NUMERICAL CALCULATION OF A FRACTIONAL PROBLEM WITH INTEGRAL CONDITION
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Date
2026
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Abstract
In this thesis, we studied a time-space fractional Partial differential equation of order α (0 < α < 1) in time
(Caputo) and order β (1 < β < 2) in space (Riemann-Liouville), with Neumann and integral boundary conditions.
For the numerical solution, we proposed an implicit finite difference scheme combining the L1 method for the
time derivative, the shifted Grünwald-Letnikov method for the space derivative, and the trapezoidal rule for the
integral condition.
We analyzed the scheme and proved:
• Consistency: The local truncation error is of order O(ht + h).
• Stability: Conditional stability under the condition:
(
ht ≤ C·hβ/α, C =
1
2βAΓ(2 −α)
)1/α
,
A=max|a(x,t)|
• Convergence: The scheme converges with first-order accuracy in time (O(ht)) and first-order accuracy in
space (O(h)).
Numerical experiments using MATLAB confirm the theoretical results.