Exact Solution for a Class of Stiff Systems by Differential Transform Method

Affiliation(s)

City University of Science and Information Technology, Peshawar, Pakistan.

Edwardes College, Peshawar, Pakistan.

University of Malakand, Chakdara Dir (Lower), Pakistan.

City University of Science and Information Technology, Peshawar, Pakistan.

Edwardes College, Peshawar, Pakistan.

University of Malakand, Chakdara Dir (Lower), Pakistan.

ABSTRACT

In this paper, Differential Transform Method (DTM) is proposed for the closed form solution of linear and non-linear stiff systems. First, we apply DTM to find the series solution which can be easily converted into exact solution. The method is described and illustrated with different examples and figures are plotted accordingly. The obtained result confirm that DTM is very easy, effective and convenient.

In this paper, Differential Transform Method (DTM) is proposed for the closed form solution of linear and non-linear stiff systems. First, we apply DTM to find the series solution which can be easily converted into exact solution. The method is described and illustrated with different examples and figures are plotted accordingly. The obtained result confirm that DTM is very easy, effective and convenient.

Cite this paper

M. Idrees, F. Mabood, A. Ali and G. Zaman, "Exact Solution for a Class of Stiff Systems by Differential Transform Method,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 440-444. doi: 10.4236/am.2013.43065.

M. Idrees, F. Mabood, A. Ali and G. Zaman, "Exact Solution for a Class of Stiff Systems by Differential Transform Method,"

References

[1] G. Hojjati, M. Y. Rahimi Arabili and S. M. Hosseini, “An Adaptive Method for Numerical Solution of Stiff System of Ordinary Differential Equations,” Mathematics and Computers in Simulation, Vol. 66, No. 1, 2004, pp. 33-41. doi:10.1016/j.matcom.2004.02.019

[2] E. Harier and G. Wanner, “Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems,” Springer-Verlag, New York, 1996. doi:10.1007/978-3-642-05221-7

[3] L. Lopidus and W. E. Schiesser, “Numerical Methods for Differential Systems,” Academic Press, New York, 1976.

[4] J. Carroll, “A Metrically Exponentially Fitted Scheme for the Numerical Solution of Stiff Initial Value Problems,” Computers & Mathematics with Applications, Vol. 26, 1993, pp. 57-64. doi:10.1016/0898-1221(93)90034-S

[5] C. H. Hsiao, “Haar Wavelet Approach to Linear Stiff Systems,” Mathematics and Computer in Simulation, Vol. 64, No. 1, 2004, pp. 561-567. doi:10.1016/j.matcom.2003.11.011

[6] C. H. Hsiao and W. J. Wang, “Haar Wavelet Approach to Non-Linear Stiff Systems,” Mathematics and Computer in Simulation, Vol. 57, No. 6, 2001, pp. 347-353. doi:10.1016/S0378-4754(01)00275-0

[7] J. K. Zhou, “Differential Transformation and Its Applications for Electrical Circuits,” Huazhong University Press, Wuhan, 1986 (in Chinese).

[8] C. K. Chen and S. H. Ho, “Solving Partial Differential Equations by Two-Dimensional Differential Transform Method,” Applied Mathematics and Computation, Vol. 106, No. 2-3, 1999, pp. 171-179. doi:10.1016/S0096-3003(98)10115-7

[9] M. J. Jang, C. L. Chen and Y. C. Liu, “Two-Dimensional Differential Transform for Partial Differential Equations,” Applied Mathematics and Computation, Vol. 121, No. 2-3, 2001, pp. 261-270. doi:10.1016/S0096-3003(99)00293-3

[10] I. H. Abdel-Halim Hassan, “Different Applications for the Differential Transformation in the Differential Equations,” Applied Mathematics and Computation, Vol. 129, No. 2-3, 2002, pp. 183-201. doi:10.1016/S0096-3003(01)00037-6

[11] F. Ayaz, “On the Two-Dimensional Differential Transform Method,” Applied Mathematics and Computation, Vol. 143, No. 2-3, 2003, pp. 361-374.

