The paper deals with the forced oscillation of the fractional differential equationm-q a x is the Riemann-Liouville fractional integral of order m -q of x, and b k (k = 1, 2, . . . , m) are/is constants/constant. We obtain some oscillation theorems for the equation by reducing the fractional differential equation to the equivalent Volterra fractional integral equation and by applying Young's inequality. We also establish some new oscillation criteria for the equation when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. The results obtained here improve and extend some existing results. An example is given to illustrate our theoretical results. MSC: 34A08; 34C10

Ladle is an important apparatus in metallurgical industry and takes charge of transferring molten steel from a converter to procedure of continuous casting or ingot casting. It not only improves production efficiency, product quality and production flexibility massively, but also decreases energy and material consumption. The ladle lifetime influences economic benefit of iron and steel enterprises directly. Distribution of its temperature field and stress field has a vital effect on the lifetime. Thermal expansion stress is one of important reasons for damage as ladles operate under the condition of high temperature and overload. Under high temperature, thermal expansion is generated in the ladle lining and shell. Thermal expansion stress is introduced when lining and shell deformation arising from thermal expansion is subjected to mutual constraint from the counterpart due to various thermal expansion coefficient. The finite element software is used to establish the ladle model and influence of working lining material thermal conductivity; thermal expansion coefficient, Young's modulus and its thickness on the ladle temperature and stress field are studied. The calculation result indicated the equivalent stress in the ladle enlarged as the thermal conductivity, thermal expansion coefficient, Young's modulus of working-lining material increased and the thickness decreased. Research in this paper plays a referring role in comprehending distribution of temperature field and stress field of the ladle and selecting appropriately physical parameters of lining. The actual operation result indicates that the research increases the ladle lifetime effectively and shows its wide popularization value.

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