### Abstract

Following Burnell et al., the dimensionless form of the steady state heat balance equation for material undergoing an exothermic reaction over the symmetrical N-dimensional unit sphere is u″(r) + (N - 1)r^{-1}u′(r) + λe^{-(1/u)} = 0. The parameter λ is held constant so that the solution structure is dependent on the bifurcation parameter U appearing in the boundary condition u(1) = U. In previous papers we discussed the existence of multiple solutions for both class A (slab, infinite cylinder, and sphere) and nonclass A geometries and showed that multiple solutions (of multiplicity greater that three) occur for 2 <N <12. In this paper, we present numerical results for some hollow geometries which indicate that similar solution structure to that of previous cases is preserved and that the multiplicity of solutions does not always depend on the size of the hollow region.

Original language | English |
---|---|

Pages (from-to) | 41-46 |

Number of pages | 6 |

Journal | Applied Mathematics Letters |

Volume | 10 |

Issue number | 5 |

Publication status | Published - Sep 12 1997 |

### Fingerprint

### Keywords

- Boundary value problems
- Combustion
- Nonlinear systems

### ASJC Scopus subject areas

- Computational Mechanics
- Control and Systems Engineering
- Applied Mathematics
- Numerical Analysis

### Cite this

*Applied Mathematics Letters*,

*10*(5), 41-46.

**Multiple solutions in hollow geometries in the theory of thermal ignition.** / Balakrishnan, E.; Swift, A.; Wake, G. C.

Research output: Contribution to journal › Article

*Applied Mathematics Letters*, vol. 10, no. 5, pp. 41-46.

}

TY - JOUR

T1 - Multiple solutions in hollow geometries in the theory of thermal ignition

AU - Balakrishnan, E.

AU - Swift, A.

AU - Wake, G. C.

PY - 1997/9/12

Y1 - 1997/9/12

N2 - Following Burnell et al., the dimensionless form of the steady state heat balance equation for material undergoing an exothermic reaction over the symmetrical N-dimensional unit sphere is u″(r) + (N - 1)r-1u′(r) + λe-(1/u) = 0. The parameter λ is held constant so that the solution structure is dependent on the bifurcation parameter U appearing in the boundary condition u(1) = U. In previous papers we discussed the existence of multiple solutions for both class A (slab, infinite cylinder, and sphere) and nonclass A geometries and showed that multiple solutions (of multiplicity greater that three) occur for 2 <N <12. In this paper, we present numerical results for some hollow geometries which indicate that similar solution structure to that of previous cases is preserved and that the multiplicity of solutions does not always depend on the size of the hollow region.

AB - Following Burnell et al., the dimensionless form of the steady state heat balance equation for material undergoing an exothermic reaction over the symmetrical N-dimensional unit sphere is u″(r) + (N - 1)r-1u′(r) + λe-(1/u) = 0. The parameter λ is held constant so that the solution structure is dependent on the bifurcation parameter U appearing in the boundary condition u(1) = U. In previous papers we discussed the existence of multiple solutions for both class A (slab, infinite cylinder, and sphere) and nonclass A geometries and showed that multiple solutions (of multiplicity greater that three) occur for 2 <N <12. In this paper, we present numerical results for some hollow geometries which indicate that similar solution structure to that of previous cases is preserved and that the multiplicity of solutions does not always depend on the size of the hollow region.

KW - Boundary value problems

KW - Combustion

KW - Nonlinear systems

UR - http://www.scopus.com/inward/record.url?scp=0011818229&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011818229&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0011818229

VL - 10

SP - 41

EP - 46

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

SN - 0893-9659

IS - 5

ER -