Time-fractional partial differential equations: a novel technique for analytical and numerical solutions

Figures & data

Figure 1. Comparison of the approximate solution of Problem 1 for distinct values ζ w.r.t. the exact solution and absolute error $|{\nu }_{\mathit{exa}.}(\eta ,\tau )-{\nu }_{\mathit{app}.}(\eta ,\tau )|$ with n = 1, $\zeta =1,,$ and $\hslash =-1..$

Figure 2. Comparison between the approximate solution at $\zeta \le 1$ and the exact solution with $\hslash =-1,$ n = 1, and η = 1 for Problem 1.

Figure 3. n-curves of $\nu (\eta ,\tau )$ for distinct values of ζ with $\hslash =-1,\eta =0.1,$ and $\tau =0.5$ for Problem 1.

Figure 4. The $\hslash$-curves of $\nu (\eta ,\tau )$ for distinct values n with $\eta =0.1,$ and $\tau =0.5$ for Problem 1.

Table 1. q-Homotopy analysis Shehu transform method for $\nu (\eta ,\tau )$ in comparison with RPSM (Wang et al., 2019), HATM (Wang et al., 2019), and HPTM (Singh & Kumar, 2018) at $\hslash$ = −1, ζ = 1, and n = 1 for Problem 1.

$(\eta ,\tau )$	$|{\nu }_{\mathit{exa}.}-{\nu }_{\mathit{RPSM}}|$	$|{\nu }_{\mathit{exa}.}-{\nu }_{\mathit{HATM}}|$	$|{\nu }_{\mathit{exa}.}-{\nu }_{\mathit{HPTM}}|$	$|{\nu }_{\mathit{exa}.}-{\nu }_{q-\mathit{HASTM}}|$
(0.25, 0.25)	2.122401E − 06	2.122401E − 06	2.122401E − 06	2.122401E − 06
(0.25, 0.50)	7.094268E − 05	7.094268E − 05	7.094268E − 05	7.094268E − 05
(0.25, 0.75)	5.634807E − 04	5.634807E − 04	5.634807E − 04	5.634807E − 04
(0.25, 1.00)	2.487124E − 03	2.487124E − 03	2.487124E − 03	2.487124E − 03
(0.50, 0.25)	4.244802E − 06	4.244802E − 06	4.244802E − 06	4.244802E − 06
(0.50, 0.50)	1.418854E − 04	1.418854E − 04	1.418854E − 04	1.418854E − 04
(0.50, 0.75)	1.126961E − 03	1.126961E − 03	1.126961E − 03	1.126961E − 03
(0.50, 1.00)	4.974248E − 03	4.974248E − 03	4.974248E − 03	4.974248E − 03
(0.75, 0.25)	6.367203E − 06	6.367203E − 06	6.367203E − 06	6.367203E − 06
(0.75, 0.50)	2.128280E − 04	2.128280E − 04	2.128280E − 04	2.128280E − 04
(0.75, 0.75)	1.690442E − 03	1.690442E − 03	1.690442E − 03	1.690442E − 03
(0.75, 1.00)	7.461371E − 03	7.461371E − 03	7.461371E − 03	7.461371E − 03

Table 2 q-Homotopy analysis Shehu transform method for $\nu (\eta ,\tau )$ in comparison with RPSM (Wang et al., 2019), HATM (Wang et al., 2019), and HPTM (Singh & Kumar, 2018) at $\hslash$ = −1, ζ = 1, and n = 1 for Problem 2.

$(\eta ,\tau )$	$|{\nu }_{\mathit{exa}.}-{\nu }_{\mathit{RPSM}}|$	$|{\nu }_{\mathit{exa}.}-{\nu }_{\mathit{HATM}}|$	$|{\nu }_{\mathit{exa}.}-{\nu }_{\mathit{HPTM}}|$	$|{\nu }_{\mathit{exa}.}-{\nu }_{q-\mathit{HASTM}}|$
(0.25, 0.25)	5.300000E − 07	5.300000E − 07	5.300000E − 07	5.300000E − 07
(0.25, 0.50)	1.773500E − 05	1.773500E − 05	1.773500E − 05	1.773500E − 05
(0.25, 0.75)	1.408702E − 04	1.408702E − 04	1.408702E − 04	1.408702E − 04
(0.25, 1.00)	6.217800E − 04	6.217800E − 04	6.217800E − 04	6.217800E − 04
(0.50, 0.25)	2.123000E − 06	2.123000E − 06	2.123000E − 06	2.123000E − 06
(0.50, 0.50)	7.094300E − 05	7.094300E − 05	7.094300E − 05	7.094300E − 05
(0.50, 0.75)	5.634830E − 04	5.634830E − 04	5.634830E − 04	5.634830E − 04
(0.50, 1.00)	2.487123E − 03	2.487123E − 03	2.487123E − 03	2.487123E − 03
(0.75, 0.25)	4.776000E − 06	4.776000E − 06	4.776000E − 06	4.776000E − 06
(0.75, 0.50)	1.596200E − 04	1.596200E − 04	1.596200E − 04	1.596200E − 04
(0.75, 0.75)	1.267830E − 03	1.267830E − 03	1.267830E − 03	1.267830E − 03
(0.75, 1.00)	5.596030E − 03	5.596030E − 03	5.596030E − 03	5.596030E − 03

Figure 5. Comparison of approximate solution of Problem 2 for distinct values ζ w.r.t. the exact solution and absolute error $|{\nu }_{\mathit{exa}.}(\eta ,\tau )-{\nu }_{\mathit{app}.}(\eta ,\tau )|$ with n = 1, $\zeta =1,,$ and $\hslash =-1..$

Figure 6. Comparison between the approximate solution at $\zeta \le 1$ and the exact solution with $\hslash =-1,$ n = 1, and η = 1 for Problem 2.

Figure 7. n-curves of $\nu (\eta ,\tau )$ for distinct values of ζ with $\hslash =-1,\eta =0.1,$ and $\tau =0.5$ for Problem 2.

Figure 8. The $\hslash$-curves of $\nu (\eta ,\tau )$ for distinct values n with $\eta =0.1,$ and $\tau =0.5$ for Problem 2.

Wang, L., Wu, Y., Ren, Y., & Chen, X. (2019). Two analytical methods for fractional partial differential equations with proportional delay. IAENG International Journal of Applied Mathematics, 49(1), 1–6.

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Singh, B. K., & Kumar, P. (2018). Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay. SeMA Journal, 75(1), 111–125. doi:10.1007/s40324-017-0117-1

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