Numerical treatment of a fractional order system of nonlinear stochastic delay differential equations using a computational scheme. (English) Zbl 1485.65012
Summary: In this article, a step-by-step collocation approach based on the shifted Legendre polynomials is presented to solve a fractional order system of nonlinear stochastic differential equations involving a constant delay. The problem is considered with suitable initial condition and the fractional derivative is in the Caputo sense. With a step-by-step process, first, the considered problem is converted into a non-delay fractional order system of nonlinear stochastic differential equations in each step and then, a shifted Legendre collocation scheme is introduced to solve this system. By collocating the obtained residual at the shifted Legendre points, we get a nonlinear system of equations in each step. The convergence analysis and rate of convergence of the proposed method are investigated. Finally, three test examples are provided to affirm the accuracy of this technique in the presence of different noise measures.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 34K50 | Stochastic functional-differential equations |
| 34K37 | Functional-differential equations with fractional derivatives |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
Keywords:
fractional calculus; stochastic delay system; step-by-step method; Legendre collocation scheme; convergence analysisReferences:
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