Application of He's Homotopy Perturbation Method for Cauchy Problem of Ill-Posed Nonlinear Diffusion Equation

Table 1

(a) Exact solution, approximate solution, absolute error, and relative error of 𝑢(𝑥,𝑡) at the time 𝑡 = 0.25

𝑥	Exact solution	Approximate solution	Absolute error	Relative error
0.1	0.2628402542	0.2628402542	0.00	0.00
0.2	0.3013610167	0.3013610083	8.20×10^−9	2.75×10^−8
0.3	0.3655622875	0.3655622312	5.62×10^−8	1.54×10^−7
0.4	0.4554440667	0.4554438471	2.20×10^−7	4.81×10^−7
0.5	0.5710063542	0.5710057031	6.51×10^−7	1.14×10^−6
0.6	0.7122491501	0.7122475258	1.62×10^−6	2.28×10^−6
0.7	0.8791724543	0.8791688716	3.58×10^−6	4.07×10^−6
0.8	1.071776267	1.071769074	7.19×10^−6	6.71×10^−6
0.9	1.290060588	1.290047189	1.34×10^−6	1.03×10^−5
1	1.534025417	1.534001959	2.34×10^−5	1.52×10^−5

(b) Exact solution, approximate solution, absolute error, and relative error of 𝑢(𝑥,𝑡) at the time 𝑡 = 0.50

𝑥	Exact solution	Approximate solution	Absolute error	Relative error
0.1	0.5164872127	0.5164872161	3.40×10^−9	6.19×10^−9
0.2	0.5659488508	0.5659488481	2.70×10^−9	4.77×10^−9
0.3	0.6483849144	0.6483848411	7.33×10^−8	1.13×10^−7
0.4	0.7637954034	0.7637950723	3.31×10^−7	4.33×10^−7
0.5	0.9121803178	0.9121793115	1.01×10^−6	1.10×10^−6
0.6	1.093539658	1.093537168	2.49×10^−6	2.27×10^−6
0.7	1.307873423	1.307868043	5.38×10^−6	4.11×10^−6
0.8	1.555181613	1.555171074	1.05×10^−5	6.77×10^−6
0.9	1.835464230	1.835445111	1.91×10^−5	1.04×10^−5
1	2.148721271	2.148688717	3.26×10^−5	1.51×10^−5

(c) Exact solution, approximate solution, absolute error, and relative error of 𝑢(𝑥,𝑡) at the time 𝑡 = 0.75

𝑥	Exact solution	Approximate solution	Absolute error	Relative error
0.1	0.7711700002	0.7711700127	1.19×10^−8	1.54×10^−8
0.2	0.8346800007	0.8346800302	2.88×10^−8	3.45×10^−8
0.3	0.9405300015	0.9405299770	2.45×10^−8	2.60×10^−8
0.4	1.088720003	1.088719693	3.11×10^−7	2.85×10^−7
0.5	1.279250004	1.279248891	1.11×10^−6	8.70×10^−7
0.6	1.512120006	1.512117104	2.90×10^−6	1.91×10^−6
0.7	1.787330008	1.787323655	6.35×10^−6	3.55×10^−6
0.8	2.104880011	2.104867651	1.24×10^−5	5.87×10^−6
0.9	2.464770014	2.464748039	2.20×10^−5	8.91×10^−6
1	2.867000017	2.866963750	3.62×10^−5	1.26×10^−5

(d) Exact solution, approximate solution, absolute error, and relative error of 𝑢(𝑥,𝑡) at the time 𝑡 = 1

𝑥	Exact solution	Approximate solution	Absolute error	Relative error
0.1	1.027182818	1.027182849	3.07×10^−8	2.98×10^−8
0.2	1.108731273	1.108731374	1.00×10^−7	9.04×10^−8
0.3	1.244645364	1.244645495	1.31×10^−7	1.04×10^−7
0.4	1.434925092	1.434925050	4.23×10^−8	2.94×10^−8
0.5	1.679570457	1.679569755	7.01×10^−7	4.17×10^−7
0.6	1.978581458	1.978579197	2.26×10^−6	1.14×10^−6
0.7	2.331958096	2.331952871	5.22×10^−6	2.24×10^−6
0.8	2.739700370	2.739690321	1.00×10^−5	3.66×10^−6
0.9	3.201808281	3.201791426	1.69×10^−5	5.26×10^−6
1	3.718281828	3.718256914	2.49×10^−5	6.70×10^−6