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Methods of functional approximation for the computer solution of initial value partial differential equation problems provide a device by which these solutions can be approximated by those of initial value problems in sets of ordinary differential equations. A number of ways to achieve this have been suggested, some of them general, some of them utilizing specific properties of the equations at hand. What all these methods have in common is the fact that the solution u(x, t) of a partial differential equation in space x and time t is approximated by a function u*(a1(t), a2(t), . . ., aN(t)), where the dependence upon x is prescribed. In most applications, u* is linear in the ai(t), i.e., u*= =ai(t)fi(x). The ai(t) satisfy a set of ordinary differential equations obtained as the result of the approximation process. This system of ordinary differential equations is then integrated by classical analog or digital computer methods.
Analog computers, computer integration, functional approximations, Hermitian problems, hybrid computers, initial value problems, integral transform methods, numerical solution, partial differential equations, Ritz?Galerkin method, Sturm?Liouville transform methods.

R. Vichnevetsky, "Use of Functional Approximation Methods in the Computer Solution of Initial Value Partial Differential Equation Problems," in IEEE Transactions on Computers, vol. 18, no. , pp. 499-512, 1969.
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