This paper develops a finite approximation approach to find a non-smooth solution of an integral equation of the second kind. The equation solutions with non-smooth kernel having a non-smooth solution have never been studied before. Such equations arise frequently when modeling stochastic systems. We construct a Banach space of (right-continuous) distribution functions and reformulate the problem into an operator equation. We provide general necessary and sufficient conditions that allow us to show convergence of the approximation approach developed in this paper. We then provide two specific choices of approximation sequences and show that the properties of these sequences are sufficient to generate approximate equation solutions that converge to the true solution assuming solution uniqueness and some additional mild regularity conditions. Our analysis is performed under the supremum norm, allowing wider applicability of our results. Worst-case error bounds are also available from solving a linear program. We demonstrate the viability and computational performance of our approach by constructing three examples. The solution of the first example can be constructed manually but demonstrates the correctness and convergence of our approach. The second application example involves stationary distribution equations of a stochastic model and demonstrates the dramatic improvement our approach provides over the use of computer simulation. The third example solves a problem involving an everywhere nondifferentiable function for which no closed-form solution is available.
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On quasisymmetric plasma equilibria sustained by small force
We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the magnetohydrostatic (steady three-dimensional incompressible Euler) equations with a small force. The solutions are also ‘nearly’ quasisymmetric. The primary idea is, given a desired quasisymmetry direction $\xi$ , to change the smooth structure on space so that the vector field $\xi$ is Killing for the new metric and construct $\xi$ –symmetric solutions of the magnetohydrostatic equations on that background by solving a generalized Grad–Shafranov equation. If $\xi$ is close to a symmetry of Euclidean space, then these are solutions on flat space up to a small forcing.
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- Award ID(s):
- 1703997
- NSF-PAR ID:
- 10271655
- Date Published:
- Journal Name:
- Journal of Plasma Physics
- Volume:
- 87
- Issue:
- 1
- ISSN:
- 0022-3778
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0022377820001610