[12] M. J. Jang, J. S. Wang and Y. C. Liu, “Applying Differential Transformation Method to Parameter Identification Problems,” Applied Mathematics and Computation, Vol. 139, No. 2, 2003, pp. 491-502. doi:10.1016/S0096-3003(02)00211-4

[13] C. K. Chen and S. P. Ju, “Application of Differential Transformation Method to Transient Adjective-Disper-sive Transport Equations,” Applied Mathematics and Computation, Vol. 155, No. 1, 2004, pp. 25-38. doi:10.1016/S0096-3003(03)00755-0

[14] F. Ayaz, “Application of Differential Transform Method to Differential-Algebraic Equations,” Applied Mathematics and Computation, Vol. 152, No. 3, 2004, pp. 649-657. doi:10.1016/S0096-3003(03)00581-2

[15] F. Ayaz, “Solution of the System of Differential Equations by Differential Transform Method,” Applied Mathematics and Computation, Vol. 147, No. 2, 2004, pp. 547-567. doi:10.1016/S0096-3003(02)00794-4

[16] A. Kurnaz, G. Oturanc and M. E. Kiris, “N-Dimensional Differential Transform Method for Solving PDEs,” International Journal of Computer Mathematics, Vol. 82, No. 3, 2005, pp. 369-380. doi:10.1080/0020716042000301725

[17] A. Arikoglu and I. Ozkol, “Solution of Boundary Value Problems for Integro Differential Equation by Using Differential Transform Method,” Applied Mathematics and Computation, Vol. 168, No. 2, 2005, pp. 1145-1158. doi:10.1016/j.amc.2004.10.009

[18] A. Arikoglu and I. Ozkol, “Solution of Fractional Differential Equation by Using Differential Transform,” Chaos Solitons & Fractials, Vol. 34, No. 5, 2006, pp. 1473-1381. doi:10.1016/j.chaos.2006.09.004

[19] I. H. Abdel-Halim Hassan, “Comparison of Differential Transformation Technique with Adomian Decomposition Method for Linear and Nonlinear Initial Value Problems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 36, No. 1, 2008, pp. 53-65.

[20] N. Bildik, A. Konuralp, F. O. Bek and S. Kucukarslan, “Solution of Different Type of the Partial Differential Equations by Differential Transform Method and Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 172, No. 1, 2006, pp. 551-567. doi:10.1016/j.amc.2005.02.037

[21] S. Catal, “Solution of Free Vibration Equations of Beam on Elastic Soil by Using Differential Transform Method,” Applied Mathematical Modelling, Vol. 32, No. 9, 2008, pp. 1744-1757. doi:10.1016/j.apm.2007.06.010

[22] A. Arikoglu and I. Ozkol, “Solution of Difference Equation by Using Differential Transform Method,” Applied Mathematics and Computation, Vol. 174, No. 2, 2006, pp. 1216-1228. doi:10.1016/j.amc.2005.06.013

[23] A. Arikoglu and I. Ozkol, “Solution of Differential-Difference Equation by Using Differential Transform Method,” Applied Mathematics and Computation, Vol. 181, No. 1, 2006, pp. 153-162. doi:10.1016/j.amc.2006.01.022

[24] A. Arikoglu and I. Ozkol, “Comments on Application of Generalized Differential Transform Method to Multi-Order Fractional Differential Equation,” Communication in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 2008, pp. 1337-1340.

[25] Z. Odibat and S. Momani, “A Generalized Differential Transform Method for Linear Partial Differential Equations of Fractional Order,” Applied Mathematics Letters, Vol. 21, No. 2, 2007, pp. 194-199.

[26] V. S. Erturk and S. Momani, “Comparing Numerical Methods for Solving Fourth-Order Boundary Value Problems,” Applied Mathematics and Computation, Vol. 188, No. 2, 2007, pp. 1963-1968. doi:10.1016/j.amc.2006.11.075

[27] M. El-Shahed, “Application of Differential Transform Method to Non-Linear Oscillatory Systems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 2007, pp. 1714-1720.

[28] M. Idrees, S. Islam, R. A. Shah and M. Zeb, “Exact Solution of Goursat Problems Using Differential Transform Method,” Journal of Advanced Research in Scientific Computing, Vol. 3, No. 3, 2011, pp. 1-13.

[29] G. A. F. Ismail and I. H. Ibrahim, “New Efficient Second Derivative Multistep Methods for Stiff Systems,” Applied Mathematical Modelling, Vol. 23, No. 4, 1999, pp. 279-288. doi:10.1016/S0307-904X(98)10086-0

[30] N. Guzel and M. Bayram, “On the Numerical Solution of Stiff Systems,” Applied Mathematics and Computation, Vol. 170, No. 1, 2005, pp. 230-236. doi:10.1016/j.amc.2004.11.035

[1] G. Hojjati, M. Y. Rahimi Arabili and S. M. Hosseini, “An Adaptive Method for Numerical Solution of Stiff System of Ordinary Differential Equations,” Mathematics and Computers in Simulation, Vol. 66, No. 1, 2004, pp. 33-41. doi:10.1016/j.matcom.2004.02.019

[2] E. Harier and G. Wanner, “Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems,” Springer-Verlag, New York, 1996. doi:10.1007/978-3-642-05221-7

[3] L. Lopidus and W. E. Schiesser, “Numerical Methods for Differential Systems,” Academic Press, New York, 1976.

[4] J. Carroll, “A Metrically Exponentially Fitted Scheme for the Numerical Solution of Stiff Initial Value Problems,” Computers & Mathematics with Applications, Vol. 26, 1993, pp. 57-64. doi:10.1016/0898-1221(93)90034-S

[5] C. H. Hsiao, “Haar Wavelet Approach to Linear Stiff Systems,” Mathematics and Computer in Simulation, Vol. 64, No. 1, 2004, pp. 561-567. doi:10.1016/j.matcom.2003.11.011

[6] C. H. Hsiao and W. J. Wang, “Haar Wavelet Approach to Non-Linear Stiff Systems,” Mathematics and Computer in Simulation, Vol. 57, No. 6, 2001, pp. 347-353. doi:10.1016/S0378-4754(01)00275-0

[7] J. K. Zhou, “Differential Transformation and Its Applications for Electrical Circuits,” Huazhong University Press, Wuhan, 1986 (in Chinese).

[8] C. K. Chen and S. H. Ho, “Solving Partial Differential Equations by Two-Dimensional Differential Transform Method,” Applied Mathematics and Computation, Vol. 106, No. 2-3, 1999, pp. 171-179. doi:10.1016/S0096-3003(98)10115-7

[9] M. J. Jang, C. L. Chen and Y. C. Liu, “Two-Dimensional Differential Transform for Partial Differential Equations,” Applied Mathematics and Computation, Vol. 121, No. 2-3, 2001, pp. 261-270. doi:10.1016/S0096-3003(99)00293-3

[10] I. H. Abdel-Halim Hassan, “Different Applications for the Differential Transformation in the Differential Equations,” Applied Mathematics and Computation, Vol. 129, No. 2-3, 2002, pp. 183-201. doi:10.1016/S0096-3003(01)00037-6

[11] F. Ayaz, “On the Two-Dimensional Differential Transform Method,” Applied Mathematics and Computation, Vol. 143, No. 2-3, 2003, pp. 361-374.

[12] M. J. Jang, J. S. Wang and Y. C. Liu, “Applying Differential Transformation Method to Parameter Identification Problems,” Applied Mathematics and Computation, Vol. 139, No. 2, 2003, pp. 491-502. doi:10.1016/S0096-3003(02)00211-4

[13] C. K. Chen and S. P. Ju, “Application of Differential Transformation Method to Transient Adjective-Disper-sive Transport Equations,” Applied Mathematics and Computation, Vol. 155, No. 1, 2004, pp. 25-38. doi:10.1016/S0096-3003(03)00755-0

[14] F. Ayaz, “Application of Differential Transform Method to Differential-Algebraic Equations,” Applied Mathematics and Computation, Vol. 152, No. 3, 2004, pp. 649-657. doi:10.1016/S0096-3003(03)00581-2

