Numerical Inversion of the Laplace Transform: The Fourier Series Approximation

This Demonstration shows how you can numerically compute the inverse of the Laplace transform of a simple function : and . The selected method is the Fourier series approximation. This method uses the following formula in order to perform the inversion of :

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You can select the appropriate values of and that give the correct inverse. This choice must be such that and , where is a measure of the maximum relative error and is the exponential order of .

The red curve is the sine function and the blue dots are the selected numerical values of the inverse of .

You can clearly see how this method may fail to give an accurate inverse if the values of and are not correctly selected. The first snapshot presents a correct inversion result. The next two snapshots show situations where the method gives erroneous data.