Application of the Hybrid Differential Transform Method to the Nonlinear Equations

Affiliation(s)

Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Samsun, Turkey.

Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Samsun, Turkey.

Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Samsun, Turkey.

Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, Samsun, Turkey.

ABSTRACT

In this paper, a hybrid method is introduced briefly to predict the behavior of the non-linear partial differential equations. The method is hybrid in the sense that different numerical methods, differential transform and finite differences, are used in different subdomains. Our aim of this approach is to combine the flexibility of differential transform and the efficiency of finite differences. An explicit hybrid method for the transient response of inhomogeneous nonlinear partial differential equations is presented; applying finite difference scheme on the fixed grid size is used to approximate the space discretisation, whereas the differential transform method is used for time operator. Comparison of the efficiency of the different approaches is a very important aspect of this study. In our test cases, the hybrid approach is faster than the corresponding highly optimized finite difference method in two dimensional computations. We compared our hybrid approach’s results with the exact and/or numerical solutions of PDE which obtained from Adomian Decomposition Method. Results show that the hybrid approach may be an important tool to reduce the execution time and memory requirements for large scale computations and get remarkable results in predicting the solutions of nonlinear initial value problems.

In this paper, a hybrid method is introduced briefly to predict the behavior of the non-linear partial differential equations. The method is hybrid in the sense that different numerical methods, differential transform and finite differences, are used in different subdomains. Our aim of this approach is to combine the flexibility of differential transform and the efficiency of finite differences. An explicit hybrid method for the transient response of inhomogeneous nonlinear partial differential equations is presented; applying finite difference scheme on the fixed grid size is used to approximate the space discretisation, whereas the differential transform method is used for time operator. Comparison of the efficiency of the different approaches is a very important aspect of this study. In our test cases, the hybrid approach is faster than the corresponding highly optimized finite difference method in two dimensional computations. We compared our hybrid approach’s results with the exact and/or numerical solutions of PDE which obtained from Adomian Decomposition Method. Results show that the hybrid approach may be an important tool to reduce the execution time and memory requirements for large scale computations and get remarkable results in predicting the solutions of nonlinear initial value problems.

Cite this paper

I. Süngü and H. Demir, "Application of the Hybrid Differential Transform Method to the Nonlinear Equations,"*Applied Mathematics*, Vol. 3 No. 3, 2012, pp. 246-250. doi: 10.4236/am.2012.33039.

I. Süngü and H. Demir, "Application of the Hybrid Differential Transform Method to the Nonlinear Equations,"

References

[1] I. H. A.-H. Hassan, “Differential Transform Technique for Solving Higher Order Initial Value Problems,” Applied Mathematics and Computation, Vol. 154, No. 2, 2004, pp. 299-311. doi:10.1016/S0096-3003(03)00708-2

[2] A. Arikoglu and I. Ozkol, “Solution of Difference Equations 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

[3] 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

[4] C. W. Bert and H. Zeng, “Analysis of Axial Vibration of Compound Bars by Differential Transform Method,” Journal of Sound and Vibration, Vol. 275, No. 3-5, 2004, pp. 641-647. doi:10.1016/j.jsv.2003.06.019

[5] C. K. Chen and S. S. Chen, “Application of the Differential Transform Method to a Non-Linear Conservative System,” Applied Mathematics and Computation, Vol. 154, No. 2, 2004, pp. 431-441. doi:10.1016/S0096-3003(03)00723-9

[6] C. L. Chen and Y. C. Liu, “Solutions of Two-BoundaryValue Problems Using the Differential Transform Method”, Journal of Optimization Theory and Application, Vol. 99, No. 1, 1998, pp. 23-35. doi:10.1023/A:1021791909142

[7] H. P. Chu and C. L. Chen, “Hybrid Differential Transform and Finite Difference Method to Solve the Nonlinear Heat Conduction Problem,” Communication in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 2008, pp. 1605-1614. doi:10.1016/j.cnsns.2007.03.002

[8] B. L. Kuo, “Applications of the Differential Transform Method to the Solutions of the Free Convection Problem,” Applied Mathematics and Computation, Vol. 165, No. 1, 2005, pp. 63-79. doi:10.1016/j.amc.2004.04.090

[9] O. U. Richardson, “The Emission of Electricity from Hot Bodies,” Longman, Green and Co., London, 1921.

