Local Polynomial Regression Solution for Partial Differential Equations with Initial and Boundary Values

Table 2

Mean square errors (MSEs) with different values of 𝑝, 𝐻 given 𝑛=30 and absolute errors (AE) given 𝑛=30, 50 in Example 4.2, 𝐼₂: unit matrix of two orders.

Parameters (ℎ, 𝐻𝑖)	MSE (𝑛=30)	AE (𝑥=0.10, 𝑛=30)	AE (𝑥=0.20, 𝑛=50)
𝑝=4, 𝐻1=(11/100)𝐼2,	2.3109×10^−4	4.7882×10^−3	9.4388×10^−5
𝑝=4, 𝐻2=(2/25)𝐼2,	4.1493×10^−4	3.1431×10^−3	4.6412×10^−5
𝑝=4, 𝐻3=(1/25)𝐼2,	3.8981×10^−5	6.9083×10^−4	1.0603×10^−5
𝑝=4, 𝐻4=(3/100)𝐼2,	9.2391×10^−6	1.1425×10^−4	7.5229×10^−6
𝑝=4, 𝐻5=(1/40)𝐼2,	9.0117×10^−6	6.8858×10^−4	9.1029×10^−7
𝑝=4, 𝐻6=(3/100)𝐼2,	7.3249×10^−6	9.4551×10^−5	2.1002×10^−6
𝑝=5, 𝐻1=(11/100)𝐼2,	7.0113×10^−7	8.9211×10^−5	6.8564×10^−6
𝑝=5, 𝐻2=(2/25)𝐼2,	4.5881×10^−7	7.8423×10^−5	5.0705×10^−6
𝑝=5, 𝐻3=(1/25)𝐼2,	2.1062×10^−7	2.6065×10^−5	9.6552×10^−7
𝑝=5, 𝐻4=(3/100)𝐼2,	9.7922×10^−8	5.8232×10^−6	8.2302×10^−7
𝑝=5, 𝐻5=(1/40)𝐼2,	3.8717×10^−6	4.2410×10^−5	3.0671×10^−7
𝑝=5, 𝐻6=(1/75)𝐼2,	8.9821×10^−6	7.9102×10^−5	6.3409×10^−6
Aziz et al. [19] (𝑝,𝑞,𝑠,𝜎,ℎ)		𝑥=0.10,steps=10	𝑥=0.20,steps=16
(0, 0, 1, 1/4, 1/20)	—	1.5000×10^−4	5.1000×10^−5
(0, 1/6, 2/3, 1/4, 1/20)		1.8000×10^−5	8.0000×10^−6
(1/144, 5/36, 17/24, 1/4, 1/20)		1.8000×10^−5	7.9000×10^−6
Evans and Yousif [20]	—	2.2000×10^−4	4.1000×10^−4
		2.5000×10^−5	4.7000×10^−5