Coupling of ¬¬Laplace Differential Transform method with Padé Approximant for the Numerical solution of Initial and Boundary value problems

Liberty Ebiwareme

doi:10.18535/ijsrm/v10i3.m01

Abstract

This paper presents a study using a novel linearization technique based on the Differential transformation method (DTM) to seek analytical solutions if it exists and approximate solutions where closed form solutions are not available. The effectiveness and accuracy of this procedure is verified by solving six problems comprising both initial and boundary value problems by a combination of DTM and Laplace transform method. The resulting solution is then treated with Padé approximation to obtain a better approximation that converges to the exact solution. Simulated results of the study reveal the proposed technique is reliable, accurate and computationally convenient even with few iterations. The result obtained is in good agreement with existing literature.

Keywords

Differential Transform method (DTM)Laplace Transform method (LTM)Padé

References

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Seyed-Habibollah, H.K., Ganji, D. D. Dynamics and Vibrations; progress in Nonlinear Analysis. Solid Mechanics and its Applications, Volume 2, Springer, New York. 2014.Google Scholar ↗
Ganji, D.D. Numerical and Analytical solutions for solving Nonlinear Equations in Heat Transfer. Advances in Mechatronics and mechanical Engineering, IGI-Global, US. 2018Google Scholar ↗
Abdelhafez, H. Solution of excited non-linear oscillators under damping effects using the modified differential transform method. Mathematics, 4, Volume 11, (2016), 1712-1725.Google Scholar ↗
Alquran, M., Al-khaled, K., Ali, M., Ta‘any, A. The combined Laplace transform-differential transform method for solving linear nonhomogeneous PDEs. Journal of Mathematics and Computer Science, 2, Volume 3, (2012), 690-701.Google Scholar ↗
Ayaz, F. Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computations, 147, (2004), 547-567.Google Scholar ↗
Ogunrinde, R.B. Comparative Study of Differential Transformation Method (DTM) and Adomian Decomposition Method (ADM) for Solving Ordinary Differential Equations. Journal of Contemporary Applied Mathematics, Vol. 9, No. 1, (2019), ISSN 2222-5498.Google Scholar ↗
Chen, C., Ho, S. (1996). Application of differential transformation to eigenvalueGoogle Scholar ↗
problems. Applied Mathematics and Computations, 79, (1996), 173-188.Google Scholar ↗
CÎrnu, M., Frumosu, F. Initial valve problems for nonlinear differential equations solved by differential transform method. Journal of Information Systems and Operations Management, 3, Volume 1, (2009), 102-107.Google Scholar ↗
Darania, P., Shali, J. A., Ivaz, K. New computational method for solvingGoogle Scholar ↗
some 2-dimensional nonlinear Volterra integro-differential equations. Numerical Algorithms, 57, Volume 1, (2011), 125-147.Google Scholar ↗
Demir, H., Süngü, Ï. Numerical solution of a class of nonlinear Emden-Fowler equations by differential transform method. Journal of Arts and Sciences, 2 (2009), 239-249.Google Scholar ↗
Haghbin, A., Hesam, S. Reduced differential transform method for solving seventh order Sawada-Kotera equations. The Journal of Mathematics and Computer Science, 5, Volume 1 (2012), 53-59.Google Scholar ↗
Hassan, I. On solving some eigenvalue problems by using differential transformation. Applied Mathematics and Computations, 127, Volume 1, (2002), 22-30.Google Scholar ↗
Hassan, I. Differential transformation technique for solving higher-order initial value problems. Applied Mathematics and Computations, 154, (2004), 299-311.Google Scholar ↗
Hassan, I. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos Solutions and Fractals. Pergamon-Elsevier Science Ltd, 36, (2008), 53-65.Google Scholar ↗
Hassan, I. Solutions of different types of the linear and nonlinear higher order boundary value problems by differential transformation method. European Journal of Pure and Applied Mathematics, 2, Volume 3, (2009), 426-447.Google Scholar ↗
Hassan, I. Application to differential transformation method for solving systems of differential equations. Applied Mathematical Modelling, 32, (2008), 2552-2559.Google Scholar ↗
