Two Kinds Of Boundary Value Problems Of Fractional Bagley-Torvik Equation

With the development of science and technology,a lot of mathematical and physical problems have attracted scholar's attention.The theory of calcu-lus lays a solid foundation for mathematical simulation of physical problems.Many scholars have investigated deeply the differential and integral equa-tions and got a great deal of results.And as a indispensable part of calculus,the fractional calculus plays a important role in the modelling of mathemati-cal problems.In this paper,we study two kinds of boundary value problems for fractional Bagley-Torvik equations,and give their numerical solutions.Furthermore,the convergence and error estimates of numerical solutions are analyzed.The main conclusions are given as follows:(1)Considering the uncertainty of boundary-value conditions,we give fuzzy boundary-value problem of fractional Bagley-Torvik equation.Under the Caputo' s H-differentiability,the fuzzy Laplace transform is introduced.Firstly,we transform Bagley-Torvik equation into the equivalent one by fuzzy Laplace transformation.Then,we give the series solution of Bagley-Torvik e-quation with the method of Laplace inverse transformation and Mittag-Leffler function.Finally,numerical results are shown to illustrate the obtained solu-tion.(2)Considering the effect of load history on physical quantity in phys-ical mechanics,we give the integral boundary-value conditions of fraction-al Bagely-Torvik equation.Based on the concept of Riemann-Liouville fractional derivative,we transform the integral boundary-value problem in-to a Fredholm integral equation of the second kind.Then,we propose the generalized piecewise Taylor-series expansion method to solve the obtained Fredholm integral equation with weakly singular kernels and the approxi-mate solution is constructed.In addition,its convergence and error estimate is made.Finally,Numerical results are reported to illustrate the obtained the approximate solutions.The obtained result develop not only the content of fractional Bagley-Torvik equations,but also the method of numerical solution for fractional Bagley-Torvik equations with boundary-value conditions.Some theoretical basis for simulating physical mechanics has been provided.