Research Article
Applications of differential transformation method to solve systems of ordinary and partial differential equations
Abstract
In this study, the numerical solutions of some systems of ordinary and partial differential equations have been analyzed by using the Differential Transformation Method (DTM) and compared with solutions of other numerical methods .This method can be used to solve some ordinary and partial differential equations in a short time by using very simple computer commands and codes. First chapter, the basic definition of its Differential Transformation Method and properties are given. In the last chapter, examples are solved by using the differential transformation method and are compared with other solutions of numerical methods.
References
- Zhou, J.K., Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, in Chinese, (1986).
- Chen, C.K. and Ho, S.H., Application of Differential Transformation to Eigenvalue Problems, Applied Mathematics and Computation, 173-188, (1996).
- Chen, C.K. and Ho, S.H., Solving Partial Differential Equations by Two Variable Differential Transform, Applied Mathematics and Computation, 106: 171-179, (1999).
- Chen, C.L. and Liu, Y.C., Differential Transformation Technique for Steady Nonlinear Heat Conduction Problems, Applied Mathematics and Computation, 95: 155-164, (1998).
- Ayaz, F., On the Two-Dimensional Differential Transform Method, Applied Mathematics and Computation, 143(2-3): 361-374, (2003).
- Ayaz, F., Applications of Differential Transform Method to Differential-algebraic Equations, Applied Mathematics and Computation, 152(3): 649-657, (2004).
- Ayaz, F., Solutions of The System of Differential Equations by Differential Transform Method, Applied Mathematics and Computation, 147(2): 547-567, (2004).
- Kangalgil, F. and Ayaz F., Solitary Wave Solutions for the Kdv and Mkdv Equations by Differential Transform Method, Chaos, Solitons and Fractals, 41(1): 464-472, (2009).
- Arikoglu, A. and Ozkol, I., Solution of Boundary Value Problems for Integro Differential Equations by Using Differential Transform Method, Applied Mathematics and Computation, 168(2): 1145-1158, (2005).
- Arikoglu A. and Ozkol, I., Solution of Differential-Difference Equations by Using Diffrential Transform Method, Applied Mathematics and Computation, 181(1): 153-162, (2006).
- Arikoglu, A. and Ozkol, I., Solution of Fractional Differential Equations by Using Differential Transform Method, Chaos, Solitons and Fractals, 34(5): 1473-1481, (2007).
- Arikoglu, A. and Ozkol, I., Solution of Fractional Integro-Differential Equations by Using Fractional Differential Transform Method, Chaos, Solitons and Fractals, 40(2): 521-529, (2009).
- Özdemir, N., and Yavuz, M., Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation. Acta Physica Polonica A, 132(3), 1050-1053, (2017). DOI: https://doi.org/10.12693/APhysPolA.132.1050
- Yavuz, M., Novel solution methods for initial boundary value problems of fractional order with conformable differentiation. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 1-7, (2018). DOI: https://doi.org/10.11121/ijocta.01.2018.00540
- Yavuz, M., and Yaşkıran, B. Approximate-Analytical Solutions of Cable Equation Using Conformable Fractional Operator. New Trends in Mathematical Sciences (NTMSCI), 5(4), 209-219, (2017). DOI: https://doi.org/10.20852/ntmsci.2017.232
- Yavuz, M., and Özdemir, N., European Vanilla Option Pricing Model of Fractional Order without Singular Kernel. Fractal and Fractional, 2(3), 1-11, (2018). DOI: https://doi.org/10.3390/fractalfract2010003
- Yavuz, M., and Özdemir, N., A Different Approach to the European Option Pricing Model with New Fractional Operator. Mathematical Modelling of Natural Phenomena, (2018). DOI: https://doi.org/10.1051/mmnp/2018009
- Wazwaz, A.M., The Variational Iteration Method for Solving Linear and Nonlinear Systems of PDEs, Computers and Mathematics with Applications, 54(7-8): 895-902, (2007).
- Khan, M., Hussain, M. and Jafari H., Application of Laplace Decomposition Method to Solve Nonlinear Coupled Partial Differential Equations, World Applied Sciences Journal, 9: 13-19, (2010).
- Wazwaz, A.M., Partial Differential Equations and Solitary Waves Theory, Higher Education Press, (2009).
- Abazari, R. Solution of Riccati Types Matrix Differential Equations Using Matrix Differential Transform Method, Journal Ofapplied Mathematics and Informatics, 27: 1133-1143, (2009).
- Al-Sawalha, M.M. and Noorani, M.S.M. Application of the Differential Transformation Method for the Solution of the Hyperchaotic Rossler System, Communications in Nonlinear Science and Numerical Simulation, 14(4): 1509-1514, (2009). Al-Sawalha, M.M. and Noorani, M.S.M. A Numeric-analytic Method for Approximating the Chaotic Chen System, Chaos, Solitons and Fractals, 42(3): 1784-1791, (2009).
