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2 Basic Definitions     continued...

The distinction between linear and nonlinear PDEs is extremely important in computational science. Many linear PDE problems can be solved exactly using techniques such as separation of variables, superposition, Fourier series, Laplace transform and Fourier transform. Exact solutions are valuable in a computational setting because they can be used to assist the computational scientist in the often difficult exercise of code validation. Generally, nonlinear PDEs do not yield to analytical solution approaches. Since most leading edge work in computational science involves nonlinear PDEs, a great deal of effort is directed toward obtaining numerical solutions. Whenever possible, computational scientists draw from the field of linear PDEs for guidance and insight in developing numerical methods for the more difficult nonlinear PDEs.