Abstract
The basic partial differential equation types are introduced as models for diffusion, wave, and equilibrium phenomena. The models are treated by the method of separation of variables in the simplest case of rectangular coordinates. Connections with discrete variable models are noted, and some of the mathematical properties of the formal series solutions are discussed.
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Notes
- 1.
In the boundary value problem motivating this discussion, there are physical and mathematical reasons why the constant is negative.
- 2.
The penalty associated with overlooking a possible λ 2-value in the separation process is usually an impasse arising several pages later in the calculation.
- 3.
There is a sleight of hand involved in the counting of solutions here. Because the T n depends only on n 2, taking the X solution as e in θ instead of the sum avoids counting the same form twice.
- 4.
Let τ = κ t, then declare τ pretentious and replace it by (a new) t.
- 5.
This relies on uniqueness of the Fourier coefficients of a function.
- 6.
This is about the limit of what can be solved by hand using elementary means. There are “tricky” ways to diagonalize coefficient matrices of the given form. See Chapter 8.7
- 7.
Note that this is purely a cosmetic effect. If the “unfortunate” sign choice is made, the problem solution procedure will force the conclusion that the function arguments are purely imaginary, and the process will “self-correct.”
- 8.
In a direct attempt, there is no restriction of the sign of the separation constant, and all possibilities (positive, zero, negative) must be entertained at once. Obtaining a “neat” solution expression is then tedious at best.
- 9.
That is, f is absolutely continuous with square-integrable derivative over [0, 2L].
Further Reading
R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics (Prentice-Hall, Englewood Cliffs, 1962)
W.E. Boyce, R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 3rd edn. (Wiley, New York, 1977)
H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd edn. (Oxford University Press, London, 1986)
R.V. Churchill, Fourier Series and Boundary Value Problems, 2nd edn. (McGraw-Hill, New York, 1963)
J.H. Davis, Differential Equations with Maple (Birkhäuser Boston, Boston, 2000)
E.C. Jordan, K.G. Balmain, Electromagnetic Waves and Radiating Systems, 2nd edn. (Prentice-Hall, Englewood Cliffs, 1968)
H. Lewy, An example of a smooth linear partial differential equation without solutions. Ann. Math. 66, 155–158 (1957)
C.C. Lin, L.A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (MacMillan, New York, 1974)
S.V. Marshall, G.G. Skitek, Electromagnetic Concepts and Applications (Prentice-Hall, Englewood Cliffs, 1990)
L.M. Milne-Thompson, Theoretical Hydrodynamics, 4th edn. (MacMillan, London, 1962)
J.A. Sommerfeld, Mechanics of Deformable Bodies (Academic, New York, 1964)
J.A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1964)
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Davis, J.H. (2016). Elementary Boundary Value Problems. In: Methods of Applied Mathematics with a Software Overview. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43370-7_3
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DOI: https://doi.org/10.1007/978-3-319-43370-7_3
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