This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.
We study two generalized versions of a system of equations which describe the time evolution of the hydrodynamic fluctuations of density and velocity in a linear viscoelastic fluid. In the first of these versions, the...We study two generalized versions of a system of equations which describe the time evolution of the hydrodynamic fluctuations of density and velocity in a linear viscoelastic fluid. In the first of these versions, the time derivatives are replaced by conformable derivatives, and in the second version left-handed Caputo’s derivatives are used. We show that the solutions obtained with these two types of derivatives exhibit significant similarities, which is an interesting (and somewhat surprising) result, taking into account that the conformable derivatives are local operators, while Caputo’s derivatives are nonlocal operators. We also show that the solutions of the generalized systems are similar to the solutions of the original system, if the order α of the new derivatives (conformable or Caputo) is less than one. On the other hand, when α is greater than one, the solutions of the generalized systems are qualitatively different from the solutions of the original system.展开更多
Because charge carriers of many organic semiconductors(OSCs)exhibit fractional drift diffusion(Fr-DD)transport properties,the need to develop a Fr-DD model solver becomes more apparent.However,the current research on ...Because charge carriers of many organic semiconductors(OSCs)exhibit fractional drift diffusion(Fr-DD)transport properties,the need to develop a Fr-DD model solver becomes more apparent.However,the current research on solving the governing equations of the Fr-DD model is practically nonexistent.In this paper,an iterative solver with high precision is developed to solve both the transient and steady-state Fr-DD model for organic semiconductor devices.The Fr-DD model is composed of two fractionalorder carriers(i.e.,electrons and holes)continuity equations coupled with Poisson’s equation.By treating the current density as constants within each pair of consecutive grid nodes,a linear Caputo’s fractional-order ordinary differential equation(FrODE)can be produced,and its analytic solution gives an approximation to the carrier concentration.The convergence of the solver is guaranteed by implementing a successive over-relaxation(SOR)mechanism on each loop of Gummel’s iteration.Based on our derivations,it can be shown that the Scharfetter–Gummel discretization method is essentially a special case of our scheme.In addition,the consistency and convergence of the two core algorithms are proved,with three numerical examples designed to demonstrate the accuracy and computational performance of this solver.Finally,we validate the Fr-DD model for a steady-state organic field effect transistor(OFET)by fitting the simulated transconductance and output curves to the experimental data.展开更多
This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomia...This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration.Moreover,with the help of fixed point theory,we study the existence and uniqueness conditions for the positive solution and prove some new results.Also,obtain the Ulam–Hyers stabilities for the proposed problem.Two gen-eralized examples are considered to show the method’s applicability and compared with other existing numerical methods.The present method performs exceptionally well in terms of efficiency and simplicity.Further,we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.展开更多
In this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method.Here we develop the discrete Adomian decomposition method to find...In this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method.Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation,nonlinear fractional discrete Schrodinger equation,fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger’s equation.The obtained solution is verified by comparison with exact solution whenα=1.展开更多
文摘This paper is concerned with the boundary value problem of a nonlinear fractional differential equation. By means of Schauder fixed-point theorem, an existence result of solution is obtained.
文摘We study two generalized versions of a system of equations which describe the time evolution of the hydrodynamic fluctuations of density and velocity in a linear viscoelastic fluid. In the first of these versions, the time derivatives are replaced by conformable derivatives, and in the second version left-handed Caputo’s derivatives are used. We show that the solutions obtained with these two types of derivatives exhibit significant similarities, which is an interesting (and somewhat surprising) result, taking into account that the conformable derivatives are local operators, while Caputo’s derivatives are nonlocal operators. We also show that the solutions of the generalized systems are similar to the solutions of the original system, if the order α of the new derivatives (conformable or Caputo) is less than one. On the other hand, when α is greater than one, the solutions of the generalized systems are qualitatively different from the solutions of the original system.
基金This work was supported in part by the National Science Foundation through Grant CNS-1726865by the USDA under Grant 2019-67021-28990.
文摘Because charge carriers of many organic semiconductors(OSCs)exhibit fractional drift diffusion(Fr-DD)transport properties,the need to develop a Fr-DD model solver becomes more apparent.However,the current research on solving the governing equations of the Fr-DD model is practically nonexistent.In this paper,an iterative solver with high precision is developed to solve both the transient and steady-state Fr-DD model for organic semiconductor devices.The Fr-DD model is composed of two fractionalorder carriers(i.e.,electrons and holes)continuity equations coupled with Poisson’s equation.By treating the current density as constants within each pair of consecutive grid nodes,a linear Caputo’s fractional-order ordinary differential equation(FrODE)can be produced,and its analytic solution gives an approximation to the carrier concentration.The convergence of the solver is guaranteed by implementing a successive over-relaxation(SOR)mechanism on each loop of Gummel’s iteration.Based on our derivations,it can be shown that the Scharfetter–Gummel discretization method is essentially a special case of our scheme.In addition,the consistency and convergence of the two core algorithms are proved,with three numerical examples designed to demonstrate the accuracy and computational performance of this solver.Finally,we validate the Fr-DD model for a steady-state organic field effect transistor(OFET)by fitting the simulated transconductance and output curves to the experimental data.
文摘This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration.Moreover,with the help of fixed point theory,we study the existence and uniqueness conditions for the positive solution and prove some new results.Also,obtain the Ulam–Hyers stabilities for the proposed problem.Two gen-eralized examples are considered to show the method’s applicability and compared with other existing numerical methods.The present method performs exceptionally well in terms of efficiency and simplicity.Further,we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.
基金to UGC New Delhi,India for financial support under the scheme”Research Fellowship in Science for Meritorious Students”vide letter No.F.4-3/2006(BSR)/11-78/2008(BSR).
文摘In this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method.Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation,nonlinear fractional discrete Schrodinger equation,fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger’s equation.The obtained solution is verified by comparison with exact solution whenα=1.
