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Zbl 1148.65099
Momani, Shaher; Odibat, Zaid
A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's formula. (English)
[J] J. Comput. Appl. Math. 220, No. 1-2, 85-95 (2008). ISSN 0377-0427

For the approximate solution of fractional partial differential equations, the authors suggest to use a method based on a (generalized) Taylor expansion of the solution. As in the classical case one can then try to compute the coefficients of the expansion, and hence the exact solution, by suitable recurrence relations based on the differential equation. This path seems to be viable only in the case of a sufficiently simple equation. In particular, the method requires that the solution can be expanded in a series of a special form, and in a typical application it is by no means clear whether such an expansion is possible. Moreover it seems (see, e.g., Example 5.2) that the computed approximate solution does not depend on any boundary conditions. Obviously the exact solution does depend on the boundary conditions, and this discrepancy raises strong concerns about the correctness of the approach.
[Kai Diethelm (Braunschweig)]

MSC 2000:: *65R20 Integral equations (numerical methods)
65M12 Stability and convergence of numerical methods (IVP of PDE)
45K05 Integro-partial differential equations
45G10 Nonsingular nonlinear integral equations
65M70 Spectral, collocation and related methods (IVP of PDE)

Keywords: fractional partial differential equation; differential transform method (DTM); Caputo derivative; generalized Taylor formula

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