A spring-damping regularization of the Fourier sine series solution to the inverse Cauchy problem for a 3D sideways heat equation

Figures & data

Figure 1. Comparing the errors of solution without regularization and one with regularization to the exact solution of an unstable second-order ODE.

Figure 2. The regularization parameter vs. n for different time step sizes.

Figure 3. For Example 1 of the inverse Cauchy problem of 3D heat equation solved by the 2D Fourier sine series method with spring-damping regularization, comparing (a) numerical and (b) exact solutions on the plane z = 1 at the final time.

Figure 4. For (a) Example 1 and (b) Example 2 of the inverse Cauchy problems of 3D heat equation solved by the 2D Fourier sine series method with spring-damping regularization, showing the maximum errors on the plane z = 1 in time.

Table 1. For Example 1 ($a=10,b=5,c=1,t_f=2,N_t=10$), the maximum errors with regularization and without considering regularization ($\alpha =0$).

$n=1,\alpha =1.3943$	$n=2,\alpha =1.0076$	$\alpha =0,n=1$	$\alpha =0,n=2$
$5.105\times {10}^{-1}$	$3.878\times {10}^{-2}$	37.150	180.463

Figure 5. For Example 2 in a larger spatial domain, showing the errors on the plane z = 5 at the final time.

Figure 6. For Example 3 of the inverse Cauchy problem of 3D heat equation solved by the 2D Fourier sine series method with spring-damping regularization, comparing (a) numerical and (b) exact solutions on the plane z = 1 at the final time.

Table 2. For Example 3 ($a=2,b=2,c=1,t_f=0.5,N_t=5$), the maximum errors with regularization and without considering regularization ($\alpha =0$).

$n=1,\alpha =1.4128$	$n=2,\alpha =0.25199$	$n=3,\alpha =0.035$	$\alpha =0,n=1$	$\alpha =0,n=2$
$4.650\times {10}^{-1}$	$6.336\times {10}^{-1}$	$9.122\times {10}^{-1}$	2.945	12.413

Figure 7. For Example 4 of the inverse Cauchy problem of 3D heat equation solved by the 2D Fourier sine series method with spring-damping regularization, comparing (a) numerical and (b) exact solutions on the plane z = 1 at the final time.