A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations
© 2017 Federacion Argentina de Cardiologia. All right reserved. In this paper we propose an e?cient and easy-to-implement nu-merical method for an a-th order Ordinary Differential Equation (ODE) when a ? (0, 1), based on a one-point quadrature rule. The quadrature point in each sub-interval of a giv...
| Main Authors: | , , |
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| Format: | Journal Article |
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American Institute of Mathematical Sciences
2017
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| Online Access: | http://hdl.handle.net/20.500.11937/56012 |
| Summary: | © 2017 Federacion Argentina de Cardiologia. All right reserved. In this paper we propose an e?cient and easy-to-implement nu-merical method for an a-th order Ordinary Differential Equation (ODE) when a ? (0, 1), based on a one-point quadrature rule. The quadrature point in each sub-interval of a given partition with mesh size h is chosen judiciously so that the degree of accuracy of the quadrature rule is 2 in the presence of the singu-lar integral kernel. The resulting time-stepping method can be regarded as the counterpart for fractional ODEs of the well-known mid-point method for 1st-order ODEs. We show that the global error in a numerical solution generated by this method is of the order O(h 2 ), independently of a. Numerical results are presented to demonstrate that the computed rates of convergence match the theoretical one very well and that our method is much more accurate than a well-known one-step method when a is small. |
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