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Abstract: The Maxey-Riley equations (MRE) describe the motion of a finite-sized, spherical particle in a fluid. Because of wake effects, the force acting on a particle depends on its past trajectory. This is modelled by an integral term in the MRE, also called Basset force, that makes its numerical solution challenging and memory intensive. A recent approach proposed by Prasath, Vasan and Govindarajan exploits connections between the integral term and fractional derivatives to reformulate the MRE as a time-dependent partial differential equation on a semi-infinite pseudo-space. They also propose a numerical algorithm based on polynomial expansions. This paper develops a numerical approach based on finite difference instead, by adopting techniques by Koleva and Fazio and Janelli to cope with the issues of having an unbounded spatial domain. We compare convergence order and computational efficiency for particles of varying size and density of the polynomial expansion by Prasath et al., our finite difference schemes and a direct integrator for the MRE based on multi-step methods proposed by Daitche.