Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders
“The theorems above have given us a large amount of information about the smoothness properties of the solutions of fractional differential equations, and in particular about the exact behaviour of the solution as , most notably the formal asymptotic expansion. […] An aspect of special significance, for example in view of the development of numerical methods, is the question for the precise values of the constants in this expansion, and most importantly the question whether certain coefficients vanish. A suitable generalisation of the Taylor expansion technique for ordinary differential equations described in [88, Chapter I.8] could be useful in this context. Precise results in this connection seem to be unknown at the moment though.”
2.1. Problem Statement
2.2. Algorithm Description
2.2.1. Multi-Order System
2.2.2. Implementation of the Differential Transform
- Transform the multi-term equation into a multi-order differential system.
- Describe an equivalent system of Volterra integral equations.
- Find the general form of the asymptotic expansion of the solution.
- Transform the multi-order differential system to a system of recurrence relations (formal application of FDT).
- Choose convenient orders of FDT.
- Describe the relation between different orders of FDT.
4.1. Equivalence and Smoothness Theorems
4.2. Fractional Differential Transformation
Conflicts of Interest
|IVP||Initial Value Problem|
|FDT||Fractional Differential Transformation|
|IFDT||Inverse Fractional Differential Transformation|
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Rebenda, J. Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders. Symmetry 2019, 11, 1390. https://doi.org/10.3390/sym11111390
Rebenda J. Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders. Symmetry. 2019; 11(11):1390. https://doi.org/10.3390/sym11111390Chicago/Turabian Style
Rebenda, Josef. 2019. "Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders" Symmetry 11, no. 11: 1390. https://doi.org/10.3390/sym11111390