Abstract
ABSTRACTWe consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with non-uniform meshes. Under the assumption that the Caputo derivative of the solution of the fractional differential equation is suitably smooth [C. Li, Q. Yi, and A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys. 316 (2016), pp. 614–631] obtained the error estimates of finite difference methods with non-uniform meshes. However, the Caputo derivative of the solution of the fractional differential equation in general has a weak singularity near the initial time. In this paper, we obtain the error estimates of finite difference methods with non-uniform meshes when the Caputo fractional derivative of the solution of the fractional differential equation has lower smoothness. The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods. Numerical results are presented that confirm the sharpness of the error analysis.
Highlights
In this paper, we will consider finite difference methods for solving the following fractional nonlinear differential equation, with α > 0, C 0 Dtα y(t) = f (t, y(t)), t >0, y(k)(0) = y0(k), k = 0, 1, . . . , α − 1, [1]where the y0(k) may be arbitrary real numbers and denotes the Caputo fractional derivative defined by Γ(
We show that the optimal convergence orders of the numerical methods may be recovered when the Caputo derivative of the solution of the fractional differential equation has lower smoothness
We see that the experimental order of convergence (EOC) of the trapezoid method (??) with non-uniform meshes is almost O(N −2) as we expected
Summary
We will consider finite difference methods for solving the following fractional nonlinear differential equation, with α > 0,. ]. Most analysis of the numerical methods for solving (??) is deduced under the assumptions that the meshes are uniform, see, for example, [? ] considered the error estimates of the rectangle formula, trapezoid formula and the predictor-corrector scheme for solving (??) with non-uniform meshes when the solution is suitably smooth, see [? We show that the optimal convergence orders of the numerical methods may be recovered when the Caputo derivative of the solution of the fractional differential equation has lower smoothness. ], where the authors applied the graded meshes to recover the convergence order of the finite difference method for solving time-fractional diffusion equation when the solution is not sufficiently smooth. Throughout, the notations C and c, with or without a subscript, denote generic constants, which may differ at different occurrences, but are always independent of the mesh size
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