[10] Y. L. Yeh, C. C. Wang and M. J. Jang, “Using Finite Difference and Differential Transformation Method to Analyze of Large Deflections of Orthotropic Rectangular Plate Problem,” Applied Mathematics and Computation, Vol. 190, No. 2, 2007, pp. 1146-1156. doi:10.1016/j.amc.2007.01.099

[11] J. K. Zhou, “Differential Transformation and its Applications for Electrical Circuits,” Huazhong University Press, Wuhan, 1986.

[12] L. Beilina, “Adaptive Hybrid Finite Element/Difference Method for Maxwell’s Equations: A Priori Error Estimate and Efficiency,” Applied and Computational Mathematics, Vol. 9, No. 2, 2010, pp. 176-197.

[13] A. M. Wazwaz, “Partial Differential Equations Methods and Applications,” Taylor & Francis, London, 2002.

[1] I. H. A.-H. Hassan, “Differential Transform Technique for Solving Higher Order Initial Value Problems,” Applied Mathematics and Computation, Vol. 154, No. 2, 2004, pp. 299-311. doi:10.1016/S0096-3003(03)00708-2

[2] A. Arikoglu and I. Ozkol, “Solution of Difference Equations 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

[3] 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

[4] C. W. Bert and H. Zeng, “Analysis of Axial Vibration of Compound Bars by Differential Transform Method,” Journal of Sound and Vibration, Vol. 275, No. 3-5, 2004, pp. 641-647. doi:10.1016/j.jsv.2003.06.019

[5] C. K. Chen and S. S. Chen, “Application of the Differential Transform Method to a Non-Linear Conservative System,” Applied Mathematics and Computation, Vol. 154, No. 2, 2004, pp. 431-441. doi:10.1016/S0096-3003(03)00723-9

[6] C. L. Chen and Y. C. Liu, “Solutions of Two-BoundaryValue Problems Using the Differential Transform Method”, Journal of Optimization Theory and Application, Vol. 99, No. 1, 1998, pp. 23-35. doi:10.1023/A:1021791909142

[7] H. P. Chu and C. L. Chen, “Hybrid Differential Transform and Finite Difference Method to Solve the Nonlinear Heat Conduction Problem,” Communication in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 2008, pp. 1605-1614. doi:10.1016/j.cnsns.2007.03.002

[8] B. L. Kuo, “Applications of the Differential Transform Method to the Solutions of the Free Convection Problem,” Applied Mathematics and Computation, Vol. 165, No. 1, 2005, pp. 63-79. doi:10.1016/j.amc.2004.04.090

[9] O. U. Richardson, “The Emission of Electricity from Hot Bodies,” Longman, Green and Co., London, 1921.

[10] Y. L. Yeh, C. C. Wang and M. J. Jang, “Using Finite Difference and Differential Transformation Method to Analyze of Large Deflections of Orthotropic Rectangular Plate Problem,” Applied Mathematics and Computation, Vol. 190, No. 2, 2007, pp. 1146-1156. doi:10.1016/j.amc.2007.01.099

[11] J. K. Zhou, “Differential Transformation and its Applications for Electrical Circuits,” Huazhong University Press, Wuhan, 1986.

[12] L. Beilina, “Adaptive Hybrid Finite Element/Difference Method for Maxwell’s Equations: A Priori Error Estimate and Efficiency,” Applied and Computational Mathematics, Vol. 9, No. 2, 2010, pp. 176-197.

[13] A. M. Wazwaz, “Partial Differential Equations Methods and Applications,” Taylor & Francis, London, 2002.