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Fractional Crank–Nicolson–Galerkin finite element scheme for the time‐fractional nonlinear diffusion equation
Author(s) -
Kumar Dileep,
Chaudhary Sudhakar,
Srinivas Kumar V.V.K.
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22399
Subject(s) - mathematics , backward euler method , crank–nicolson method , discretization , gronwall's inequality , quadrature (astronomy) , nonlinear system , finite element method , uniqueness , mathematical analysis , burgers' equation , partial differential equation , physics , quantum mechanics , thermodynamics , electrical engineering , inequality , engineering
Abstract This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L 2 (Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation.