A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor’s formula. (English) Zbl 1148.65099

Momani, Shaher; Odibat, Zaid

J. Comput. Appl. Math. 220, No. 1-2, 85-95 (2008).

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.

Reviewer: Kai Diethelm (Braunschweig)

MSC:

65R20	Numerical methods for integral equations
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
45K05	Integro-partial differential equations
45G10	Other nonlinear integral equations
65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs

Keywords:

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

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