In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit n...In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.展开更多
In this article,natural convection of a temperature-sensitive magnetic fluid in a porous media is studied numerically by using lattice Boltzmann method.Results show that the heat transfer decreases when the ball numbe...In this article,natural convection of a temperature-sensitive magnetic fluid in a porous media is studied numerically by using lattice Boltzmann method.Results show that the heat transfer decreases when the ball numbers increase.When the magnetic field is increased,the heat transfer is enhanced;however the average wall Nusselt number increases at small ball numbers but decreases at large ball numbers due to the induced flow being more likely confined near the bottom walls with a high number of obstacles.展开更多
基金supported by the National Natural Science Foundation of China(Grants 11472161,11102102,and 91130017)the Independent Innovation Foundation of Shandong University(Grant 2013ZRYQ002)the Natural Science Foundation of Shandong Province(Grant ZR2014AQ015)
文摘In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.
基金This work was supported by a grant-in-aid for Scientific Research(C)from the Ministry of Education,Culture,Sports,Science and Technology,Japan。
文摘In this article,natural convection of a temperature-sensitive magnetic fluid in a porous media is studied numerically by using lattice Boltzmann method.Results show that the heat transfer decreases when the ball numbers increase.When the magnetic field is increased,the heat transfer is enhanced;however the average wall Nusselt number increases at small ball numbers but decreases at large ball numbers due to the induced flow being more likely confined near the bottom walls with a high number of obstacles.
