Fractional partial differential equations are generalisations of classical partial differential equations, which relax the requirement that the derivatives are of integer order. A computational cost inherent to fractional derivatives is their non-local nature, and so they naturally benefit from the global approximation functions that characterise spectral methods. Using Jacobi polynomials integrated under Gaus-sian quadrature, several spectral collocation schemes are developed and tested on a

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... riety of partial differential equations, fractional in both the time and space dimensions. The methods are tested on problems of varying degree, dimension, and linearity, as well as problems with derivative boundary conditions. Numerical results are compared to those obtained with both similar and dissimilar methods investigated in prior literature, where it is found that the methods implemented in this project compare generally favourably, and occasionally present the best known approximation. Declaration