An Explicit Numerical Method for the Fractional Cable Equation

Figure 5

Exact solution (lines) and numerical solution (symbols) provided by our method for the fractional cable equation with 𝛾₁=0.5 and 𝛾₂=0.5 for different numbers of timesteps when Δ𝑥=1/10, Δ𝑡=10⁻⁵, 𝐾=𝜇=1, 𝐿=5, and 𝑆=(Δ𝑡)^𝛾1/(Δ𝑥)²=0.316. This case is inside the stability region because 𝑆 is smaller than the stability limit 𝑆_×=(2^𝛾2−𝜇²(Δ𝑡)^𝛾2)/(2^{2+𝛾2−𝛾1})≃0.352… provided by (3.7). The inset shows the results on logarithmic scale.