- Borhanifar, A. and Abazari, R., Exact Solutions for Non-linear Schrödinger Equations by Differential Transformation Method, Journal of Applied Mathematics and Computing, 35(1-2): 37-51, (2009).
- Abdel-Halim Hassan, I.H., Differential Transformation Technique for Solving Higher-Order Initial Value Problems, Applied Mathematics and Computation, 154(2): 299-311, (2004).
- Hesam, S., Nazemi, A.R. and Haghbin, A., Analytical Solution for The Fokker-Planck Equation by Differential Transform Method, Scientia Iranica, 19(4): 1140-1145, (2012).
- Jang, M.J. and Chen, C.L., Analysis of The Response of A Strongly Nonlinear Damped System Using A Differential Transformation Technique, Applied Mathematics and Computation, 88: 137-151, (1997).
- Kurnaz, A. and Oturanc G., the Differential Transform Approximation for the System of Ordinary Differential Equations, International Journal of Computer Mathematics, 82(6): 709-719, (2005).
- Odibat, Z.M., Differential Transform Method for Solving Volterra Integral Equation With Separable Kernels, Mathematical and Computer Modelling, 48(7-8): 1144-1149, (2008).
- Jang, M.J., Chen, C.L. and Liu, Y.C., Two-variable Differential Transform for Partial Differential Equations, Applied Mathematics and Computation, 121: 261-270, (2001).
- Kurnaz, A., Oturanc, G. and Kiris E.M., n-variable Differential Transformation Method for Solving PDE, International Journal of Computer Mathematics, 3(82): 369-380, (2005).
- Berwal, N., Panchal, D., and Parihar, C.L., Solving System of Linear Differential Equations Using Haar Wavelet, Applied Mathematics and Computational Intelligence, 2(2): 183-193, (2013).
- Gustafson, G. B., De and Linear Algebra Manuscripts, Engineering Math Spring, Mathematics Department University Of Utah, (2006).
- Olayiwola, M.O., Variational Iteration Method: A Computational Tool for Solving Coupled System of Nonlinear Partial Differential Equations, Journal of Science and Arts, 3(36): 243-248, (2016).
Adi ve kısmi diferansiyel denklem sistemlerinin çözümü için diferansiyel dönüşüm yönteminin uygulamaları
Abstract
Bu çalışmada Diferansiyel Dönüşüm Yöntemi ile bazı adi ve kısmi diferansiyel denklem sitemleri incelenmiş ve diğer çözüm yöntemleri ile yapılmış sonuçlarla karşılaştırılmıştır. Basit bilgisayar kodları ile bu yöntem birçok adi ve kısmi diferansiyel denklem için kısa sürede çözüme ulaşabilen bir yöntemdir. Giriş kısmında Diferansiyel Dönüşüm Yöntemi ile ilgili temel tanım ve ifadeler verilmiş olup son başlıkta uygulamalara yer verilmiştir.
References
- Zhou, J.K., Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, in Chinese, (1986).
- Chen, C.K. and Ho, S.H., Application of Differential Transformation to Eigenvalue Problems, Applied Mathematics and Computation, 173-188, (1996).
- Chen, C.K. and Ho, S.H., Solving Partial Differential Equations by Two Variable Differential Transform, Applied Mathematics and Computation, 106: 171-179, (1999).
- Chen, C.L. and Liu, Y.C., Differential Transformation Technique for Steady Nonlinear Heat Conduction Problems, Applied Mathematics and Computation, 95: 155-164, (1998).
- Ayaz, F., On the Two-Dimensional Differential Transform Method, Applied Mathematics and Computation, 143(2-3): 361-374, (2003).
- Ayaz, F., Applications of Differential Transform Method to Differential-algebraic Equations, Applied Mathematics and Computation, 152(3): 649-657, (2004).
- Ayaz, F., Solutions of The System of Differential Equations by Differential Transform Method, Applied Mathematics and Computation, 147(2): 547-567, (2004).
- Kangalgil, F. and Ayaz F., Solitary Wave Solutions for the Kdv and Mkdv Equations by Differential Transform Method, Chaos, Solitons and Fractals, 41(1): 464-472, (2009).
- Arikoglu, A. and Ozkol, I., Solution of Boundary Value Problems for Integro Differential Equations by Using Differential Transform Method, Applied Mathematics and Computation, 168(2): 1145-1158, (2005).
- Arikoglu A. and Ozkol, I., Solution of Differential-Difference Equations by Using Diffrential Transform Method, Applied Mathematics and Computation, 181(1): 153-162, (2006).
- Arikoglu, A. and Ozkol, I., Solution of Fractional Differential Equations by Using Differential Transform Method, Chaos, Solitons and Fractals, 34(5): 1473-1481, (2007).
