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

Table 1

Mean square errors (MSEs) with different values of 𝑝, 𝐻 given 𝑛=20, 30, 50 in Example 4.1, 𝐼₂: unit matrix of two orders.

Parameters (𝑝,𝐻𝑖)	MSE (𝑛=20)	MSE (𝑛=30)	MSE (𝑛=50)
𝑝= 3, 𝐻1=(11/100)𝐼2	8.5223×10^−3	6.5223×10^−4	6.3252×10^−5
𝑝=3, 𝐻2=(2/25)𝐼2	3.0309×10^−3	4.2541×10^−4	9.7206×10^−6
𝑝=3, 𝐻3=(1/25)𝐼2	1.1268×10^−4	8.3418×10^−5	6.0211×10^−6
𝑝=3, 𝐻4=(3/100)𝐼2	5.3082×10^−4	5.2087×10^−5	1.1528×10^−6
𝑝=3, 𝐻5=(1/40)𝐼2	6.9123×10^−4	5.6021×10^−5	3.9107×10^−7
𝑝=3, 𝐻6=(1/75)𝐼2	3.3082×10^−3	2.8339×10^−5	1.8815×10^−6
𝑝=4, 𝐻1=(11/100)𝐼2	4.1004×10^−4	6.6033×10^−6	5.8311×10^−7
𝑝=4, 𝐻2=(2/25)𝐼2	1.8791×10^−4	2.2416×10^−6	3.4525×10^−7
𝑝=4, 𝐻3=(1/25)𝐼2	6.6568×10^−5	9.5066×10^−7	3.0373×10^−7
𝑝=4, 𝐻4=(3/100)𝐼2	9.0893×10^−5	7.3137×10^−6	9.7815×10^−8
𝑝=4, 𝐻5=(1/40)𝐼2	5.9803×10^−4	5.6401×10^−6	4.1266×10^−8
𝑝=4, 𝐻6=(1/75)𝐼2	5.7911×10^−4	7.3452×10^−6	2.3098×10^−7