Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions

Figures & data

Figure 1. General representation of the domain D and boundary Γ = Γ_U ∪ Γ_S, with unspecified initial and boundary conditions (·····) on Γ_U, collocation points () on Γ_S and source points (- - -) placed on external to the domain D.

Figure 2. Particularization of Figure 1 for s(t) given by Equation (17) in Example 1.

Figure 3. L-curve plots for δ = 1% (—–), δ = 3% () and δ = 5% () when M = 30 in (12), for Example 1.

Figure 4. The exact solutions (—–) and MFS approximations for: (a) u(0, t), (b) u_x(0, t) and (c) u(x, 0). All MFS approximations have been generated for noise levels δ = 1% (•) with λ = 10⁻⁶, δ = 3% (▪) with λ = 10⁻⁵ and δ = 5% (▴) with λ = 10⁻⁵, and obtained with h = 2, and M = 30, for Example 1.

Figure 5. Plots of the exact solution (—–) and the best (*) and least (+) accurate MFS approximations from 10 different sets of noisy data with noise level δ = 5% for (a) u(0, t) and (b) u(x, 0). Both plots are obtained with h = 2, M = 30 and λ = 10⁻⁵, for Example 1.

Figure 6. (a) The exact solution u(x, t) for all (x, t) ∈ D and (b) the absolute error for all (x, t) ∈ D for noise level δ = 5%, obtained with h = 2, λ = 10⁻⁵, M = 30, for Example 1.

Figure 7. Plots of the absolute error for all (x, t) ∈ D for noise level δ = 5% obtained with h = 2.5, λ = 10⁻⁶ and (a) M = 30 and (b) M = 16, for Example 1.

Figure 8. Particularization of Figure 1 for s(t) given by Equation (25) in Example 2.

Figure 9. L-curve plots for δ = 1% (), δ = 3% () and δ = 5% () when M = 30, for Example 2.

Figure 10. The exact solutions (—–) and MFS approximations for: (a) u(0, t), (b) u_x(0, t) and (c) u(x, 0). All MFS approximations have been generated for noise levels δ = 1% (•) with λ = 10⁻⁶, δ = 3% (▪) with λ = 10⁻⁵ and δ = 5% (▴) with λ = 10⁻⁵, and obtained with h = 2, and M = 30, for Example 2.

Figure 11. Plots of the exact solution (—–) and the best (*) and least (+) accurate MFS approximations from 10 different sets of noisy data with noise level δ = 5% for (a) u(0, t) and (b) u(x, 0). Both plots are obtained with h = 2, M = 30 and λ = 10⁻⁴, for Example 2.

Figure 12. (a) The exact solution u(x, t) for all and (b) the absolute error for all (x, t) ∈ D for noise level δ = 5%, obtained with h = 2, λ = 10⁻⁵, M = 30, for Example 2.

Figure 13. Particularization of Figure 1 for s(t) given by Equation (32) in Example 3.

Figure 14. (a) The exact solution u(0, t) (—–) and the MFS approximation, and (b) the exact solution u(x, 0) (—–) and the MFS approximation. Both plots are obtained with λ = 10⁻⁸, h = 2 and M = 30, for Example 3.

Figure 15. L-curve plots with the singularity removed for δ = 1% (), δ = 3% () and δ = 5% () when M = 30, for Example 3.

Figure 16. The exact solutions (—–) and MFS approximations with singularity removed for: (a) u(0, t), (b) u_x(0, t) and (c) u(x, 0). All MFS approximations have been generated for noise levels δ = 1% (•) with λ = 10⁻⁶, δ = 3% (▪) with λ = 10⁻⁵ and δ = 5% (▴) with λ = 10⁻⁵, and obtained with h = 2, and M = 30, for Example 3.

Figure 17. Plots of the exact solution (—–) and the best (*) and least (+) accurate MFS approximations from 10 different sets of noisy data with noise level δ = 5% for (a) u(0, t) and (b) u(x, 0). Both plots are obtained with the singularity removed, h = 2, M = 30 and λ = 10⁻⁴, for Example 3.

Figure 18. (a) The exact solution u(x, t) for all and (b) the absolute error for all (x, t) ∈ D for noise level δ = 5%, obtained with the singularity removed, h = 2, λ = 10⁻⁵, M = 30, for Example 3.