Haziqah, C., Hussin, C., Kiliçman, A. On the solutions of nonlinear higher-order boundary value problems by using differential transformation method and Adomian decomposition method. Hindawi Publishing Corporation. 2011Google Scholar ↗
Hussin, C., Kiliçman, A., Mandangan, A. General differential transformation method for higher order of linear boundary value problem. Borneo Science, 27,(2010),35-46.Google Scholar ↗
Iftikhar, M., Rehman, H., Younis, M. Solution of thirteen order boundary value problems by differential transformation method. Asian Journal of Mathematics and Applications, 2014.Google Scholar ↗
Jang, M., Chen, C., Liy, Y. On solving the initial-value problems using the differential transformation method. Applied Mathematics and Computations, 115(2000), 145-160.Google Scholar ↗
Karakoç, F., Bereketoglu, H. Solutions of delay differential equations by using differential transform method. International Journal of Computer Mathematics, 86, Volume 5, (2010), 914-923.Google Scholar ↗
Keskin, Y., Oturanc, G. Reduced differential transform method for solving linear and nonlinear wave equations. Iranian Journal of Science and Technology, 34, Volume 2, (2010).Google Scholar ↗
Khambayat, A., Patil, N. The numerical solution of differential transform method and the Laplace transform method for second order differential equations. International Journal of Mathematics and Computer Research, 3, Volume 2, (2015), 871-875.Google Scholar ↗
Kumari, K., Gupta, P., Shanker, G. Coupling of Laplace transform and differential transform for wave equation. Physical Science International Journal, 9, Volume 4, (2016), 1-10.Google Scholar ↗
Mirzaee, F. Differential transform method for solving linear and nonlinear systems of ordinary differential equations. Applied Mathematical Sciences, 5(7), 3465-3472Google Scholar ↗
Momani, S., Ertürk, V. Solution of nonlinear oscillator by the modified differential transform method. Computer and Mathematics with Applications, 55, (2008), 833-842.Google Scholar ↗
Moon, S., Bhosale, A., Gajbhiye, P., Lonare, G. Solution of non-linear equations by using differential transform method. International Journal of Mathematics and Statistics Invention, 2, Volume 3, (2004), 78-82.Google Scholar ↗
Nourazar, S., Mirzabeigy, A. Approximate solution for nonlinear duffing oscillator with damping effect using the modified differential transform method Scientia Iranica, 20, (2011), 364-368.Google Scholar ↗
Othman, M., Mahdy, A. Differential transformation method and variation iteration method for Cauchy reaction-diffusion problems. Journal of Mathematics and Computer Science, 1, Volume 2, (2010), 61-75.Google Scholar ↗
Patil, N., Khambayat, A. Differential transform method for system of linear differential equations. Research Journal of Mathematical and Statistical Sciences, 2, Volume 3, (2014), 4-6.Google Scholar ↗
Soltanalizadeh, B. (2011). Differential transformation method for solving one-space dimensional telegraph equation. Computational and Applied Mathematics, 30(3), 639-653.Google Scholar ↗
Kajani, M., Shehni, N. Differential transform method: an effective tool for solving nonlinear Volterra integro-differential equations. Australian Journal of Basic and Applied Sciences, 5, Volume 9, (2011), 30-39.Google Scholar ↗
Lin, Y., Tang, H., Chen, C. Modified differential transform method for two singular boundary value problems. Journal of Applied Mathematics, ID 138087. 2014Google Scholar ↗
Xie, L., Zhou, C., Xu, S. An effective numerical method to solve a class of nonlinear singular boundary value problem using improved differential transform method. Journal of Numerical Analysis, 3, Volume 4, (2016), 201-215.Google Scholar ↗
Ebiwareme, L. Application of Semi-Analytical Iteration Techniques for the Numerical Solution of Linear and Nonlinear Differential Equations. International Journal of Mathematics Trends and Technology, Volume 67, Issue 2, (2021), 146-158.Google Scholar ↗
Abdel-Halim Hassan, I.H. Application to differential transformation method for solving systems of differential equations. Applied Mathematical Modelling, 32, (2008), 2552-2559.Google Scholar ↗

