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2015
This chapter describes a classical technique for constructing series solutions of linear PDE problems. Classical examples like the heat equation, the wave equation and Laplace’s equations are studied in detail.
David F. Griffiths +2 more
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This chapter describes a classical technique for constructing series solutions of linear PDE problems. Classical examples like the heat equation, the wave equation and Laplace’s equations are studied in detail.
David F. Griffiths +2 more
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2016
A large number of real-life problems considered in a variety of disciplines including engineering, physics, economics and biology, are modeled by differential equations .
Marco Baronti +3 more
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A large number of real-life problems considered in a variety of disciplines including engineering, physics, economics and biology, are modeled by differential equations .
Marco Baronti +3 more
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The American Mathematical Monthly, 1968
(1968). Separation of Variables. The American Mathematical Monthly: Vol. 75, No. 8, pp. 844-847.
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(1968). Separation of Variables. The American Mathematical Monthly: Vol. 75, No. 8, pp. 844-847.
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1999
In the previous chapter, some of the mathematical preliminaries were discussed. The principal physical equations which we will consider are those for heat flow, the wave equation and the potential equation of Laplace and are derived in the companion volume, Numerical methods for partial differential equations. These equations are very typical of second-
Gwynne A. Evans +2 more
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In the previous chapter, some of the mathematical preliminaries were discussed. The principal physical equations which we will consider are those for heat flow, the wave equation and the potential equation of Laplace and are derived in the companion volume, Numerical methods for partial differential equations. These equations are very typical of second-
Gwynne A. Evans +2 more
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HUYGENS' PRINCIPLE AND SEPARATION OF VARIABLES
Reviews in Mathematical Physics, 2000We demonstrate a close relation between the algebraic structure of the (local) group of conformal transformations on a smooth Lorentzian manifold [Formula: see text] and the existence of nontrivial hierarchies of wave-type hyperbolic operators satisfying Huygens' principle on [Formula: see text].
Berest, Yuri, Winternitz, Pavel
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2013
Three cardinal linear second-order partial differential equations have collectively driven the development of the entire subject. The first two we have already encountered: The wave equation describes vibrations and waves in continuous media, including sound waves, water waves, elastic waves, electromagnetic waves, and so on.
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Three cardinal linear second-order partial differential equations have collectively driven the development of the entire subject. The first two we have already encountered: The wave equation describes vibrations and waves in continuous media, including sound waves, water waves, elastic waves, electromagnetic waves, and so on.
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On a General Concept for Separation of Variables
SIAM Journal on Mathematical Analysis, 1982A new definition of (generalized) separability for a linear operator L of arbitrary order is given which, when specialized to the Helmholtz operator, includes ordinary separability in nonorthogonal and “heat type” coordinates. On the basis of this definition, a set of commuting operators is explicitly constructed and various relationships between ...
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Separation of Variables in Electromagnetic Theory
Journal of Applied Physics, 1951Separability conditions are obtained for the partial differential equations of electromagnetic theory. Wave guide and antenna problems are expressed in terms of the vector Helmholtz equation, and solutions are indicated by use of the simple method of separation of variables without recourse to Green's functions.
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