Crank–Nicolson method for solving time-fractional singularly perturbed delay partial differential equations

Figures & data

Figure 1. Surface plots of the numerical solution of example 4.1 for different values of β with $\mathit{ϵ}={10}^{-10},\mathit{M}=80$ and N = 64.

Figure 2. Surface plots of the numerical solution of example 4.1 for different values of ϵ with $\mathit{\beta }=0.4,\mathit{M}=80$ and N = 64.

Figure 3. Surface plots of the numerical solution of example 4.2 for different values of β with $\mathit{ϵ}={10}^{-10},\mathit{M}=80$ and N = 64.

Figure 4. Surface plots of the numerical solution of example 4.2 for different values of ϵ with $\mathit{\beta }=0.6,\mathit{M}=80$ and N = 64.

Figure 5. Loglog plot of maximum pointwise error for examples 4.1 and 4.2 for β = 0.8.

Table 1. Maximum pointwise error $\left(E^{\mathit{M},\mathit{N}}\right)$ and rate of convergence $\left(R^{\mathit{M},\mathit{N}}\right)$ for example 4.1 using fractional order β = 0.8 with different values of $\mathit{ϵ}.$

$\mathit{ϵ}\downarrow /(\mathit{M},\mathit{N})\to$	$(20,16)$	$(40,32)$	$(80,64)$	$(160,128)$
1	$2.0241\times {10}^{-3}$	$1.5604\times {10}^{-3}$	$1.0961\times {10}^{-3}$	$6.8509\times {10}^{-4}$
	0.9840	0.9944	0.9970	0.9986
10^−2	$2.2312\times {10}^{-2}$	$1.3099\times {10}^{-2}$	$6.6770\times {10}^{-3}$	$2.8843\times {10}^{-3}$
	1.1029	1.0546	1.0191	1.0057
10^−4	$2.2905\times {10}^{-2}$	$1.4199\times {10}^{-2}$	$7.1512\times {10}^{-3}$	$3.0362\times {10}^{-3}$
	0.9696	0.9885	0.9955	0.9962
10^−6	$2.2905\times {10}^{-2}$	$1.4199\times {10}^{-2}$	$7.1512\times {10}^{-3}$	$3.0362\times {10}^{-3}$
	0.9696	0.9885	0.9955	0.9962
10^−8	$2.2905\times {10}^{-2}$	$1.4199\times {10}^{-2}$	$7.1512\times {10}^{-3}$	$3.0362\times {10}^{-3}$
	0.9696	0.9885	0.9955	0.9962
10^−10	$2.2905\times {10}^{-2}$	$1.4199\times {10}^{-2}$	$7.1512\times {10}^{-3}$	$3.0362\times {10}^{-3}$
	0.9696	0.9885	0.9955	0.9962
10^−15	$2.2905\times {10}^{-2}$	$1.4199\times {10}^{-2}$	$7.1512\times {10}^{-3}$	$3.0362\times {10}^{-3}$
	0.9696	0.9885	0.9955	0.9962
10^−20	$2.2905\times {10}^{-2}$	$1.4199\times {10}^{-2}$	$7.1512\times {10}^{-3}$	$3.0362\times {10}^{-3}$
	0.9696	0.9885	0.9955	0.9962
$E^{\mathit{M},\mathit{N}}$	$2.2905\times {10}^{-2}$	$1.4199\times {10}^{-2}$	$7.1512\times {10}^{-3}$	$3.0362\times {10}^{-3}$
$R^{\mathit{M},\mathit{N}}$	0.9696	0.9885	0.9955	0.9962

Table 2. Maximum pointwise error $\left(E^{\mathit{M},\mathit{N}}\right)$ and rate of convergence $\left(R^{\mathit{M},\mathit{N}}\right)$ for example 4.1 using $\mathit{ϵ}={10}^{-10}$ with different values of β

$\mathit{\beta }\downarrow /(\mathit{M},\mathit{N})\to$	$(20,16)$	$(40,32)$	$(80,64)$	$(160,128)$
β = 0.2	$6.9776\times {10}^{-3}$	$4.0080\times {10}^{-3}$	$2.1218\times {10}^{-3}$	$1.0750\times {10}^{-3}$
	0.8072	0.9098	0.9573	0.9798
β = 0.4	$7.8970\times {10}^{-3}$	$4.3810\times {10}^{-3}$	$2.1834\times {10}^{-3}$	$9.9822\times {10}^{-4}$
	0.8549	0.9374	0.9738	0.9900
β = 0.6	$1.1736\times {10}^{-2}$	$6.5010\times {10}^{-3}$	$3.0851\times {10}^{-3}$	$1.3040\times {10}^{-3}$
	0.9320	0.9760	0.9936	0.9970
β = 0.8	$2.2905\times {10}^{-2}$	$1.4199\times {10}^{-2}$	$7.1512\times {10}^{-3}$	$3.0362\times {10}^{-3}$
	0.9696	0.9885	0.9955	0.9962

