The current paper is concerned with a modified Homotopy perturbation technique.This modification allows achieving an exact solution of an initial value problem of the fractional differential equation.The approach is p...The current paper is concerned with a modified Homotopy perturbation technique.This modification allows achieving an exact solution of an initial value problem of the fractional differential equation.The approach is powerful,effective,and promising in analyzing some classes of fractional differential equations for heat conduction problems and other dynamical systems.To crystallize the new approach,some illustrated examples are introduced.展开更多
There are few numerical techniques available to solve the Bagley-Torvik equation which occurs considerably frequently in various offshoots of applied mathematics and mechanics. In this paper, we show that Chelyshkov-t...There are few numerical techniques available to solve the Bagley-Torvik equation which occurs considerably frequently in various offshoots of applied mathematics and mechanics. In this paper, we show that Chelyshkov-tau method is a very effective tool in numerically solving this equation. To show the accuracy and the efficiency of the method, several problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that Chelyshkov-tau method is superior to other existing ones and is highly accurate.展开更多
In this paper,two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect...In this paper,two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function.Using the composite Boole's rule the distributedorder Bagley-Torvik equation is approximated by a multi-term time-fractional equation,which is then solved by the GrunwaldLetnikov method(GLM)and the fractional differential transform method(FDTM).Finally,we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.展开更多
In this study,we present a numerical scheme similar to the Galerkin method in order to obtain numerical solutions of the Bagley Torvik equation of fractional order 3/2.In this approach,the approximate solution is assu...In this study,we present a numerical scheme similar to the Galerkin method in order to obtain numerical solutions of the Bagley Torvik equation of fractional order 3/2.In this approach,the approximate solution is assumed to have the form of a polynomial in the variable t=xα,whereαis a positive real parameter of our choice.The problem is firstly expressed in vectoral form via substituting the matrix counterparts of the terms present in the equation.After taking inner product of this vector with nonnegative integer powers of t up to a selected positive parameter N,a set of linear algebraic equations is obtained.After incorporation of the boundary conditions,the approximate solution of the problem is then computed from the solution of this linear system.The present method is illustrated with two examples.展开更多
In this paper, a new methodology of fractional derivatives based upon Hermite polynomial is projected. The fractional derivatives are demonstrated according to Caputo sense. Hermite collocation technique is introduced...In this paper, a new methodology of fractional derivatives based upon Hermite polynomial is projected. The fractional derivatives are demonstrated according to Caputo sense. Hermite collocation technique is introduced to express the definite results of Bagley-Torvik Equations. The appropriateness and straightforwardness of numerical plan is presented by graphs and error tables.展开更多
In this study,nonhomogeneous differential equation of the second order is considered,which contains fractional derivative(Bagley-Torvik equation),where the derivative order ranges within 1 and 2.This equation is appli...In this study,nonhomogeneous differential equation of the second order is considered,which contains fractional derivative(Bagley-Torvik equation),where the derivative order ranges within 1 and 2.This equation is applied in mechanics of oscillation processes.To study the equation,we use the Laplace transform,which allows us to obtain an image of the solution in an explicit form.Two typical kinds of functions of the right-hand side of the equation are considered.Numerical solutions are constructed for each of them.The solutions obtained are compared with experimental information on polymer concrete samples.The comparison allows for the conclusion about the adequacy of the numerical and analytical solutions to the nonhomogeneous Bagley-Torvik equation.展开更多
基金The National Natural Science Foundations of China(11601409)the Natural Science Foundation of Shaanxi Province(2016JM1009)+3 种基金the Science Foundation of the Education Department of Shaanxi Province(17JK0344)the National Natural Science Foundation of Zhejiang Province(LY14A010007LQ14G010002)the Natural Science Foundation of Ningbo(2015A610173)
文摘The current paper is concerned with a modified Homotopy perturbation technique.This modification allows achieving an exact solution of an initial value problem of the fractional differential equation.The approach is powerful,effective,and promising in analyzing some classes of fractional differential equations for heat conduction problems and other dynamical systems.To crystallize the new approach,some illustrated examples are introduced.
文摘There are few numerical techniques available to solve the Bagley-Torvik equation which occurs considerably frequently in various offshoots of applied mathematics and mechanics. In this paper, we show that Chelyshkov-tau method is a very effective tool in numerically solving this equation. To show the accuracy and the efficiency of the method, several problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that Chelyshkov-tau method is superior to other existing ones and is highly accurate.
文摘In this paper,two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function.Using the composite Boole's rule the distributedorder Bagley-Torvik equation is approximated by a multi-term time-fractional equation,which is then solved by the GrunwaldLetnikov method(GLM)and the fractional differential transform method(FDTM).Finally,we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.
文摘In this study,we present a numerical scheme similar to the Galerkin method in order to obtain numerical solutions of the Bagley Torvik equation of fractional order 3/2.In this approach,the approximate solution is assumed to have the form of a polynomial in the variable t=xα,whereαis a positive real parameter of our choice.The problem is firstly expressed in vectoral form via substituting the matrix counterparts of the terms present in the equation.After taking inner product of this vector with nonnegative integer powers of t up to a selected positive parameter N,a set of linear algebraic equations is obtained.After incorporation of the boundary conditions,the approximate solution of the problem is then computed from the solution of this linear system.The present method is illustrated with two examples.
文摘In this paper, a new methodology of fractional derivatives based upon Hermite polynomial is projected. The fractional derivatives are demonstrated according to Caputo sense. Hermite collocation technique is introduced to express the definite results of Bagley-Torvik Equations. The appropriateness and straightforwardness of numerical plan is presented by graphs and error tables.
文摘In this study,nonhomogeneous differential equation of the second order is considered,which contains fractional derivative(Bagley-Torvik equation),where the derivative order ranges within 1 and 2.This equation is applied in mechanics of oscillation processes.To study the equation,we use the Laplace transform,which allows us to obtain an image of the solution in an explicit form.Two typical kinds of functions of the right-hand side of the equation are considered.Numerical solutions are constructed for each of them.The solutions obtained are compared with experimental information on polymer concrete samples.The comparison allows for the conclusion about the adequacy of the numerical and analytical solutions to the nonhomogeneous Bagley-Torvik equation.
