ANALYSING PDE USING SVD, DMD, SEPARATION OF VARIABLES, ROM AND ITS APPLICATION ON LANDSLIDE PROBLEM

Abstract

Nature has given a wide variety of things to observe and permits to develop a logical and in depth analysis of the phenomenon that can take even a life time to understand the iota of its significance. Scientists over the years have dedicated their life in concluding their observations and illustrate their findings of similar phenomenon to the world. As a baby step towards analysing such natural existence, the Kortewegh – da – vries equation that led to the phenomenon of the so called “Soliton” or Solitary Waves was first undertaken. The thesis is based on the analysis and solving the KDV equations as step one and then escalating the level with solution of wave equations, Berger’s equation and other PDE’s. We during the course of our thesis have applied the Artificial Intelligence and Machine Learning Models for solving the Partial Differential equations, but this time not with the FFT but through SVD, Dimensionality Reduction etc. As a Step two of the thesis, the separation of variables method for finding out the solution to the PDE’s was explored and compared with SVD. The legacy models for solving the partial differential equations utilized Fourier modes including Separation of variables that although discretized the data but still had the complexity of the order of (nlogn). We then explored how to numerically solve the complex Partial differential equations using SVD that proved be a great enabler in finding out the solution with fewer number of modes. We used python and MATLAB ODE45 for stepping the solution along the time axis for a given initial condition.