Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM

Table 5

Comparisons of the exact solution 𝑢(𝑥,𝑡)=sinh(𝑡−𝑥), with 𝑈_5,5(𝑥,𝑡) obtained by classic DTM [5] and 𝑈₅(𝑥,𝑡) obtained by reduced DTM at some test points (𝑥,𝑡) in Example 11.

𝑥	𝑡	𝑢(𝑥,𝑡)	Classic DTM [5]		Reduced DTM
			𝑈5,5(𝑥,𝑡)	|𝑢(𝑥,𝑡)−𝑈5,5(𝑥,𝑡)|	𝑈5(𝑥,𝑡)	|𝑢(𝑥,𝑡)−𝑈5(𝑥,𝑡)|
0.2	0.1	−0.1001667500	−0.1001667561	6.0968226301𝑒−09	−0.1001667498	2.5944103810𝑒−10
	0.4	+0.2013360025	+0.2013367851	7.8252557270𝑒−07	+0.2013368189	8.1631586063𝑒−07
	0.7	+0.5210953055	+0.5211116472	1.6341756253𝑒−05	+0.5211117115	1.6406043554𝑒−05
	1	+0.8881059822	+0.8881853333	7.9351145710𝑒−05	+0.8881854339	7.9451747271𝑒−05
0.5	0.1	−0.4107523258	−0.4107529453	6.1950968455𝑒−07	−0.4107523251	7.0149602793𝑒−10
	0.4	−0.1001667500	−0.1001714167	4.6666468226𝑒−06	−0.1001641445	2.6055545215𝑒−06
	0.7	+0.2013360025	+0.2013887643	5.2761781823𝑒−05	+0.2014033478	1.7345289013𝑒−05
	1	+0.5210953055	+0.5215820313	4.8672575625𝑒−04	+0.5216052465	5.0994098756𝑒−04
0.8	0.1	−0.7585837018	−0.7585783977	5.3041062001𝑒−06	−0.7585837006	1.2071608158𝑒−09
	0.4	−0.4107523258	−0.4108535808	1.0125499718𝑒−04	−0.4107476947	4.6310571234𝑒−06
	0.7	−0.1001667500	−0.1002690359	1.4228584682𝑒−04	−0.1000423588	1.2439120515𝑒−04
	1	+0.2013360025	+0.2019546667	6.1866412557𝑒−04	+0.2023226727	9.8667016118𝑒−04
1	0.1	−1.0265167257	−1.0264561563	6.0569458175𝑒−05	−1.0265167241	1.6018943949𝑒−09
	0.4	−0.6366535821	−0.6370106667	3.5708451843𝑒−04	−0.6366473802	6.2019846800𝑒−06
	0.7	−0.3045202934	−0.3051720521	6.5175863619𝑒−04	−0.3043519610	1.6833243259𝑒−04
	1	+0.0000000000	−0.0000000000	−0.0000000000	+0.0013512390	1.3512390404𝑒−03