Hadamard transform

Scheme (mathematics)

Norm (philosophy)

Service (business)

Applied mathematics

Remainder

Economics

Computer science

Thermodynamics

Physics

Diffusion

Mathematics

Political science

Mathematical analysis

Uniqueness

Economy

Diffusion equation

Finance

Artificial neural network

Fractional calculus

Order (exchange)

Machine learning

Arithmetic

Convolution (computer science)

Journal of Computational and Applied Mathematics

In this paper, a second-order scheme with nonuniform time meshes for Caputo–Hadamard fractional sub-diffusion equations with initial singularity is investigated. Firstly, a Taylor-like formula with integral remainder is proposed, which is crucial to studying the discrete convolution kernels and error estimate. Secondly, we come up with an error convolution structure (ECS) analysis for Llog,2−1σ interpolation approximation to the Caputo–Hadamard fractional derivative. The core result in this paper is an ECS bound and a global consistency analysis established at an offset point. By virtue of this result, we obtain a sharp L2-norm error estimate of a second-order Crank–Nicolson-like scheme for Caputo–Hadamard fractional differential equations. Ultimately, an example is presented to show the sharpness of our analysis.

A second-order scheme with nonuniform time grids for Caputo–Hadamard fractional sub-diffusion equations