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A higher‐order unconditionally stable scheme for the solution of fractional diffusion equation
Author(s) -
Ghaffar Fazal,
Ullah Saif,
Badshah Noor,
Khan Najeeb Alam
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6406
Subject(s) - mathematics , compact finite difference , truncation error , order (exchange) , scheme (mathematics) , matrix difference equation , multigrid method , mathematical analysis , matrix (chemical analysis) , fractional calculus , stability (learning theory) , finite difference method , diffusion equation , finite difference , space (punctuation) , partial differential equation , materials science , composite material , riccati equation , linguistics , philosophy , finance , machine learning , computer science , economics , economy , service (business)
In this paper, a higher‐order compact finite difference scheme with multigrid algorithm is applied for solving one‐dimensional time fractional diffusion equation. The second‐order derivative with respect to space is approximated by higher‐order compact difference scheme. Then, Grünwald–Letnikov approximation is used for the Riemann–Liouville time derivative to get an implicit scheme. The scheme is based on a heptadiagonal matrix with eighth‐order accurate local truncation error. Fourier analysis is used to analyze the stability of higher‐order compact finite difference scheme. Matrix analysis is used to show that the scheme is convergent with the accuracy of eighth‐order in space. Numerical experiments confirm our theoretical analysis and demonstrate the performance and accuracy of our proposed scheme.