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- W2912155798 abstract "Every applied mathematician has used separation of variables. So begins “Synthesis, as Opposed to Separation, of Variables” by A. S. Fokas and E. A. Spence. As the authors note, separation of variables dates back to the 1750s and is a standard staple within any undergraduate partial differential equation course. Its name, separation of variables, captures its essence and at first glance evokes a sense of simplicity. The central idea is to look for solutions which are products of functions of individual variables. In this way, a partial differential equation is converted to a pair of ordinary differential equations. Of course, the devil is always in the details and in this context the details revolve around boundary conditions, initial values, non-self-adjoint operators, and the convergence of series or integrals. The linearized Korteweg–de Vries (KdV) equation and the Helmholtz equation are both considered. Limitations associated with a classical separation of variables approach are first presented as well as some attempts to address these for Helmholtz operators. For high frequencies, the article concludes that “the angular series expansion obtained by Lord Rayleigh is correct but useless, and the radial series expansion obtained by Watson and Sommerfeld is useful but incorrect!” The paper describes an alternative transform method that can be used for certain nonseparable and non-self-adjoint problems. This transform is motivated by observations associated with classical approaches and complex variables. Specifically, “the classical approach begins in the complex plane $ldots,$ abandons the complex plane $ldots,$ and sometimes returns to the complex plane.” The new method is instead based on remaining in the complex plane while solving the boundary value problem and is more a synthesis of variables (via parametrization) than a separation. This paper provides a truly interesting perspective on a classical topic, separation of variables, as well as highlighting an intriguing alternative which can address some of its limitations." @default.
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- W2912155798 date "2012-01-01" @default.
- W2912155798 modified "2023-09-26" @default.
- W2912155798 title "Research Spotlights" @default.
- W2912155798 doi "https://doi.org/10.1137/siread000054000002000289000001" @default.
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