The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced.Exact analytical solutions are obtained for a class of unsteady flows for the generalized secon...The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced.Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus.The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates.In addition,the solutions of the shear stresses at the plates are also determined.展开更多
Some quadrature formulae for the numerical evaluation of singular integrals of arbitrary order are established and both the estimate of remainder and the convergence of each quadrature formula derived here are also gi...Some quadrature formulae for the numerical evaluation of singular integrals of arbitrary order are established and both the estimate of remainder and the convergence of each quadrature formula derived here are also given.展开更多
This paper presents a study of nonlinear waves in shallow water.The Korteweg-de Vries(KdV)equa-tion has a canonical version based on oceanography theory,the shallow water waves in the oceans,and the internal ion-acous...This paper presents a study of nonlinear waves in shallow water.The Korteweg-de Vries(KdV)equa-tion has a canonical version based on oceanography theory,the shallow water waves in the oceans,and the internal ion-acoustic waves in plasma.Indeed,the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method(HPTM)to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness.This approach is connected with the fuzzy generalized integral transform and HPTM.Besides that,two novel results for fuzzy generalized integral transforma-tion concerning fuzzy partial gH-derivatives are presented.Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method.Furthermore,2D and 3D simulations de-pict the comparison analysis between two fractional derivative operators(Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense)under generalized gH-differentiability.The projected method(GHPTM)demonstrates a diverse spectrum of applications for dealing with nonlinear wave equa-tions in scientific domains.The current work,as a novel use of GHPTM,demonstrates some key differ-ences from existing similar methods.展开更多
In this work,we use a Predictor–Corrector method to implement and derive an iterative solution of an existing Tuberculosis(TB)model with two fractional derivatives,namely,Caputo–Fabrizio fractional derivative and th...In this work,we use a Predictor–Corrector method to implement and derive an iterative solution of an existing Tuberculosis(TB)model with two fractional derivatives,namely,Caputo–Fabrizio fractional derivative and the new generalized Caputo fractional derivative.We begin by recalling some existing results such as the basic reproduction number R0 and the equilibrium points of the model.Then,we study the global asymptotic stability of disease-free equilibrium of the fractional models.We also prove,for each fractional model,the existence and uniqueness of solutions.An iterative solution of the two models is computed using the Predictor–Corrector method.Using realistic parameter values,we perform numerical simulations for different values of the fractional order.Simulation results permit to conclude that the new generalized Caputo fractional derivative provides a more realistic analysis than the Caputo–Fabrizio fractional derivative and the classical integer-order TB model.展开更多
In this paper,a sufficient conditions to guarantee the existence and stability of solutions for generalized nonlinear fractional differential equations of orderα(1<α<2)are given.The main results are obtained b...In this paper,a sufficient conditions to guarantee the existence and stability of solutions for generalized nonlinear fractional differential equations of orderα(1<α<2)are given.The main results are obtained by using Krasnoselskii's fixed point theorem in a weighted Banach space.Two examples are given to demonstrate the validity of the proposed results.展开更多
In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are ap...In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.展开更多
As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized...As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized diffusion equations with perturbation. Complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is obtained. As a result, the corresponding approximate derivative-dependent functional separable solutions to some resulting perturbed equations are derived by way of examples.展开更多
In this paper, we investigate the optimality conditions of a class of special nonsmoothprogramming min F(x) = which arises from Li-norm optimization,where ci ∈ R is constant and fi ∈ C1,i = 1,2, ...,m. These conditi...In this paper, we investigate the optimality conditions of a class of special nonsmoothprogramming min F(x) = which arises from Li-norm optimization,where ci ∈ R is constant and fi ∈ C1,i = 1,2, ...,m. These conditions can easily be tested by computer.展开更多