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[refR-2] Seyed-Habibollah, H.K., Ganji, D. D. Dynamics and Vibrations; progress in Nonlinear Analysis. Solid Mechanics and its Applications, Volume 2, Springer, New York. 2014.Google Scholar ↗

[refR-3] Ganji, D.D. Numerical and Analytical solutions for solving Nonlinear Equations in Heat Transfer. Advances in Mechatronics and mechanical Engineering, IGI-Global, US. 2018Google Scholar ↗

[refR-4] Abdelhafez, H. Solution of excited non-linear oscillators under damping effects using the modified differential transform method. Mathematics, 4, Volume 11, (2016), 1712-1725.Google Scholar ↗

[refR-5] Alquran, M., Al-khaled, K., Ali, M., Ta‘any, A. The combined Laplace transform-differential transform method for solving linear nonhomogeneous PDEs. Journal of Mathematics and Computer Science, 2, Volume 3, (2012), 690-701.Google Scholar ↗

[refR-6] Ayaz, F. Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computations, 147, (2004), 547-567.Google Scholar ↗

[refR-7] Ogunrinde, R.B. Comparative Study of Differential Transformation Method (DTM) and Adomian Decomposition Method (ADM) for Solving Ordinary Differential Equations. Journal of Contemporary Applied Mathematics, Vol. 9, No. 1, (2019), ISSN 2222-5498.Google Scholar ↗

[refR-8] Chen, C., Ho, S. (1996). Application of differential transformation to eigenvalueGoogle Scholar ↗

[refR-9] problems. Applied Mathematics and Computations, 79, (1996), 173-188.Google Scholar ↗

[refR-10] CÎrnu, M., Frumosu, F. Initial valve problems for nonlinear differential equations solved by differential transform method. Journal of Information Systems and Operations Management, 3, Volume 1, (2009), 102-107.Google Scholar ↗

[refR-11] Darania, P., Shali, J. A., Ivaz, K. New computational method for solvingGoogle Scholar ↗

[refR-12] some 2-dimensional nonlinear Volterra integro-differential equations. Numerical Algorithms, 57, Volume 1, (2011), 125-147.Google Scholar ↗

[refR-13] Demir, H., Süngü, Ï. Numerical solution of a class of nonlinear Emden-Fowler equations by differential transform method. Journal of Arts and Sciences, 2 (2009), 239-249.Google Scholar ↗

[refR-14] Haghbin, A., Hesam, S. Reduced differential transform method for solving seventh order Sawada-Kotera equations. The Journal of Mathematics and Computer Science, 5, Volume 1 (2012), 53-59.Google Scholar ↗

[refR-15] Hassan, I. On solving some eigenvalue problems by using differential transformation. Applied Mathematics and Computations, 127, Volume 1, (2002), 22-30.Google Scholar ↗

[refR-16] Hassan, I. Differential transformation technique for solving higher-order initial value problems. Applied Mathematics and Computations, 154, (2004), 299-311.Google Scholar ↗

[refR-17] Hassan, I. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos Solutions and Fractals. Pergamon-Elsevier Science Ltd, 36, (2008), 53-65.Google Scholar ↗

[refR-18] Hassan, I. Solutions of different types of the linear and nonlinear higher order boundary value problems by differential transformation method. European Journal of Pure and Applied Mathematics, 2, Volume 3, (2009), 426-447.Google Scholar ↗

[refR-19] Hassan, I. Application to differential transformation method for solving systems of differential equations. Applied Mathematical Modelling, 32, (2008), 2552-2559.Google Scholar ↗

[refR-20] Haziqah, C., Hussin, C., Kiliçman, A. On the solutions of nonlinear higher-order boundary value problems by using differential transformation method and Adomian decomposition method. Hindawi Publishing Corporation. 2011Google Scholar ↗

[refR-21] Hussin, C., Kiliçman, A., Mandangan, A. General differential transformation method for higher order of linear boundary value problem. Borneo Science, 27,(2010),35-46.Google Scholar ↗

