In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatmen...In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.展开更多
This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities...This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.展开更多
The numerical simulation of the steady incompressible viscous flows in a no-slip channel is considered. A discrete artificial boundary condition on a given segmental artificial boundary is designed by the method of li...The numerical simulation of the steady incompressible viscous flows in a no-slip channel is considered. A discrete artificial boundary condition on a given segmental artificial boundary is designed by the method of lines. Then the original problem is reduced to a boundary value problem of Navier-Stokes equations on a bounded domain. The numerical examples show that this artificial boundary condition is very effective and more accurate than Dirichlet and Neumann boundary conditions used in engineering literature.展开更多
In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractiona...In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractional derivatives are described in the Caputo sense.The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations.展开更多
Using Hartogs'fundamental theorem for analytic functions in several complex variables,we establish a multiple q-exponential differential operational identity for the analytic functions in several variables,which c...Using Hartogs'fundamental theorem for analytic functions in several complex variables,we establish a multiple q-exponential differential operational identity for the analytic functions in several variables,which can be regarded as a multiple q-translation formula.This multiple q-translation formula is a fundamental result and play a pivotal role in q-mathematics.Using this q-translation formula,we can easily recover many classical conclusions in q-mathematics and derive some new q-formulas.Our work reveals some profound connections between the theory of complex functions in several variables and q-mathematics.展开更多
In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear...In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The performance of presented method has been compared with other methods, namely Sinc-Galerkin, quadratic spline collocation and LiuLin method. Numerical examples show better accuracy of the proposed method. Moreover, the computation cost decreases at least by a factor of 6 in this method.展开更多
We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are in （0, t）, that is, the largest eigenvalue distribution. Associated with this probability, in the ladder operator...We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are in （0, t）, that is, the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely, an（t） and/3r, （t）, as well as three auxiliary quantities, denoted by rn（t）, Rn（t）, σn（t）. We establish the second order differential equations for both βn（t） and rn（t）. By investigating the soft edge scaling limit when a - O（n） as n→ ∞ or a is finite, we derive a PH, the σ-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials Pn （z） corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n, asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy＇s equation for （t） = （t） with （t） satisfying the a-form of PIV or PV.展开更多
The Chebyshev polynomials are harnessed as functions of the one parameter of the nondimensionalized differential equation for trinomial homogeneous linear differential equations of arbitrary order n that have constant...The Chebyshev polynomials are harnessed as functions of the one parameter of the nondimensionalized differential equation for trinomial homogeneous linear differential equations of arbitrary order n that have constant coefficients and exhibit vibration. The use of the Chebyshev polynomials allows calculation of the analytic solutions for arbitrary n in terms of the orthogonal Chebyshev polynomials to provide a more stable solution form and natural sensitivity analysis in terms of one parameter and the initial conditions in 6n + 7 arithmetic operations and one square root.展开更多
文摘In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.
基金partially supported by the NSFC(12061042)the NSF of Jiangxi(20202BAB201003)+3 种基金the support of the National Science Center(Poland)via grant 2017/25/B/ST1/00931partially supported by the Project PID2021-124472NB-I00funded by MCIN/AEI/10.13039/501100011033by"EFDF A way of making Europe"。
文摘This paper is devoted to considering the quasiperiodicity of complex differential polynomials,complex difference polynomials and complex delay-differential polynomials of certain types,and to studying the similarities and differences of quasiperiodicity compared to the corresponding properties of periodicity.
文摘The numerical simulation of the steady incompressible viscous flows in a no-slip channel is considered. A discrete artificial boundary condition on a given segmental artificial boundary is designed by the method of lines. Then the original problem is reduced to a boundary value problem of Navier-Stokes equations on a bounded domain. The numerical examples show that this artificial boundary condition is very effective and more accurate than Dirichlet and Neumann boundary conditions used in engineering literature.
文摘In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractional derivatives are described in the Caputo sense.The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations.
基金Supported by the National Natural Science Foundation of China(Grant No.11971173)Science and Technology Commission of Shanghai Municipality(Grant No.22DZ2229014)。
文摘Using Hartogs'fundamental theorem for analytic functions in several complex variables,we establish a multiple q-exponential differential operational identity for the analytic functions in several variables,which can be regarded as a multiple q-translation formula.This multiple q-translation formula is a fundamental result and play a pivotal role in q-mathematics.Using this q-translation formula,we can easily recover many classical conclusions in q-mathematics and derive some new q-formulas.Our work reveals some profound connections between the theory of complex functions in several variables and q-mathematics.
文摘In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The performance of presented method has been compared with other methods, namely Sinc-Galerkin, quadratic spline collocation and LiuLin method. Numerical examples show better accuracy of the proposed method. Moreover, the computation cost decreases at least by a factor of 6 in this method.
基金The financial support of the Macao Science and Technology Development Fund under grant number FDCT 077/2012/A3, FDCT 130/2014/A3the University of Macao for generous support: MYRG 2014–00011 FST, MYRG 2014–00004 FST
文摘We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are in （0, t）, that is, the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely, an（t） and/3r, （t）, as well as three auxiliary quantities, denoted by rn（t）, Rn（t）, σn（t）. We establish the second order differential equations for both βn（t） and rn（t）. By investigating the soft edge scaling limit when a - O（n） as n→ ∞ or a is finite, we derive a PH, the σ-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials Pn （z） corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n, asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy＇s equation for （t） = （t） with （t） satisfying the a-form of PIV or PV.
文摘The Chebyshev polynomials are harnessed as functions of the one parameter of the nondimensionalized differential equation for trinomial homogeneous linear differential equations of arbitrary order n that have constant coefficients and exhibit vibration. The use of the Chebyshev polynomials allows calculation of the analytic solutions for arbitrary n in terms of the orthogonal Chebyshev polynomials to provide a more stable solution form and natural sensitivity analysis in terms of one parameter and the initial conditions in 6n + 7 arithmetic operations and one square root.
