Tests for accuracy of den Iseger’s algorithm
To ensure usefulness for the considered problem of den Iseger’s algorithm as well as its precision we made tests for the following test functions:
\(y\left(s\right)=\frac{1}{s^{2}+s+1}\) (A1)
\(y\left(s\right)=\frac{2\bullet s}{{(s^{2}+1)}^{2}}\) (A2)
\(y\left(s\right)=\frac{1}{\sqrt{s}+\sqrt{s+1}}\) (A3)
\(y\left(s\right)=\frac{1}{4}\frac{25\bullet sinh(4\bullet\sqrt{\frac{s}{2}})}{sinh(5\sqrt{\frac{s}{2}})\bullet s}\)(A4)
Corresponding to them functions in time domain are given by eq. (A5)-(A8):
\(y\left(t\right)=\frac{2}{3}\bullet\sqrt{3}\bullet exp(-\frac{1}{2}\bullet\sqrt{3}\bullet t)\bullet sin(\frac{1}{2}\bullet\sqrt{3}\bullet t)\)(A5)
\(y\left(t\right)=t\bullet\sin{(t)}\) (A6)
\(y\left(t\right)=\frac{1-exp(-t)}{\sqrt{4\bullet\pi\bullet t^{3}}}\)(A7)
\(y\left(t\right)=5+\frac{50}{4\pi}\bullet\sum_{n=1}^{\infty}\left(\frac{\left(-1\right)^{n}}{n}\bullet\sin\left(\frac{4n\bullet\pi}{5}\right)\bullet\exp\left(\frac{{2n}^{2}\bullet\pi^{2}}{25}\bullet t\right)\right)\)(A8)
A comment is required for equations (A4) and (A8). Eq. (A4) is a Laplace domain solution the following diffusion equation:
\(\frac{\partial C(r,t)}{\partial t}=a\bullet\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\bullet\frac{\partial C\left(r,t\right)}{\partial r})\)(A9)
with appropriate initial and boundary conditions
\(C\left(r,0\right)=0\) (A10)
\(\frac{\partial C(r,t)}{\partial r}|_{r=0}=0\) (A11)
\(C\left(r=R,t\right)=C_{0}\) (A12)
Analytic solution of the presented initial-boundary value problem is as follows:
\(C\left(r,t\right)=C_{0}+\frac{2\bullet R\bullet C_{0}}{\pi\bullet r}\bullet\sum_{n=1}^{\infty}\left(\frac{\left(-1\right)^{n}}{n}\bullet\sin\left(\frac{n\bullet\pi\bullet r}{R}\right)\bullet\exp\left(\frac{n^{2}\bullet\pi^{2}}{R^{2}}\bullet a\bullet t\right)\right)\)(A13)
It is easily to check that eq.(A8) for a=2, R=5, C0=5 and r=4 corresponds to eq.(A13)
The tests were carried out for two different time steps\(\frac{t_{m}}{N}\) (N=32 or N=128). Results are presented in Table A1.
Table A1. Standard deviation between numerical and analytic solutions and relative time of computations