- Arikoglu, A. and Ozkol, I., Solution of Fractional Integro-Differential Equations by Using Fractional Differential Transform Method, Chaos, Solitons and Fractals, 40(2): 521-529, (2009).
- Özdemir, N., and Yavuz, M., Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation. Acta Physica Polonica A, 132(3), 1050-1053, (2017). DOI: https://doi.org/10.12693/APhysPolA.132.1050
- Yavuz, M., Novel solution methods for initial boundary value problems of fractional order with conformable differentiation. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 1-7, (2018). DOI: https://doi.org/10.11121/ijocta.01.2018.00540
- Yavuz, M., and Yaşkıran, B. Approximate-Analytical Solutions of Cable Equation Using Conformable Fractional Operator. New Trends in Mathematical Sciences (NTMSCI), 5(4), 209-219, (2017). DOI: https://doi.org/10.20852/ntmsci.2017.232
- Yavuz, M., and Özdemir, N., European Vanilla Option Pricing Model of Fractional Order without Singular Kernel. Fractal and Fractional, 2(3), 1-11, (2018). DOI: https://doi.org/10.3390/fractalfract2010003
- Yavuz, M., and Özdemir, N., A Different Approach to the European Option Pricing Model with New Fractional Operator. Mathematical Modelling of Natural Phenomena, (2018). DOI: https://doi.org/10.1051/mmnp/2018009
- Wazwaz, A.M., The Variational Iteration Method for Solving Linear and Nonlinear Systems of PDEs, Computers and Mathematics with Applications, 54(7-8): 895-902, (2007).
- Khan, M., Hussain, M. and Jafari H., Application of Laplace Decomposition Method to Solve Nonlinear Coupled Partial Differential Equations, World Applied Sciences Journal, 9: 13-19, (2010).
- Wazwaz, A.M., Partial Differential Equations and Solitary Waves Theory, Higher Education Press, (2009).
- Abazari, R. Solution of Riccati Types Matrix Differential Equations Using Matrix Differential Transform Method, Journal Ofapplied Mathematics and Informatics, 27: 1133-1143, (2009).
- Al-Sawalha, M.M. and Noorani, M.S.M. Application of the Differential Transformation Method for the Solution of the Hyperchaotic Rossler System, Communications in Nonlinear Science and Numerical Simulation, 14(4): 1509-1514, (2009). Al-Sawalha, M.M. and Noorani, M.S.M. A Numeric-analytic Method for Approximating the Chaotic Chen System, Chaos, Solitons and Fractals, 42(3): 1784-1791, (2009).
- Borhanifar, A. and Abazari, R., Exact Solutions for Non-linear Schrödinger Equations by Differential Transformation Method, Journal of Applied Mathematics and Computing, 35(1-2): 37-51, (2009).
- Abdel-Halim Hassan, I.H., Differential Transformation Technique for Solving Higher-Order Initial Value Problems, Applied Mathematics and Computation, 154(2): 299-311, (2004).
- Hesam, S., Nazemi, A.R. and Haghbin, A., Analytical Solution for The Fokker-Planck Equation by Differential Transform Method, Scientia Iranica, 19(4): 1140-1145, (2012).
- Jang, M.J. and Chen, C.L., Analysis of The Response of A Strongly Nonlinear Damped System Using A Differential Transformation Technique, Applied Mathematics and Computation, 88: 137-151, (1997).
- Kurnaz, A. and Oturanc G., the Differential Transform Approximation for the System of Ordinary Differential Equations, International Journal of Computer Mathematics, 82(6): 709-719, (2005).
- Odibat, Z.M., Differential Transform Method for Solving Volterra Integral Equation With Separable Kernels, Mathematical and Computer Modelling, 48(7-8): 1144-1149, (2008).
- Jang, M.J., Chen, C.L. and Liu, Y.C., Two-variable Differential Transform for Partial Differential Equations, Applied Mathematics and Computation, 121: 261-270, (2001).
- Kurnaz, A., Oturanc, G. and Kiris E.M., n-variable Differential Transformation Method for Solving PDE, International Journal of Computer Mathematics, 3(82): 369-380, (2005).
- Berwal, N., Panchal, D., and Parihar, C.L., Solving System of Linear Differential Equations Using Haar Wavelet, Applied Mathematics and Computational Intelligence, 2(2): 183-193, (2013).
- Gustafson, G. B., De and Linear Algebra Manuscripts, Engineering Math Spring, Mathematics Department University Of Utah, (2006).
- Olayiwola, M.O., Variational Iteration Method: A Computational Tool for Solving Coupled System of Nonlinear Partial Differential Equations, Journal of Science and Arts, 3(36): 243-248, (2016).
There are 33 citations in total.
Details
| Primary Language | English |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Publication Date | December 1, 2018 |
| Submission Date | February 5, 2018 |
| Published in Issue | Year 2018 Volume: 20 Issue: 2 |