A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities
It is well-known that distributional solutions to the Cauchy problem for $u_t + (b(t,x)u)_{x} = 0$ with $b(t,x) = 2H(x-t)$, where H is the Heaviside function, are non-unique. However, it has a unique generalized solution in the sense of Colombeau. The relationship between its generalized solutio...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2011-06-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2011/76/abstr.html |
| _version_ | 1811257854033657856 |
|---|---|
| author | Hideo Deguchi |
| author_facet | Hideo Deguchi |
| author_sort | Hideo Deguchi |
| collection | DOAJ |
| description | It is well-known that distributional solutions to the Cauchy problem for $u_t + (b(t,x)u)_{x} = 0$ with $b(t,x) = 2H(x-t)$, where H is the Heaviside function, are non-unique. However, it has a unique generalized solution in the sense of Colombeau. The relationship between its generalized solutions and distributional solutions is established. Moreover, the propagation of singularities is studied. |
| first_indexed | 2024-04-12T18:04:02Z |
| format | Article |
| id | doaj.art-1f5e682a2724413896b1670e688b4b3b |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-12T18:04:02Z |
| publishDate | 2011-06-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-1f5e682a2724413896b1670e688b4b3b2022-12-22T03:22:02ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912011-06-01201176,125A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularitiesHideo DeguchiIt is well-known that distributional solutions to the Cauchy problem for $u_t + (b(t,x)u)_{x} = 0$ with $b(t,x) = 2H(x-t)$, where H is the Heaviside function, are non-unique. However, it has a unique generalized solution in the sense of Colombeau. The relationship between its generalized solutions and distributional solutions is established. Moreover, the propagation of singularities is studied.http://ejde.math.txstate.edu/Volumes/2011/76/abstr.htmlFirst-order hyperbolic equationdiscontinuous coefficientgeneralized solutions |
| spellingShingle | Hideo Deguchi A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities Electronic Journal of Differential Equations First-order hyperbolic equation discontinuous coefficient generalized solutions |
| title | A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities |
| title_full | A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities |
| title_fullStr | A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities |
| title_full_unstemmed | A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities |
| title_short | A linear first-order hyperbolic equation with discontinuous coefficient: distributional shadows and propagation of singularities |
| title_sort | linear first order hyperbolic equation with discontinuous coefficient distributional shadows and propagation of singularities |
| topic | First-order hyperbolic equation discontinuous coefficient generalized solutions |
| url | http://ejde.math.txstate.edu/Volumes/2011/76/abstr.html |
| work_keys_str_mv | AT hideodeguchi alinearfirstorderhyperbolicequationwithdiscontinuouscoefficientdistributionalshadowsandpropagationofsingularities AT hideodeguchi linearfirstorderhyperbolicequationwithdiscontinuouscoefficientdistributionalshadowsandpropagationofsingularities |