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

Table 4

Comparisons of the exact solution 𝑢(𝑥,𝑡)=sin(𝑥+𝑡), 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 10.

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	0.2955202067	0.2955202184	1.1688660428𝑒−08	0.2955202070	2.9532337686𝑒−10
	0.4	0.5646424734	0.5646439552	1.4818049646𝑒−06	0.5646439183	1.4448767682𝑒−06
	0.7	0.7833269096	0.7833750550	4.8145422517𝑒−05	0.7833749959	4.8086256376𝑒−05
	1	0.9320390860	0.9325020000	4.6291403277𝑒−04	0.9325019239	4.6283789648𝑒−04
0.5	0.1	0.5646424734	0.5646461680	3.6945737146𝑒−06	0.5646424741	6.8315986201𝑒−10
	0.4	0.7833269096	0.7833397500	1.2840372517𝑒−05	0.7833299139	3.0042708277𝑒−06
	0.7	0.9320390860	0.9321460859	1.0699997027𝑒−04	0.9321309857	9.1899716402𝑒−05
	1	0.9974949866	0.9983398437	8.4485714595𝑒−04	0.9983208230	8.2583639762𝑒−04
0.8	0.1	0.7833269096	0.7834038828	7.6973172517𝑒−05	0.7833269106	1.0099717729𝑒−09
	0.4	0.9320390860	0.9322215424	1.8245643277𝑒−04	0.9320433813	4.2953023213𝑒−06
	0.7	0.9974949866	0.9978859380	3.9095139595𝑒−04	0.9976224907	1.2750404846𝑒−04
	1	0.9738476309	0.9752880000	1.4403691218𝑒−03	0.9749626963	1.1150653929𝑒−03
1	0.1	0.8912073601	0.8915382743	3.3091424412𝑒−04	0.8912073612	1.1792198329𝑒−09
	0.4	0.9854497300	0.9861662222	7.1649223376𝑒−04	0.9854546786	4.9486333631𝑒−06
	0.7	0.9916648105	0.9928385451	1.1737346864𝑒−03	0.9918098802	1.4506974814𝑒−04
	1	0.9092974268	0.9118055556	2.5081287299𝑒−03	0.9105512242	1.2537973843𝑒−03