This study presents a numerical solution of Fisher’s equation. For time integration, Crank–Nicolson and fourth-order one-step implicit schemes are used and for space discretization, quintic B-spline Collocation and quintic B-spline Galerkin methods are employed. The truncation error is analyzed and the stability of the suggested methods is discussed matrix stability analysis. Three examples are studied to compare the present results with existing numerical results by computing error norm L∞ and the order of convergence. The obtained results show that the proposed methods are satisfactorily efficient in terms of accuracy.