In this paper we have proved that the solution of parabolic equation and its Fast Fourier Transform generate continuous wavelet transforms. Indeed, we have solved the parabolic equation using PDETool, exported its solution and coefficients to Matlab workspace. We have then imported the solution from workspace to signal processing tool. We have sampled the imported solution with the sampling frequency of 8192Hz and applied the band pass filter with that frequency. The convolution of the sampled PDE solution with the impulse response of the band pass filter has generated wavelet transform. This algorithm computes the wavelet transform either directly of via Faster Fourier Transform. The computation of the FFT of the PDE solution has produced complex wavelet.

Significant work has already been done for complex quadratics. However, the dynamics of rational functions and their properties are equally interesting. In this paper we have generated computer images from a C++ computer program. We have then developed an artificial neural network model using predictive modeling software based on RMS type of error out of two samples of points obtained from the generated images. The imaginary part of sample II was predicted by applied the real parts of sample I and sample II to the artificial neural network. The real part of sample II was more important than the real part of sample I in predicting the imaginary part of sample II. The predicted imaginary part of sample II was then imported to Matlab Signal Processing Tool (SPTool) via Matlab workspace. We have applied a stable band pass filter to the model to eliminate noise from it for its analysis. A modulated signal produced reveals that the methodology used shall be applied to explore properties of computer generated images from the generated wavelet. We have further imported the predicted imaginary part of sample II to autoSIGNAL software for time and frequency range analysis of the continuous wavelet transform.