A meshless method for solving 1D time-dependent heat source problem

Figures & data

Figure 1. Graph of $g_{1_{no}}^{\prime }$ and ${\mathit{\varphi }}_{0_{no}}^{″}$ with various values of polynomials $(m_1)$ and scattered points $(n_1)$ for Example 1.

Figure 2. Graph of $g_{1_{no}}^{\prime }$ and $k_{no}^{\prime }$ and absolute errors $|g_{1_{Exact}}^{\prime }-g_{1_{{no}_{Numeric}}}^{\prime }|$ and $|k_{Exact}^{\prime }-k_{{no}_{Numeric}}^{\prime }|$ for Example 1.

Figure 3. Graph of $g_{1_{no}}^{\prime }$ and $k_{no}^{\prime }$ and absolute errors $|g_{1_{Exact}}^{\prime }-g_{1_{{no}_{Numeric}}}^{\prime }|$ and $|k_{Exact}^{\prime }-k_{{no}_{Numeric}}^{\prime }|$ for Example 2.

Figure 4. Graph of ${\mathit{\varphi }}_{0_{no}}^{″}$ and ${\mathit{\varphi }}_{T_{no}}^{″}$ and absolute errors $|{\mathit{\varphi }}_{0_{Exact}}^{″}-{\mathit{\varphi }}_{0_{{no}_{Numeric}}}^{″}|$ and $|{\mathit{\varphi }}_{T_{Exact}}^{″}-{\mathit{\varphi }}_{T_{{no}_{Numeric}}}^{″}|$ for Example 1.

Figure 5. Graph of ${\mathit{\varphi }}_{0_{no}}^{″}$ and ${\mathit{\varphi }}_{T_{no}}^{″}$ and absolute errors $|{\mathit{\varphi }}_{0_{Exact}}^{″}-{\mathit{\varphi }}_{0_{{no}_{Numeric}}}^{″}|$ and $|{\mathit{\varphi }}_{T_{Exact}}^{″}-{\mathit{\varphi }}_{T_{{no}_{Numeric}}}^{″}|$ for Example 2.

Figure 6. Graph of f(t) and absolute error $|f_{{}_{Exact}}-f_{{}_{Numeric}}|$ with noiseless and noisy data when $n_1=n_2=n_3=4,s_1=s_2=s_3=5,m=10$ and $T_0=1.2$ for Example 1.

Figure 7. Graph of $g_0(t)$ and absolute error $|g_{0_{Exact}}-g_{0_{Numeric}}|$ with noiseless and noisy data when $n_1=n_2=n_3=4,s_1=s_2=s_3=5,m=10$ and $T_0=1.2$ for Example 1.

Figure 8. Graph of h(t) and absolute error $|h_{Exact}-h_{Numeric}|$ with noiseless and noisy data when $n_1=n_2=n_3=4,s_1=s_2=s_3=5,m=10$ and $T_0=1.2$ for Example 1.

Figure 9. Graph of $RMS(f),RMS(g_0),RMS(h)$ and RES(f),  RES(h) with noiseless data when $n_1=n_2=n_3=4,s_1=s_2=s_3=5,m=10$ and various values of $T_0$ for Example 1.

Figure 10. Graph of f(t) with noiseless and noisy data when $n_1=n_2=n_3=4,s_1=s_2=s_3=5,m=10$ and $T_0=1.5$ for Example 2.

Figure 11. Graph of $g_0(t)$ with noiseless and noisy data when $n_1=n_2=n_3=4,s_1=s_2=s_3=5,m=10$ and $T_0=1.5$ for Example 2.

Figure 12. Graph of h(t) with noiseless and noisy data when $n_1=n_2=n_3=4,s_1=s_2=s_3=5,m=10$ and $T_0=1.5$ for Example 2.

Figure 13. Graph of $RMS(f),RMS(g_0)RMS(h)$ and RES(f),  RES(h) with noiseless data when $n_1=n_2=n_3=4,s_1=s_2=s_3=5,m=10$ and various values of $T_0$ for Example 2.

Table 1. The values of RMS(f), RES(f),  $RMS(g_0)$, $RES(g_0)$, RMS(h) and RES(h) with different choose of n and m when $s_1=s_2=s_3=6$ and $\mathit{\sigma }=1\%$ for Example 1.

