Inverse laplace transform in a physics problem.

Question

This came up during a physics problem, where we need to find the inverse laplace transform of $$X(s) = \left( 1+ \frac{k}{ms^{3/2}}\right)^{-1} \left( \frac{c_1}{s^2} + \frac{c_2}{s} \right)$$ to arrive at a closed form of $x(t)$. Any ideas?

Answer 1

Using partial fractions and Maple I am obtaining

$$x \left( t \right) =c_{{2}}\sqrt [3]{-{\frac {k}{m}}} \left( a-\sqrt [ 3]{-{\frac {k}{m}}} \right) ^{-1} \left( -b+\sqrt [3]{-{\frac {k}{m}}} \right) ^{-1}{\frac {1}{\sqrt {\pi \,t}}}+ \left( \left( b-\sqrt [3] {-{\frac {k}{m}}} \right) \left( c_{{1}}+c_{{2}}{a}^{2} \right) {\it erfc} \left( -a\sqrt {t} \right) {{\rm e}^{t{a}^{2}}}+{{\rm e}^{t{b}^{ 2}}}{\it erfc} \left( -b\sqrt {t} \right) \left( c_{{1}}+c_{{2}}{b}^{ 2} \right) \left( -a+\sqrt [3]{-{\frac {k}{m}}} \right) + \left( a-b \right) \left( c_{{1}}{{\rm e}^{ \left( -{\frac {k}{m}} \right) ^{2/ 3}t}}{\it erfc} \left( -\sqrt [3]{-{\frac {k}{m}}}\sqrt {t} \right) + \\ c _{{2}} \left( \sqrt [3]{-{\frac {k}{m}}}{\frac {1}{\sqrt {\pi \,t}}}+ \left( -{\frac {k}{m}} \right) ^{2/3}{{\rm e}^{ \left( -{\frac {k}{m} } \right) ^{2/3}t}}{\it erfc} \left( -\sqrt [3]{-{\frac {k}{m}}}\sqrt {t} \right) \right) \right) \right) \left( -a+\sqrt [3]{-{\frac {k }{m}}} \right) ^{-1} \left( -b+\sqrt [3]{-{\frac {k}{m}}} \right) ^{-1 } \left( a-b \right) ^{-1} $$

where

$$a=-(1/2)\,\sqrt [3]{-{\frac {k}{m}}}+(1/2)\,\sqrt {-3\, \left( -{\frac {k} {m}} \right) ^{2/3}} $$

$$b=-(1/2)\,\sqrt [3]{-{\frac {k}{m}}}-(1/2)\,\sqrt {-3\, \left( -{\frac {k} {m}} \right) ^{2/3}} $$

A numerical example with $k=1,m=1,c_1=1,c_2=1$ is given by

$$x \left( t \right) ={\frac {- 0.097+ 0.16\,i}{\sqrt {t}}}+ \left( 0.0062- 0.20\,i \right) \left( \left( - 1.5+ 0.86\,i \right) {\it erfc} \left( \left( - 0.49+ 0.87\,i \right) \sqrt {t} \right) { {\rm e}^{ \left( - 0.52- 0.85\,i \right) t}}+ \left( 0.054+ 3.4\,i \right) {{\rm e}^{ \left( 1.0- 0.020\,i \right) t}}{\it erfc} \left( \left( 1.0- 0.01\,i \right) \sqrt {t} \right) + \left( 1.5- 0.87\,i \right) \left( {{\rm e}^{ \left( - 0.51+ 0.86\,i \right) t}} {\it erfc} \left( \left( - 0.51- 0.86\,i \right) \sqrt {t} \right) +{ \frac { 0.29+ 0.48\,i}{\sqrt {t}}}- \left( 0.51- 0.86\,i \right) { {\rm e}^{ \left( - 0.51+ 0.86\,i \right) t}}{\it erfc} \left( \left( - 0.51- 0.86\,i \right) \sqrt {t} \right) \right) \right) $$

and the corresponding curve is