Abstract
Integration is just a special case of the solution of differential equations. Similarly to numerical integration, the approximate solution of differential equations is of great importance in, among other subjects, physics, engineering and chemistry. We give a short introduction to this subject.
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Notes
- 1.
The results of this chapter remain valid for τ′ < τ with obvious modifications.
- 2.
We assume that the one-sided derivatives exist and are continuous at the exceptional points.
- 3.
It is sufficient to assume the mere continuity if \(\mathop{\mathrm{dim}}\nolimits X < \infty \) because we may assume U to be compact and then Heine’s theorem implies the uniform continuity.
- 4.
We recall that nh = τ′ −τ and that (1 + x∕n)n < e x for all x > 0.
- 5.
In fact, it is exactly one: see Exercise 12.1 below, p. 295.
- 6.
In fact, it is exactly two: see Exercise 12.2 below, p. 295.
- 7.
- 8.
This proposition is closely related to the definition of the spectral radius in Exercise 12.3, p. 295.
- 9.
See Sect. 12.3.
- 10.
Using the linear operator A h of Sect. 12.3, now defined on L h , we have u(k τ, x) = (A h k g)(x).
- 11.
This assumption is justified by physical motivations; a 2 is called the diffusion coefficient.
- 12.
Note that ρ(A) ≤ A by definition.
- 13.
There is a natural physical interpretation of this property: if the boundary of a body is maintained at a constant temperature, then its temperature at any point always remains between the minimal and maximal temperatures of the body at the initial moment.
Bibliography
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L.C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1999.
R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, Addison Wesley Longman, 1970.
I.G. Petrovsky, Lectures on Partial Differential Equations, Dover, New York, 1991.
S.L. Sobolev, Partial Differential Equations of Mathematical Physics, Dover, New York, 1989.
A.N. Tikhonov, A.A. Samarskii, Equations of Mathematical Physics, Dover, New York, 1990.
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Komornik, V. (2017). Numerical Solution of Differential Equations. In: Topology, Calculus and Approximation. Springer Undergraduate Mathematics Series. Springer, London. https://doi.org/10.1007/978-1-4471-7316-8_12
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