The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis

Zhen Wang, Luhan Sun

2023Communications in Analysis and Mechanics16 citationsDOI Open Access PDF

Abstract

<abstract><p>In this paper, we investigate the local discontinuous Galerkin (LDG) finite element method for the fractional Allen-Cahn equation with Caputo-Hadamard derivative in the time domain. First, the regularity of the solution is analyzed, and the results indicate that the solution of this equation generally possesses initial weak regularity in the time dimension. Due to this property, a logarithmic nonuniform L1 scheme is adopted to approximate the Caputo-Hadamard derivative, while the LDG method is used for spatial discretization. The stability and convergence of this fully discrete scheme are proven using a discrete fractional Gronwall inequality. Numerical examples demonstrate the effectiveness of this method.</p></abstract>

Topics & Concepts

Hadamard transformMathematicsDiscretizationStability (learning theory)Convergence (economics)Applied mathematicsMathematical analysisGronwall's inequalityFractional calculusDomain (mathematical analysis)Discontinuous Galerkin methodDerivative (finance)Finite element methodInequalityComputer sciencePhysicsEconomic growthMachine learningFinancial economicsThermodynamicsEconomicsDifferential Equations and Numerical MethodsFractional Differential Equations SolutionsSolidification and crystal growth phenomena