Solving and Analyzing Partial Differential Equations Using the Laplace Transform

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University of New Brunswick
Given a partial differential equation in two variables, it is possible to derive whether the solutions to said differential equation exhibit exponential unboundedness. In order to determine this, one may find the Laplace transform of the solution — which may be easier to find than the solution itself — and then find the singularities of the analytic continuation of this transform. If any of the singularities have positive real part, there will exist an instability for generic initial conditions. This thesis is dedicated to deriving an algorithm to find the Laplace transforms of solutions to partial differential equations of arbitrary order with constant coefficients. This algorithm allows for the question of boundedness of solutions to a particular class of partial differential equations to be reduced to the question of invertibility of an infinite-dimensional matrix.