Computational methods for delay parabolic and time-fractional partial differential equations.

*(English)*Zbl 1204.65114Summary: This article is concerned with \(\vartheta\)-methods for delay parabolic partial differential equations. The methodology is extended to time-fractional-order parabolic partial differential equations in the sense of Caputo. The fully implicit scheme preserves delay-independent asymptotic stability and the solution continuously depends on the time-fractional order. Several numerical examples of interest are included to demonstrate the effectiveness of the method.

Reviewer: Li Changpin (Logan)

##### MSC:

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

35R10 | Functional partial differential equations |

35R11 | Fractional partial differential equations |

35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

method of lines; diffusion equations; Runge-Kutta method; \(\vartheta\)-methods; delay parabolic partial differential equations; time-fractional-order parabolic partial differential equations; asymptotic stability; numerical examples##### Software:

RADAR5
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\textit{F. A. Rihan}, Numer. Methods Partial Differ. Equations 26, No. 6, 1556--1571 (2010; Zbl 1204.65114)

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