$n_1$	4	5	6	7	10
$n_2$	4	5	6	7	10
$n_3$	4	5	6	7	10
m	0	0	0	0	0
RMS(f)	0.9346294	0.8176032	5.8213482	2.8554492	0.7765079
RES(f)	0.2664936	0.2331256	1.6598583	0.8141828	0.2214080
$RMS(g_0)$	0.2306375	0.2201657	2.7895291	1.0043515	0.2635528
$RES(g_0)$	Inf	Inf	Inf	Inf	Inf
RMS(h)	0.6876904	0.6363895	4.6353827	2.0763144	0.5000441
RES(h)	1.0377705	0.9603541	6.9951010	3.1332970	0.7545999
m	5	5	5	5	5
RMS(f)	$1.1708181\times {10}^{-4}$	0.0020049	1.2009122	0.8593210	0.5491819
RES(f)	$3.3383883\times {10}^{-5}$	$5.7165785\times {10}^{-4}$	0.3424197	0.2450208	0.1565899
$RMS(g_0)$	$5.7108347\times {10}^{-5}$	$5.1326584\times {10}^{-4}$	0.2980472	0.2751808	0.3038408
$RES(g_0)$	Inf	Inf	Inf	Inf	Inf
RMS(h)	$1.3195569\times {10}^{-4}$	0.0014865	0.9101555	0.4405244	0.2311222
RES(h)	$1.9912992\times {10}^{-4}$	0.0022432	1.3734852	0.6647807	0.3487788
m	10	10	10	10	10
RMS(f)	$7.7663001\times {10}^{-4}$	$3.6622293\times {10}^{-4}$	1.1895994	1.2881577	0.7591267
RES(f)	$2.2144282\times {10}^{-4}$	$1.0442224\times {10}^{-4}$	0.3391940	0.3672962	0.2164520
$RMS(g_0)$	$1.9886186\times {10}^{-4}$	$1.0914386\times {10}^{-4}$	0.2959987	0.5151874	0.1970341
$RES(g_0)$	Inf	Inf	Inf	Inf	Inf
RMS(h)	$5.3880119\times {10}^{-4}$	$2.4684676\times {10}^{-4}$	0.9011449	0.9751129	0.2777104
RES(h)	$8.1308687\times {10}^{-4}$	$3.7250822\times {10}^{-4}$	1.3598877	1.4715103	0.4190836

Table 2. The values of RMS(f), RES(f),  $RMS(g_0)$, $RES(g_0)$, RMS(h) and RES(h) with different choose of $s_1,s_2,s_3$ when $n_1=n_2=n_3=4,m=10$ and $\mathit{\sigma }=1\%$ for Example 1.

$s_1$	4	5	8	10	12	15
$s_2$	4	5	8	10	12	15
$s_3$	4	5	8	10	12	15
RMS(f)	2.3748682	$4.6850674\times {10}^{-4}$	$8.6736970\times {10}^{-4}$	$8.9234946\times {10}^{-4}$	$9.5057167\times {10}^{-4}$	$8.5945433\times {10}^{-4}$
RES(f)	0.6771532	$1.3358673\times {10}^{-4}$	$2.4731571\times {10}^{-4}$	$2.5443826\times {10}^{-4}$	$2.5375868\times {10}^{-4}$	$2.4505876\times {10}^{-4}$
$RMS(g_0)$	0.3374261	$1.1987426\times {10}^{-4}$	$2.2341940\times {10}^{-4}$	$2.3043220\times {10}^{-4}$	$2.3035733\times {10}^{-4}$	$2.2339808\times {10}^{-4}$
$RES(g_0)$	Inf	Inf	Inf	Inf	Inf	Inf
RMS(h)	1.4943393	$3.0130733\times {10}^{-4}$	$6.1014690\times {10}^{-4}$	$6.2990701\times {10}^{-4}$	$6.2858599\times {10}^{-4}$	$6.0600135\times {10}^{-4}$
RES(h)	2.2550576	$4.5469284\times {10}^{-4}$	$9.2075230\times {10}^{-4}$	$9.5057167\times {10}^{-4}$	$9.4857818\times {10}^{-4}$	$9.1449637\times {10}^{-4}$

Table 3. The values of RMS(f) and RES(f) for various values of $T_0$, $\mathit{\sigma }=5\%,m=10$, $n_1=n_2=n_3=4$ and $s_1=s_2=s_3=6$ for Example 1.