Table 3. Maximum pointwise error $\left(E^{\mathit{M},\mathit{N}}\right)$ and rate of convergence $\left(R^{\mathit{M},\mathit{N}}\right)$ for example 4.2 using fractional order β = 0.8 with different values of ϵ

$\mathit{ϵ}\downarrow /(\mathit{M},\mathit{N})\to$	$(20,16)$	$(40,32)$	$(80,64)$	$(160,128)$
1	$1.4720\times {10}^{-3}$	$1.1663\times {10}^{-3}$	$8.4357\times {10}^{-4}$	$5.4721\times {10}^{-4}$
	0.9755	0.9914	0.9963	0.9983
10^−2	$1.0522\times {10}^{-2}$	$6.6778\times {10}^{-3}$	$3.8007\times {10}^{-3}$	$1.8941\times {10}^{-3}$
	1.0234	1.0244	1.0063	1.0013
10^−4	$1.0698\times {10}^{-2}$	$6.9495\times {10}^{-3}$	$3.9883\times {10}^{-3}$	$1.9748\times {10}^{-3}$
	0.9409	0.9741	0.9859	0.9927
10^−6	$1.0698\times {10}^{-2}$	$6.9495\times {10}^{-3}$	$3.9883\times {10}^{-3}$	$1.9748\times {10}^{-3}$
	0.9409	0.9741	0.9859	0.9927
10^−8	$1.0698\times {10}^{-2}$	$6.9495\times {10}^{-3}$	$3.9883\times {10}^{-3}$	$1.9748\times {10}^{-3}$
	0.9409	0.9741	0.9859	0.9927
10^−10	$1.0698\times {10}^{-2}$	$6.9495\times {10}^{-3}$	$3.9883\times {10}^{-3}$	$1.9748\times {10}^{-3}$
	0.9409	0.9741	0.9859	0.9927
10^−15	$1.0698\times {10}^{-2}$	$6.9495\times {10}^{-3}$	$3.9883\times {10}^{-3}$	$1.9748\times {10}^{-3}$
	0.9409	0.9741	0.9859	0.9927
10^−20	$1.0698\times {10}^{-2}$	$6.9495\times {10}^{-3}$	$3.9883\times {10}^{-3}$	$1.9748\times {10}^{-3}$
	0.9409	0.9741	0.9859	0.9927
$E^{\mathit{M},\mathit{N}}$	$1.0698\times {10}^{-2}$	$6.9495\times {10}^{-3}$	$3.9883\times {10}^{-3}$	$1.9748\times {10}^{-3}$
$R^{\mathit{M},\mathit{N}}$	0.9409	0.9741	0.9859	0.9927

Table 4. Maximum pointwise error $\left(E^{\mathit{M},\mathit{N}}\right)$ and rate of convergence $\left(R^{\mathit{M},\mathit{N}}\right)$ for example 4.2 using $\mathit{ϵ}={10}^{-10}$ with different values of β

$\mathit{\beta }\downarrow /(\mathit{M},\mathit{N})\to$	$(20,16)$	$(40,32)$	$(80,64)$	$(160,128)$
β = 0.2	$3.9808\times {10}^{-3}$	$2.1409\times {10}^{-3}$	$1.0701\times {10}^{-3}$	$5.1691\times {10}^{-4}$
	0.8438	0.9369	0.9738	0.9890
β = 0.4	$4.0522\times {10}^{-3}$	$2.1378\times {10}^{-3}$	$1.0272\times {10}^{-3}$	$4.8148\times {10}^{-4}$
	0.8480	0.9502	0.9833	0.9923
β = 0.6	$5.5952\times {10}^{-3}$	$3.1798\times {10}^{-3}$	$1.6561\times {10}^{-2}$	$8.0421\times {10}^{-4}$
	0.9041	0.9751	0.9892	0.9949
β = 0.8	$1.0698\times {10}^{-2}$	$6.9495\times {10}^{-3}$	$3.9883\times {10}^{-3}$	$1.9748\times {10}^{-3}$
	0.9409	0.9741	0.9859	0.9927