It was suggested by Pantanen that the mean squared error may be used to measure the inefficiency of the least squares estimator. Styan[2] and Rao[3] et al. discussed this inefficiency and it's bound later. In this...It was suggested by Pantanen that the mean squared error may be used to measure the inefficiency of the least squares estimator. Styan[2] and Rao[3] et al. discussed this inefficiency and it's bound later. In this paper we propose a new inefficiency of the least squares estimator with the measure of generalized variance and obtain its bound.展开更多
We present a generalization of the linear one-dimensional diffusion equation by combining the fractional derivatives and the internal degrees of freedom. The solutions are constructed from those of the scalar fraction...We present a generalization of the linear one-dimensional diffusion equation by combining the fractional derivatives and the internal degrees of freedom. The solutions are constructed from those of the scalar fractional diffusion equation. We analyze the interpolation between the standard diffusion and wave equations defined by the fractional derivatives. Our main result is that we can define a diffusion process depending on the internal degrees of freedom associated to the system.展开更多
In this paper, we proposed a Extension Definition to derive, simultaneously, the first, second and high order generalized derivatives for non-smooth functions, in which the involved functions are Riemann integrable bu...In this paper, we proposed a Extension Definition to derive, simultaneously, the first, second and high order generalized derivatives for non-smooth functions, in which the involved functions are Riemann integrable but not necessarily locally Lipschitz or continuous. Indeed, we define a functional optimization problem corresponding to smooth functions where its optimal solutions are the first and second derivatives of these functions in a domain. Then by applying these functional optimization problems for non-smooth functions and using this method we obtain generalized first derivative (GFD) and generalized second derivative (GSD). Here, the optimization problem is approximated with a linear programming problem that by solving of which, we can obtain these derivatives, as simple as possible. We extend this approach for obtaining generalized high order derivatives (GHODs) of non-smooth functions, simultaneously. Finally, for efficiency of our approach some numerical examples have been presented.展开更多
In this paper,we study univalent functions f for which log f’belongs to the analytic Morrey spaces.By using the characterization of higher order derivatives of functions in analytic Morrey spaces,we establish some ne...In this paper,we study univalent functions f for which log f’belongs to the analytic Morrey spaces.By using the characterization of higher order derivatives of functions in analytic Morrey spaces,we establish some new descriptions for the analytic Morrey domains in terms of two kinds of generalized Schwarzian derivatives.展开更多
基金The project supported by the National Natural Science Foundation of China (10372007,10002003) and CNPC Innovation Fund
文摘The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced.Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus.The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates.In addition,the solutions of the shear stresses at the plates are also determined.
基金Supported by NNSF and RFDP of Higher Education of China.
文摘Some quadrature formulae for the numerical evaluation of singular integrals of arbitrary order are established and both the estimate of remainder and the convergence of each quadrature formula derived here are also given.
文摘This paper presents a study of nonlinear waves in shallow water.The Korteweg-de Vries(KdV)equa-tion has a canonical version based on oceanography theory,the shallow water waves in the oceans,and the internal ion-acoustic waves in plasma.Indeed,the main goal of this investigation is to employ a semi-analytical method based on the homotopy perturbation transform method(HPTM)to obtain the numerical findings of nonlinear dispersive and fifth order KdV models for investigating the behaviour of magneto-acoustic waves in plasma via fuzziness.This approach is connected with the fuzzy generalized integral transform and HPTM.Besides that,two novel results for fuzzy generalized integral transforma-tion concerning fuzzy partial gH-derivatives are presented.Several illustrative examples are illustrated to show the effectiveness and supremacy of the proposed method.Furthermore,2D and 3D simulations de-pict the comparison analysis between two fractional derivative operators(Caputo and Atangana-Baleanu fractional derivative operators in the Caputo sense)under generalized gH-differentiability.The projected method(GHPTM)demonstrates a diverse spectrum of applications for dealing with nonlinear wave equa-tions in scientific domains.The current work,as a novel use of GHPTM,demonstrates some key differ-ences from existing similar methods.