[15] F. Ayaz, “Solution of the System of Differential Equations by Differential Transform Method,” Applied Mathematics and Computation, Vol. 147, No. 2, 2004, pp. 547-567. doi:10.1016/S0096-3003(02)00794-4

[16] A. Kurnaz, G. Oturanc and M. E. Kiris, “N-Dimensional Differential Transform Method for Solving PDEs,” International Journal of Computer Mathematics, Vol. 82, No. 3, 2005, pp. 369-380. doi:10.1080/0020716042000301725

[17] A. Arikoglu and I. Ozkol, “Solution of Boundary Value Problems for Integro Differential Equation by Using Differential Transform Method,” Applied Mathematics and Computation, Vol. 168, No. 2, 2005, pp. 1145-1158. doi:10.1016/j.amc.2004.10.009

[18] A. Arikoglu and I. Ozkol, “Solution of Fractional Differential Equation by Using Differential Transform,” Chaos Solitons & Fractials, Vol. 34, No. 5, 2006, pp. 1473-1381. doi:10.1016/j.chaos.2006.09.004

[19] I. H. Abdel-Halim Hassan, “Comparison of Differential Transformation Technique with Adomian Decomposition Method for Linear and Nonlinear Initial Value Problems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 36, No. 1, 2008, pp. 53-65.

[20] N. Bildik, A. Konuralp, F. O. Bek and S. Kucukarslan, “Solution of Different Type of the Partial Differential Equations by Differential Transform Method and Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 172, No. 1, 2006, pp. 551-567. doi:10.1016/j.amc.2005.02.037

[21] S. Catal, “Solution of Free Vibration Equations of Beam on Elastic Soil by Using Differential Transform Method,” Applied Mathematical Modelling, Vol. 32, No. 9, 2008, pp. 1744-1757. doi:10.1016/j.apm.2007.06.010

[22] A. Arikoglu and I. Ozkol, “Solution of Difference Equation by Using Differential Transform Method,” Applied Mathematics and Computation, Vol. 174, No. 2, 2006, pp. 1216-1228. doi:10.1016/j.amc.2005.06.013

[23] A. Arikoglu and I. Ozkol, “Solution of Differential-Difference Equation by Using Differential Transform Method,” Applied Mathematics and Computation, Vol. 181, No. 1, 2006, pp. 153-162. doi:10.1016/j.amc.2006.01.022

[24] A. Arikoglu and I. Ozkol, “Comments on Application of Generalized Differential Transform Method to Multi-Order Fractional Differential Equation,” Communication in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 2008, pp. 1337-1340.

[25] Z. Odibat and S. Momani, “A Generalized Differential Transform Method for Linear Partial Differential Equations of Fractional Order,” Applied Mathematics Letters, Vol. 21, No. 2, 2007, pp. 194-199.

[26] V. S. Erturk and S. Momani, “Comparing Numerical Methods for Solving Fourth-Order Boundary Value Problems,” Applied Mathematics and Computation, Vol. 188, No. 2, 2007, pp. 1963-1968. doi:10.1016/j.amc.2006.11.075

[27] M. El-Shahed, “Application of Differential Transform Method to Non-Linear Oscillatory Systems,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 2007, pp. 1714-1720.

[28] M. Idrees, S. Islam, R. A. Shah and M. Zeb, “Exact Solution of Goursat Problems Using Differential Transform Method,” Journal of Advanced Research in Scientific Computing, Vol. 3, No. 3, 2011, pp. 1-13.

[29] G. A. F. Ismail and I. H. Ibrahim, “New Efficient Second Derivative Multistep Methods for Stiff Systems,” Applied Mathematical Modelling, Vol. 23, No. 4, 1999, pp. 279-288. doi:10.1016/S0307-904X(98)10086-0

[30] N. Guzel and M. Bayram, “On the Numerical Solution of Stiff Systems,” Applied Mathematics and Computation, Vol. 170, No. 1, 2005, pp. 230-236. doi:10.1016/j.amc.2004.11.035