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Theory and Modern Applications

Table 1 Comparison between analytical and semi-analytical solutions of the Cahn–Allen equation to calculate the values of absolute error, which explains the accuracy of both kinds of solutions

From: Analytical, semi-analytical, and numerical solutions for the Cahn–Allen equation

Value of

Value of exact solution

Value of semi-analytical solution

Value of absolute error

0.001

0.5010000000

0.5010000000

2.661339773964523×10−9

0.002

0.5020000000

0.5020000000

2.1248204400363×10−8

0.003

0.5030000000

0.5030000000

7.15695510822817×10−8

0.004

0.5040000000

0.5040000000

1.6930786361275573×10−7

0.005

0.5050000000

0.5050000000

3.3001991137803534×10−7

0.006

0.5060000000

0.5060000000

5.691375020145864×10−7

0.007

0.5070000000

0.5070000000

9.019682321171425×10−7

0.008

0.5079990000

0.5080000000

1.3436962324583925×10−6

0.009

0.5089990000

0.5090000000

1.9093829096897144×10−6

0.010

0.5099990000

0.5100000000

2.6139676849378895×10−6