Scientific Reports

Table 5 Numerical results for \(\Phi (\chi ,\tau )\) and \(\Psi (\chi ,\tau )\) in Problem 1 with respect to Caputo, \(\texttt{CF}\) and \(\texttt{AB}\) derivatives and different values of \(\mu\).

From: Analytical solutions for the Noyes Field model of the time fractional Belousov Zhabotinsky reaction using a hybrid integral transform technique

\(\chi\)	\(\tau\)	\(\mu =0.25\)			\(\mu =0.5\)			\(\mu =0.75\)			\(\mu =1\)
\(\chi\)	\(\tau\)	\(\Phi ^{\texttt{C}}(\chi ,\tau )\)	\(\Phi ^{\texttt{CF}}(\chi ,\tau )\)	\(\Phi ^{\texttt{AB}}(\chi ,\tau )\)	\(\Phi ^{\texttt{C}}(\chi ,\tau )\)	\(\Phi ^{\texttt{CF}}(\chi ,\tau )\)	\(\Phi ^{\texttt{AB}}(\chi ,\tau )\)	\(\Phi ^{\texttt{C}}(\chi ,\tau )\)	\(\Phi ^{\texttt{CF}}(\chi ,\tau )\)	\(\Phi ^{\texttt{AB}}(\chi ,\tau )\)	\(\Phi ^{\texttt{C}}(\chi ,\tau )\)	\(\Phi ^{\texttt{CF}}(\chi ,\tau )\)	\(\Phi ^{\texttt{AB}}(\chi ,\tau )\)	Exact
0	0.01	0.49636	0.04699	− 0.33421	0.32729	0.50549	0.47244	0.27207	0.43867	0.45131	0.25629	0.25629	0.25629	0.25629
	0.03	0.53178	0.02843	− 0.49065	0.38934	0.50083	0.4291	0.30183	0.44907	0.47258	0.26909	0.26909	0.26909	0.26909
	0.05	0.52374	0.0095	-0.58715	0.43211	0.4955	0.38948	0.32767	0.4591	0.48851	0.28218	0.28218	0.28218	0.28218
1	0.01	0.36338	0.98654	1.10328	0.16017	0.59615	0.67451	0.12233	0.27886	0.297	0.11276	0.11276	0.11276	0.11276
	0.03	0.49639	0.99393	1.13589	0.21114	0.61122	0.73721	0.14143	0.29357	0.33074	0.12043	0.12043	0.12043	0.12043
	0.05	0.57952	1.00122	1.15294	0.25308	0.6262	0.77981	0.15919	0.30846	0.35975	0.12849	0.12849	0.12849	0.12848
2	0.01	0.23224	1.99389	2.78488	0.0623	0.63036	0.81349	0.04397	0.1521	0.16937	0.03981	0.03981	0.03981	0.03981
	0.03	0.41410	2.03510	3.08512	0.09162	0.6632	0.98328	0.05267	0.16597	0.20353	0.04311	0.04311	0.04311	0.04311
	0.05	0.56328	2.07661	3.26549	0.11996	0.69671	1.11284	0.06125	0.18044	0.23508	0.04665	0.04665	0.04665	0.04665
3	0.01	0.10794	1.36783	1.97655	0.02012	0.36926	0.49644	0.01339	0.0644	0.07387	0.01198	0.01198	0.01198	0.01198
	0.03	0.21988	1.39918	2.21076	0.03225	0.39181	0.61678	0.01644	0.07197	0.09303	0.01309	0.01309	0.01309	0.01309
	0.05	0.31780	1.43081	2.35214	0.04520	0.41491	0.70988	0.01958	0.07996	0.11119	0.0143	0.01430	0.01430	0.01430
4	0.01	0.03797	0.56269	0.82203	0.00573	0.14370	0.19613	0.00367	0.02165	0.02522	0.00326	0.00326	0.00326	0.00326
	0.03	0.08210	0.57600	0.92221	0.00970	0.15296	0.24607	0.00458	0.02450	0.03252	0.00359	0.00359	0.00359	0.00359
	0.05	0.12162	0.58943	0.98278	0.01416	0.16246	0.28486	0.00553	0.02752	0.03950	0.00394	0.00394	0.00394	0.00394

