüm metodu,adi diferansiyel denklemlerin sayısal çözümü,MAPLE

References

1. [1]Amodio,P., Mazzia F., Numerical solution of differential-algebraic equations and Computation of consistent initial/boundary conditions. Journal of Computational and Applied Mathematics. 87(1997) 135-146.
2. [2]Ascher, UM., Petzold, LR., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, (1998), Philadelphia.
3. [3]Ayaz, F., On the two dimentional differential transform method, Applied Mathematics and Computation, 143, (2003), 361-374.
4. [4]Blue , JL., Gummel, HK., Rational approximations to matrix exponential for systems of stiff differential equations, Journal of Computational Physics, Vol.5,Issue 1,( 1970),70-83.
5. [5]Brenan, K.E., Campbell, S.L., Petzold L.R., Numerical solution of Initial-value problems in Differential-Algebraic Equations, North-Holland, Amsterdam ,(19899.
6. [6]Brydon D., Pearson J., Marder M., Solving Stiff Differential Equations with the Method of Patches, Journal of Computational Physics 144,(1998)280-298.
7. [7]Burden,RL., Faires J.D., Numerical Analysis, Fifth Edition, PWS Publishing Company,(1993),Boston.
8. [8]Chen, C.L, Liu, Y.C., Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method, Journal of Optimization and Applications,Vol.99,No.1, Oct.(1998),23-35.

9. [9]Chen,C.K., Ho,S.H., Application of differential transformation to eigenvalue problems, Appl. Math. Comp., 79, (1996), 173-188.
10. [10]Chen,C.K., Ho,S.H., Solving partial differential equations by two dimensional differential transform method, Applied Mathematics and Computation, 106, (1999), 171-179.
11. [11]Cooper,GJ., The numerical solution of stiff differential equations, FEBS Letters, Vol.2, Supplement 1, March 1969, S22-S29.
12. [12]Corliss G. And Chang Y. F., Solving Ordinary Differential Equations Using Taylor Series, ACM Trans. Math. Soft. 8(1982), 114-144.
13. [13]Frank G., MAPLE V:CRC Press Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431, (1996).
14. [14]Garfinkel, D.,Marbach, C.B, Stiff Differential Equations, Ann. Rev. BioPhys. Bioeng. 6,( 1977),525-542.
15. [15]Guzel, N., Bayram,M., On the numerical solution of stiff systems, Applied Mathematics and Computation, 170, (2005), 230-236.
16. [16]Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebreis Problems, Springer-Verlag,( 1991).
17. [17]Hairer,E., Wanner,G., Stiff differential equations solved by Radau Methods, Journal of Computational and Applied Mathematics, 111, 1999,93-111.
18. [18]Hassan,I.H.Abdel-Halim, Different applications for the differential transformation in the differential equations, Applied Mathematics and Computation, 129, (2002), 183-201.
19. [19]Henrici, P., Applied Computational Complex Analysis, Vol.1, John Wiely&Sons, New York, Chap.1, (1974).
20. [20]Hojjati,G., Ardabili MYR., .Hosseini, SM, Newsecond derivative multistep methods for stiff systems, Applied Mathematical Modelling,30,(2006)466-476.
21. [21]Hull ,TE., Enright,WH., Fellen BM., Sedgwick AE., Comparing numerical methods for ordinary differential equations, SIAM J, Numer.Anal. 9 (1972) 603.
22. [22]Jackiewicz, Z., Implementation of DIMSIMs for stiff differential systems, Applied Numerical Mathematics 42 (2002) 251-267.
23. [23]Jang,C. Chen C.L.,Liu,YC. On Solving The initial-value problems using the differential transformation method, Applied Mathematics and Computation, 115,(2000), 145-160.
24. [24]Preiser,DL. , A Class of nonlinear multistep A-stable numerical methods for solving stiff differential equations, Computer&Mathematics with Applications,Vol.4,Issue 1, 1978,43-51.
25. [25]Press, WH., Flannery BP., Teukolsky S.A., Vetterling W.T., Numerical Recipes, Cambridge University Press,(1988), Cambridge.
26. [26]Rangaiah,G.P., Comparson of two algorithms for solving STIFF differential equations, Computer&Structures, Vol. 20, Issue 6,(1985), 915-920.
27. [27]Sottas,G., Rational Runge-Kutta methods are not suitable for stiff systems of ODEs, Journal of Computational and Applied Mathematics, Vol.10, Issue 2, April 1984,169-174.
28. [28]Tripathy,SC., S. and Balasubramanian, R., Semi-İmplicit Runge-Kutta methods for power system transient stability studies, International journal of Electrical Power&Energi Systems,Vol.10,Issue 4,Oct 1988,253-259.
29. [29]Ypma,T.J., Relaxed Newton-Like Methods for stiff differential systems, Journal of Computational and Applied Mathematics, Vol.16, Issue 1, September 1986,95-103.
30. [30]Zhou,JK., Differential Transformation and Its Applications for Electrical Circuits,Huazhong Un.Press,Wuhan,China,(1986).