Long time numerical behaviors of fractional pantograph equations. (English) Zbl 1482.34189
Summary: This paper is concerned with long time numerical behaviors of nonlinear fractional pantograph equations. The L1 method with the linear interpolation procedure is applied to solve these nonlinear problems. It is proved that the proposed numerical scheme can inherit the long time behavior of the underlying problems without any stepsize restrictions. After that, the fast evaluation is presented to speed up the calculation of the Caputo fractional derivative. Numerical examples are shown to confirm the theoretical results. Besides, several counter-examples are also given to show that not all the numerical methods can inherit the long time behavior of the underlying problems.
MSC:
| 34K37 | Functional-differential equations with fractional derivatives |
| 65L03 | Numerical methods for functional-differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L07 | Numerical investigation of stability of solutions to ordinary differential equations |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
fractional delay differential equations; fractional differential equations with proportional delay; long time behavior; stability; fractional pantograph equationsReferences:
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