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

Table 3

Comparisons of the exact solution 𝑢(𝑥,𝑡)=2(𝑥−𝑡)(2+sin(𝑥+𝑡)), with 𝑈_5,5(𝑥,𝑡) obtained by classic DTM [5] and 𝑈₅(𝑥,𝑡) obtained by RDTM at some test points (𝑥,𝑡) in Example 9.

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	+0.4591040413	+0.4591115320	7.4906677321𝑒−06	+0.4591040577	1.6393507773𝑒−08
	0.4	−1.0258569894	−1.0257575253	9.9464024681𝑒−05	−1.0257906614	6.6327927673𝑒−05
	0.7	−2.7833269096	−2.7813944067	1.9325029608𝑒−03	−2.7814532650	1.8736446239𝑒−03
	1	−4.6912625375	−4.6755840000	1.5678537548𝑒−02	−4.6756686352	1.5593902330𝑒−02
0.5	0.1	+2.0517139787	+2.0516661667	4.7812049361𝑒−05	+2.0517139939	1.5172903822𝑒−08
	0.4	+0.5566653819	+0.5573213333	6.5595140784𝑒−04	+0.5567258924	6.0510490391𝑒−05
	0.7	−1.1728156344	−1.1698785000	2.9371343869𝑒−03	−1.1711316158	1.6840186272𝑒−03
	1	−2.9974949866	−2.9817708333	1.5724153271𝑒−02	−2.9836944470	1.3800539634𝑒−02
0.8	0.1	+3.8966576735	+3.8929643413	3.6933321451𝑒−03	+3.8966576865	1.3025739598𝑒−08
	0.4	+2.3456312688	+2.3439851520	1.6461167738𝑒−03	+2.3456822669	5.0998086549𝑒−05
	0.7	+0.5994989973	+0.6014245867	1.9255893459𝑒−03	+0.6008906123	1.3916149372𝑒−03
	1	−1.1895390524	−1.1754026667	1.4136385685𝑒−02	−1.1783733000	1.1165752332𝑒−02
1	0.1	+5.2041732481	+5.1881855000	1.5987748111𝑒−02	+5.2041732592	1.1127633925𝑒−08
	0.4	+3.5825396760	+3.5680853333	1.4454342653𝑒−02	+3.5825824990	4.2822997450𝑒−05
	0.7	+1.7949988863	+1.7840151667	1.0983719605𝑒−02	+1.7961453619	1.1464756153𝑒−03
	1	+0.0000000000	−0.0000000000	−0.0000000000	+0.0090050384	9.0050384311𝑒−03