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This study aims to use new formula derived based on the shifted Jacobi functions have been defined and some theorems of the left- and right-sided fractional derivative for them have been presented.

In this article, the authors apply the method of lines (MOL) together with the pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative (SFPDEs). Then, using the collocation nodes to reduce the SFPDEs to the system of ordinary differential equations, which can be solved by the ode45 MATLAB toolbox.

Applying the MOL method together with the pseudospectral discretization method converts the space-dependent on fractional partial differential equations to the system of ordinary differential equations.

This paper contributes to gain choosing the shifted Jacobi functions basis with special parameters a, b and give the authors this opportunity to obtain the left- and right-sided fractional differentiation matrices for this basis exactly. The results of the examples are presented in this article. The authors found that the method is efficient and provides accurate results, and the authors found significant implications for success in the science, technology, engineering and mathematics domain.