New numerical methods for solving the partial fractional differential equations with uniform and non-uniform meshes

In this paper, we design and develop some algorithms by using the piecewise linear interpolation polynomial for solving the partial fractional differential equations involving Caputo derivative, with uniform and non-uniform meshes. For designing new methods, we select the mesh points based on the two equal-height and equal-area distribution. Furthermore, the error bounds of proposed methods with uniform and equidistributing meshes are obtained. We also show that our numerical method is stable and convergent with the accuracy of O(kappa(2) + h). Also, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical methods. Finally, a comparative study for different values of parameters is also presented.

Automated summary

In this paper, algorithms using piecewise linear interpolation polynomial are designed and developed to solve partial fractional differential equations involving Caputo derivative. The algorithms are applied to both uniform and non-uniform meshes, and new methods for selecting mesh points are proposed. Error bounds for the proposed methods with uniform and equidistributing meshes are obtained. The numerical method is stable and convergent with an accuracy of O(kappa(2) + h), as demonstrated through numerical examples and a comparative study for different parameter values.