Brahimi, Saadoune2026-10-062026-10-062026http://dspace.univ-khenchela.dz:4000/handle/123456789/11622In 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.enThesis