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Theory and Modern Applications

Table 1 Absolute error of the expectation of \(\pmb{X(t)}\) with the Euler, RK2, and RK4 methods and \(\pmb{h=\frac{1}{20}}\) , \(\pmb{h=\frac{1}{50}}\)

From: Mean square numerical solution of stochastic differential equations by fourth order Runge-Kutta method and its application in the electric circuits with noise

t

Euler

RK2

RK4

\(\boldsymbol {h=\frac{1}{20}}\)

\(\boldsymbol {h=\frac{1}{50}}\)

\(\boldsymbol {h=\frac{1}{20}}\)

\(\boldsymbol {h=\frac{1}{50}}\)

\(\boldsymbol {h=\frac{1}{20}}\)

\(\boldsymbol {h=\frac{1}{50}}\)

0.1

7.548 × 10−3

4.463 × 10−3

5.315 × 10−5

2.421 × 10−5

1.253 × 10−8

1.007 × 10−8

0.2

9.157 × 10−3

6.128 × 10−3

1.173 × 10−4

6.548 × 10−5

2.427 × 10−8

1.568 × 10−8

0.3

1.253 × 10−2

8.421 × 10−3

2.074 × 10−4

1.104 × 10−4

3.523 × 10−8

2.123 × 10−8

0.4

2.107 × 10−2

1.058 × 10−2

3.433 × 10−4

2.352 × 10−4

4.602 × 10−8

3.312 × 10−8

0.5

3.257 × 10−2

2.108 × 10−2

5.541 × 10−4

3.425 × 10−4

5.928 × 10−8

4.986 × 10−8

0.6

4.369 × 10−2

3.249 × 10−2

8.845 × 10−4

5.124 × 10−4

8.100 × 10−8

6.253 × 10−8

0.7

5.578 × 10−2

4.823 × 10−2

1.405 × 10−3

7.461 × 10−4

1.241 × 10−7

8.159 × 10−8

0.8

8.457 × 10−2

6.467 × 10−2

2.229 × 10−3

1.253 × 10−3

2.180 × 10−7

1.109 × 10−7

0.9

1.253 × 10−1

8.812 × 10−2

3.539 × 10−3

1.895 × 10−3

4.209 × 10−7

2.542 × 10−7

1.0

2.346 × 10−1

1.439 × 10−1

5.637 × 10−3

2.764 × 10−3

8.506 × 10−7

4.864 × 10−7