Pure mathematics

Applied mathematics

Variable (mathematics)

Economics

Thermodynamics

Physics

Kernel (algebra)

Advection

Diffusion

Mathematics

Legendre polynomials

Mathematical analysis

Reaction–diffusion system

Finance

Fractional calculus

Piecewise

Order (exchange)

Thermal Science

This paper structures some new reproductive kernel spaces based on Legendre polynomials to solve time variable order fractional advection-reaction-diffusion equations. Some examples are given to show the effectiveness and reliability of the method.

The piecewise reproducing kernel method for the time variable fractional order advection-reaction-diffusion equations