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SO 42-/TiO 2 -ZrO 2 (STZ) solid superacid catalysts were prepared using the coprecipitation-impregnation method and characterized by FT-IR, XRD, NH 3 -TPD, N 2 adsorption-desorption isotherms, and SEM. It was observed that the sulfur content and the specific surface area of STZ changes with the ZrO 2 content, and passes through a maximum at Zr/Ti = 1/4. The results indicate that the molar ratio of Zr/Ti and nonionic surfactant, PEG, were helpful for increasing the surface areas and stabilizing the sulfate species on the surfaces of the catalysts. The catalytic activity results were supported by the hydrolysis reaction rate of 2-methyl-1,3-dioxane (2MD) that was strongly dependent on the sulfur content in the zirconia promoted sulfated titania catalyst.

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<abstract><p>This paper discusses a class of fractional Langevin equations on a star graph with mixed boundary conditions. Using Schaefer's fixed point theorem and Banach contraction mapping principle, the existence and uniqueness of solutions are established. Finally, two examples are constructed to illustrate the application of the obtained results. This study provides new results that enrich the existing literature on the fractional boundary value problem for graphs.</p></abstract>