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**Solving linear and nonlinear initial value problems by the projected differential transform method.**
*(English)*
Zbl 1205.65205

Summary: We propose a novel computational algorithm for solving linear and nonlinear initial value problems by using the modified version of differential transform method (DTM), which is called the projected differential transform method (PDTM). The PDTM can be easily applied to the initial value problems with less computational work. For the several illustrative examples, the computational results are compared with those obtained by many other methods; the Adomian decomposition, the variational iteration and the spline method. For all examples, the PDTM provides exact solutions. It has been shown that the PDTM is a reliable algorithm in obtaining analytic as well as approximate solution for the initial value problems.

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |

34A34 | Nonlinear ordinary differential equations and systems |

### Keywords:

Taylor series; initial value problems; differential transform method; numerical examples; algorithm; Adomian decomposition; variational iteration; spline method
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\textit{B. Jang}, Comput. Phys. Commun. 181, No. 5, 848--854 (2010; Zbl 1205.65205)

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### References:

[1] | Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer: Kluwer Boston · Zbl 0802.65122 |

[2] | Ayaz, Fatma, On the two-dimensional differential transform method, Appl. Math. Comput., 143, 2-3, 361-374 (2003) · Zbl 1023.35005 |

[3] | Dehghab, Mehdi; Shokri, Ali, Numerical solution of the nonlinear Klein-Fordon equation using radial basis functions, J. Comput. Appl. Math., 230, 400-410 (2009) · Zbl 1168.65398 |

[4] | Kangalgil, F.; Ayaz, Fatma, Solitary wave solutions for the KdV and mKdV equations by differential transform method, Chaos Solitons Fractals, 1, 464-472 (2009) · Zbl 1198.35222 |

[5] | Chen, Cha’o Kuang; Ho, Shing Huei, Solving partial differential equations by two-dimensional differential transform method, Appl. Math. Comput., 106, 171-179 (1999) · Zbl 1028.35008 |

[6] | Hassan, I. H.A., Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos Solitons Fractals, 36, 53-65 (2008) · Zbl 1152.65474 |

[7] | He, Ji-Huan, Variational iteration method – a kind of non-linear analytical technique: Some examples, Int. J. Nonlinear Mech., 34, 4, 699-708 (1999) · Zbl 1342.34005 |

[8] | He, Ji-Huan, Variational iteration method – some recent results and new interpretations, J. Comput. Appl. Math., 207, 3-17 (2007) · Zbl 1119.65049 |

[9] | Shoua, Da-Hua; He, Ji-Huan, Beyond Adomian method: The variational iteration method for solving heat-like and wave-like equations with variable coefficients, Phys. Lett. A, 372, 3, 233-237 (2008) · Zbl 1217.35091 |

[10] | Jang, M. J.; Chen, C. L.; Liu, Y. C., Two-dimensional differential transform for partial differential equations, Appl. Math. Comput., 121, 261-270 (2001) · Zbl 1024.65093 |

[11] | Jang, Bongsoo, Solutions to the non-homogeneous parabolic problems by the extended HADM, Appl. Math. Comput., 191, 2, 466-483 (2007) · Zbl 1193.65182 |

[12] | Jang, Bongsoo, Two-point boundary value problems by the extended Adomian decomposition method, J. Comput. Appl. Math., 219, 1, 253-262 (2008) · Zbl 1145.65049 |

[13] | Mohanty, R. K., An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients, Appl. Math. Comput., 165, 229-236 (2005) · Zbl 1070.65076 |

[14] | Rashidinia, J.; Mohammadi, R.; Jalilian, R., Spline methods for the solution of hyperbolic equation with variable coefficients, Numer. Methods Partial Differential Equations, 23, 6, 1411-1419 (2007) · Zbl 1131.65078 |

[15] | Soufyane, A.; Boulmalf, M., Solution of linear and nonlinear parabolic equations by the decomposition method, Appl. Math. Comput., 162, 687-693 (2005) · Zbl 1063.65111 |

[16] | Wazwaz, Abdul-Majid; Gorguisb, Alice, Exact solutions for heat-like and wave-like equations with variable coefficients, Appl. Math. Comput., 149, 15-29 (2004) · Zbl 1038.65103 |

[17] | Zhou, J. K., Differential Transformation and its Application for Electrical Circuits (1986), Huazhong University Press: Huazhong University Press China, (in Chinese) |

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