文摘In this work,we use a Predictor–Corrector method to implement and derive an iterative solution of an existing Tuberculosis(TB)model with two fractional derivatives,namely,Caputo–Fabrizio fractional derivative and the new generalized Caputo fractional derivative.We begin by recalling some existing results such as the basic reproduction number R0 and the equilibrium points of the model.Then,we study the global asymptotic stability of disease-free equilibrium of the fractional models.We also prove,for each fractional model,the existence and uniqueness of solutions.An iterative solution of the two models is computed using the Predictor–Corrector method.Using realistic parameter values,we perform numerical simulations for different values of the fractional order.Simulation results permit to conclude that the new generalized Caputo fractional derivative provides a more realistic analysis than the Caputo–Fabrizio fractional derivative and the classical integer-order TB model.
文摘In this paper,a sufficient conditions to guarantee the existence and stability of solutions for generalized nonlinear fractional differential equations of orderα(1<α<2)are given.The main results are obtained by using Krasnoselskii's fixed point theorem in a weighted Banach space.Two examples are given to demonstrate the validity of the proposed results.
文摘In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.
基金Supported by the National Natural Science Foundation of China under Grant No.10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.SJ08A05
文摘As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized diffusion equations with perturbation. Complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is obtained. As a result, the corresponding approximate derivative-dependent functional separable solutions to some resulting perturbed equations are derived by way of examples.
基金The project was supported by NNSFC(No. 19771047) and NSFJS (BK97059).
文摘In this paper, we investigate the optimality conditions of a class of special nonsmoothprogramming min F(x) = which arises from Li-norm optimization,where ci ∈ R is constant and fi ∈ C1,i = 1,2, ...,m. These conditions can easily be tested by computer.
文摘It was suggested by Pantanen that the mean squared error may be used to measure the inefficiency of the least squares estimator. Styan[2] and Rao[3] et al. discussed this inefficiency and it's bound later. In this paper we propose a new inefficiency of the least squares estimator with the measure of generalized variance and obtain its bound.
基金This work has been partially supported by the Project "The Sciences of Complexity (ZiF,Bielefeld Universitat),the Comision Interministerial de Ciecia y Tecnologia of Spain(grant PB98-0850) and the European Project COSIC of SENS(HPRN-CT-2000-00158)
文摘We present a generalization of the linear one-dimensional diffusion equation by combining the fractional derivatives and the internal degrees of freedom. The solutions are constructed from those of the scalar fractional diffusion equation. We analyze the interpolation between the standard diffusion and wave equations defined by the fractional derivatives. Our main result is that we can define a diffusion process depending on the internal degrees of freedom associated to the system.
文摘In this paper, we proposed a Extension Definition to derive, simultaneously, the first, second and high order generalized derivatives for non-smooth functions, in which the involved functions are Riemann integrable but not necessarily locally Lipschitz or continuous. Indeed, we define a functional optimization problem corresponding to smooth functions where its optimal solutions are the first and second derivatives of these functions in a domain. Then by applying these functional optimization problems for non-smooth functions and using this method we obtain generalized first derivative (GFD) and generalized second derivative (GSD). Here, the optimization problem is approximated with a linear programming problem that by solving of which, we can obtain these derivatives, as simple as possible. We extend this approach for obtaining generalized high order derivatives (GHODs) of non-smooth functions, simultaneously. Finally, for efficiency of our approach some numerical examples have been presented.
基金supported by National Natural Science Foundation of China(Grant No.11601100)the PhD research startup foundation of Guizhou Normal University(Grant No.11904-05032130006)supported by National Natural Science Foundation of China(Grant No.11501157)
文摘In this paper,we study univalent functions f for which log f’belongs to the analytic Morrey spaces.By using the characterization of higher order derivatives of functions in analytic Morrey spaces,we establish some new descriptions for the analytic Morrey domains in terms of two kinds of generalized Schwarzian derivatives.