$T_0$	1.1	1.5	2.1	10	20
RMS(f)	$9.7665878\times {10}^{-4}$	$5.8738177\times {10}^{-4}$	$9.5202576\times {10}^{-4}$	0.0012657	0.0017081
RES(f)	$2.7847762\times {10}^{-4}$	$1.6748191\times {10}^{-4}$	$2.7145393\times {10}^{-4}$	$3.6088924\times {10}^{-4}$	$4.8702289\times {10}^{-4}$
RMS(f)[23]	0.3322	0.6894	0.2308	0.1774	0.1074
RES(f)[23]	0.0954	0.1979	0.0663	0.0510	0.0308

Table 4. The values of RES(f) for various values of $\mathit{\sigma }$ and $x^{\ast }$ when $n_1=n_2=n_3=4,s_1=s_2=s_3=6,m=10$ and $T_0=2.1$ for Example 1.

$x^{\ast }$	MMFS	MMFS	MMFS	MFS [23]	MFS [23]	MFS [23]
	$\mathit{\sigma }=10\%$	$\mathit{\sigma }=1\%$	$\mathit{\sigma }=0.1\%$	$\mathit{\sigma }=10\%$	$\mathit{\sigma }=1\%$	$0.1\%$
0.1	0.0877574	$5.6502939\times {10}^{-4}$	$3.9930249\times {10}^{-4}$	1.3852	0.1445	$1.631\times {10}^{-2}$
0.2	0.0535205	$1.5203652\times {10}^{-5}$	$1.1297643\times {10}^{-5}$	0.8343	0.0869	$8.821\times {10}^{-3}$
0.3	0.0069395	$2.4644146\times {10}^{-4}$	$7.4674617\times {10}^{-6}$	0.6340	0.0645	$6.560\times {10}^{-3}$
0.4	0.0015191	$2.5093960\times {10}^{-4}$	$5.7029333\times {10}^{-6}$	0.5540	0.0542	$5.966\times {10}^{-3}$
0.5	0.0030382	$2.2111907\times {10}^{-4}$	$7.4674617\times {10}^{-6}$	0.5256	0.0507	$5.850\times {10}^{-3}$
0.6	0.0068312	$5.6227131\times {10}^{-5}$	$6.6374241\times {10}^{-6}$	0.5455	0.0539	$6.123\times {10}^{-3}$
0.7	0.0016227	$4.3797139\times {10}^{-5}$	$6.4720234\times {10}^{-6}$	0.6029	0.0648	$7.123\times {10}^{-3}$
0.8	$8.3356403\times {10}^{-4}$	$4.0891921\times {10}^{-4}$	$5.3830217\times {10}^{-6}$	0.9311	0.0933	$1.003\times {10}^{-2}$
0.9	0.0010245	$3.9911678\times {10}^{-4}$	$3.5354251\times {10}^{-6}$	1.8509	0.1870	$1.947\times {10}^{-2}$

Table 5. The values of RMS(f), RES(f),  $RMS(g_0)$, $RES(g_0)$, RMS(h) and RES(h) with different choose of n and m when $\mathit{\sigma }=1\%$ and $s_1=s_2=s_3=6$ for Example 2.