[refR-22] Iftikhar, M., Rehman, H., Younis, M. Solution of thirteen order boundary value problems by differential transformation method. Asian Journal of Mathematics and Applications, 2014.Google Scholar ↗

[refR-23] Jang, M., Chen, C., Liy, Y. On solving the initial-value problems using the differential transformation method. Applied Mathematics and Computations, 115(2000), 145-160.Google Scholar ↗

[refR-24] Karakoç, F., Bereketoglu, H. Solutions of delay differential equations by using differential transform method. International Journal of Computer Mathematics, 86, Volume 5, (2010), 914-923.Google Scholar ↗

[refR-25] Keskin, Y., Oturanc, G. Reduced differential transform method for solving linear and nonlinear wave equations. Iranian Journal of Science and Technology, 34, Volume 2, (2010).Google Scholar ↗

[refR-26] Khambayat, A., Patil, N. The numerical solution of differential transform method and the Laplace transform method for second order differential equations. International Journal of Mathematics and Computer Research, 3, Volume 2, (2015), 871-875.Google Scholar ↗

[refR-27] Kumari, K., Gupta, P., Shanker, G. Coupling of Laplace transform and differential transform for wave equation. Physical Science International Journal, 9, Volume 4, (2016), 1-10.Google Scholar ↗

[refR-28] Mirzaee, F. Differential transform method for solving linear and nonlinear systems of ordinary differential equations. Applied Mathematical Sciences, 5(7), 3465-3472Google Scholar ↗

[refR-29] Momani, S., Ertürk, V. Solution of nonlinear oscillator by the modified differential transform method. Computer and Mathematics with Applications, 55, (2008), 833-842.Google Scholar ↗

[refR-30] Moon, S., Bhosale, A., Gajbhiye, P., Lonare, G. Solution of non-linear equations by using differential transform method. International Journal of Mathematics and Statistics Invention, 2, Volume 3, (2004), 78-82.Google Scholar ↗

[refR-31] Nourazar, S., Mirzabeigy, A. Approximate solution for nonlinear duffing oscillator with damping effect using the modified differential transform method Scientia Iranica, 20, (2011), 364-368.Google Scholar ↗

[refR-32] Othman, M., Mahdy, A. Differential transformation method and variation iteration method for Cauchy reaction-diffusion problems. Journal of Mathematics and Computer Science, 1, Volume 2, (2010), 61-75.Google Scholar ↗

[refR-33] Patil, N., Khambayat, A. Differential transform method for system of linear differential equations. Research Journal of Mathematical and Statistical Sciences, 2, Volume 3, (2014), 4-6.Google Scholar ↗

[refR-34] Soltanalizadeh, B. (2011). Differential transformation method for solving one-space dimensional telegraph equation. Computational and Applied Mathematics, 30(3), 639-653.Google Scholar ↗

[refR-35] Kajani, M., Shehni, N. Differential transform method: an effective tool for solving nonlinear Volterra integro-differential equations. Australian Journal of Basic and Applied Sciences, 5, Volume 9, (2011), 30-39.Google Scholar ↗

[refR-36] Lin, Y., Tang, H., Chen, C. Modified differential transform method for two singular boundary value problems. Journal of Applied Mathematics, ID 138087. 2014Google Scholar ↗

[refR-37] Xie, L., Zhou, C., Xu, S. An effective numerical method to solve a class of nonlinear singular boundary value problem using improved differential transform method. Journal of Numerical Analysis, 3, Volume 4, (2016), 201-215.Google Scholar ↗

[refR-38] Ebiwareme, L. Application of Semi-Analytical Iteration Techniques for the Numerical Solution of Linear and Nonlinear Differential Equations. International Journal of Mathematics Trends and Technology, Volume 67, Issue 2, (2021), 146-158.Google Scholar ↗

[refR-39] Abdel-Halim Hassan, I.H. Application to differential transformation method for solving systems of differential equations. Applied Mathematical Modelling, 32, (2008), 2552-2559.Google Scholar ↗