\(\chi\)	\(\tau\)	\(\mu =0.25\)			\(\mu =0.5\)			\(\mu =0.75\)			\(\mu =1\)
\(\chi\)	\(\tau\)	\(\Psi ^{\texttt{C}}(\chi ,\tau )\)	\(\Psi ^{\texttt{CF}}(\chi ,\tau )\)	\(\Psi ^{\texttt{AB}}(\chi ,\tau )\)	\(\Psi ^{\texttt{C}}(\chi ,\tau )\)	\(\Psi ^{\texttt{CF}}(\chi ,\tau )\)	\(\Psi ^{\texttt{AB}}(\chi ,\tau )\)	\(\Psi ^{\texttt{C}}(\chi ,\tau )\)	\(\Psi ^{\texttt{CF}}(\chi ,\tau )\)	\(\Psi ^{\texttt{AB}}(\chi ,\tau )\)	\(\Psi ^{\texttt{C}}(\chi ,\tau )\)	\(\Psi ^{\texttt{CF}}(\chi ,\tau )\)	\(\Psi ^{\texttt{AB}}(\chi ,\tau )\)
0	0.01	−0.50364	−0.95301	−1.33421	−0.67271	−0.49451	−0.52756	−0.72793	−0.56133	−0.54869	−0.74371	−0.74371	−0.74371	−0.74371
	0.03	−0.46822	−0.97157	−1.49065	−0.61066	−0.49917	−0.5709	−0.69817	−0.55093	−0.52742	−0.73091	−0.73091	−0.73091	−0.73091
	0.05	−0.47626	−0.99050	−1.58715	−0.56789	−0.50450	−0.61052	−0.67233	−0.54090	−0.51149	−0.71782	−0.71782	−0.71782	−0.71782
1	0.01	−0.63662	−0.01346	0.10328	−0.83983	−0.40385	−0.32549	−0.87767	−0.72114	−0.70300	−0.88724	−0.88724	−0.88724	−0.88724
	0.03	−0.50361	−0.00607	0.13589	−0.78886	−0.38878	−0.26279	−0.85857	−0.70643	−0.66926	−0.87957	−0.87957	−0.87957	−0.87957
	0.05	−0.42048	0.00122	0.15294	−0.74692	−0.3738	−0.22019	−0.84081	−0.69154	−0.64025	0.87151	−0.87151	−0.87151	−0.87152
2	0.01	−0.76776	0.99389	1.78488	−0.9377	−0.36964	−0.18651	−0.95603	−0.8479	−0.83063	−0.96019	−0.96019	−0.96019	−0.96019
	0.03	−0.58590	1.03510	2.08512	−0.90838	−0.3368	−0.01672	−0.94733	−0.83403	−0.79647	−0.95689	−0.95689	−0.95689	−0.95689
	0.05	−0.43672	1.07661	2.26549	−0.88004	−0.30329	0.11284	−0.93875	−0.81956	−0.76492	−0.95335	−0.95335	−0.95335	−0.95335
3	0.01	−0.89206	0.36783	0.97655	−0.97988	−0.63074	−0.50356	−0.98661	−0.9356	−0.92613	−0.98802	−0.98802	−0.98802	−0.98802
	0.03	−0.78012	0.39918	1.21076	−0.96775	−0.60819	−0.38322	−0.98356	−0.92803	−0.90697	−0.98691	−0.98691	−0.98691	−0.98691
	0.05	−0.68220	0.43081	1.35214	−0.9548	−0.58509	−0.29012	−0.98042	−0.92004	−0.88881	−0.98570	−0.98570	−0.98570	−0.98570
4	0.01	−0.96203	−0.43731	−0.17797	−0.99427	−0.8563	−0.80387	−0.99633	−0.97835	−0.97478	−0.99674	−0.99674	−0.99674	−0.99674
	0.03	−0.91790	−0.42400	−0.07779	−0.99030	−0.84704	−0.75393	−0.99542	−0.97550	−0.96748	−0.99641	−0.99641	−0.99641	−0.99641
	0.05	−0.87838	−0.41057	−0.01722	−0.98584	−0.83754	−0.71514	−0.99447	−0.97248	−0.96050	−0.99606	−0.99606	−0.99606	−0.99606

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