$n_1$	4	5	6	7	10
$n_2$	4	5	6	7	10
$n_3$	4	5	6	7	10
m	0	0	0	0	0
RMS(f)	0.5653877	0.5236957	1.4769239	0.4444294	0.0122931
RES(f)	0.1243311	0.1151629	0.3247817	0.0977319	0.0027033
$RMS(g_0)$	0.1454090	0.1373134	0.2101423	0.2589738	0.0120869
$RES(g_0)$	0.1304997	0.1232342	0.1885958	0.2324204	0.0108476
RMS(h)	0.4290696	0.4016710	0.8378655	0.4415788	0.0176552
RES(h)	Inf	Inf	Inf	Inf	Inf
m	5	5	5	5	5
RMS(f)	0.0013772	0.0065025	0.9937791	0.0916205	0.0169339
RES(f)	$3.0284532\times {10}^{-5}$	0.0014299	0.2185362	0.0201477	0.0037238
$RMS(g_0)$	0.0056976	0.0054787	0.2153635	0.0423792	0.0048539
$RES(g_0)$	0.0051134	0.0049170	0.1932816	0.0380339	0.0043562
RMS(h)	$5.3510792\times {10}^{-4}$	0.0048394	0.7070144	0.0462939	0.0055545
RES(h)	Inf	Inf	Inf	Inf	Inf
m	10	10	10	10	10
RMS(f)	0.0012349	0.0013419	0.8455426	0.2068440	0.0041247
RES(f)	$2.7156682\times {10}^{-4}$	$2.9509986\times {10}^{-4}$	0.1859383	0.0454859	$9.0703566\times {10}^{-4}$
$RMS(g_0)$	0.0057432	0.0056993	0.2217135	0.0787748	0.0071372
$RES(g_0)$	0.0051543	0.0051149	0.1989805	0.0706977	0.0064054
RMS(h)	$3.3999927\times {10}^{-5}$	$4.8030741\times {10}^{-4}$	0.6530542	0.1691280	0.0050551
RES(h)	Inf	Inf	Inf	Inf	Inf

Table 6. The values of RMS(f), RES(f),  $RMS(g_0)$, $RES(g_0)$, RMS(h) and RES(h) with different choose of $s_1,s_2,s_3$ when $\mathit{\sigma }=1\%,m=10$ and $n_1=n_2=n_3=4$ for Example 2.

$s_1$	4	5	8	10	12	15
$s_2$	4	5	8	10	12	15
$s_3$	4	5	8	10	12	15
RMS(f)	1.1994413	0.0011669	0.0065513	0.0058406	0.0021611	0.0066031
RES(f)	0.2637621	$2.5661022\times {10}^{-4}$	0.0014407	0.0012844	$4.7523476\times {10}^{-4}$	0.0014521
$RMS(g_0)$	0.2502911	0.0056858	0.0053042	0.0052823	0.0054196	0.0062030
$RES(g_0)$	0.2246280	0.0051028	0.0047604	0.0047407	0.0048639	0.0055670
RMS(h)	0.8555651	$4.7487152\times {10}^{-5}$	0.0049015	0.0043500	0.0013703	0.0050395
RES(h)	Inf	Inf	Inf	Inf	Inf	Inf

Table 7. The values of RES(f) for various values of $\mathit{\sigma }$ and $x^{\ast }$ when $n_1=n_2=n_3=4,s_1=s_2=s_3=5,m=10$ and $T_0=1.5$ for Example 2.

$x^{\ast }$	$\mathit{\sigma }=10\%$	$\mathit{\sigma }=1\%$	$\mathit{\sigma }=0.1\%$	$\mathit{\sigma }=0.01\%$
0.1	0.0053800	$3.0558216\times {10}^{-4}$	$2.5224462\times {10}^{-4}$	$2.5183178\times {10}^{-4}$
0.2	$3.5759946\times {10}^{-4}$	$2.5873937\times {10}^{-4}$	$2.5187456\times {10}^{-4}$	$2.5183224\times {10}^{-4}$
0.3	0.0059561	$2.5704555\times {10}^{-4}$	$2.5189546\times {10}^{-4}$	$2.5183227\times {10}^{-4}$
0.4	0.0034747	$2.5668941\times {10}^{-4}$	$2.5189138\times {10}^{-4}$	$2.5183227\times {10}^{-4}$
0.5	$7.9545926\times {10}^{-4}$	$2.5661022\times {10}^{-4}$	$2.5188664\times {10}^{-4}$	$2.5183227\times {10}^{-4}$
0.6	$9.4023044\times {10}^{-4}$	$2.5668996\times {10}^{-4}$	$2.5188448\times {10}^{-4}$	$2.5183224\times {10}^{-4}$
0.7	0.0013511	$2.5682483\times {10}^{-4}$	$2.5188469\times {10}^{-4}$	$2.5183227\times {10}^{-4}$
0.8	0.0011087	$2.5699136\times {10}^{-4}$	$2.5188451\times {10}^{-4}$	$2.5183227\times {10}^{-4}$
0.9	$4.9337547\times {10}^{-4}$	$2.5718092\times {10}^{-4}$	$2.5188690\times {10}^{-4}$	$2.5183221\times {10}^{-4}$