In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouv...In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouville sense. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical results are introduced to satisfy the accuracy and the applicability of the proposed method.展开更多
We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness wФ^r（f, t） where Ф is an admissible step-weight function. An equivalen...We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness wФ^r（f, t） where Ф is an admissible step-weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.展开更多
Many problems in engineering sciences can be described by linear,inhomogeneous,m-th order ordinary differential equations(ODEs)with variable coefficients.For this wide class of problems,we here present a new,simple,fl...Many problems in engineering sciences can be described by linear,inhomogeneous,m-th order ordinary differential equations(ODEs)with variable coefficients.For this wide class of problems,we here present a new,simple,flexible,and robust solution method,based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals.The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus.Based on cubic approximation polynomials,the presented method can be expected to perform(i)similar to the Runge-Kutta method,when applied to stiff initial value problems,and(ii)significantly better than the finite difference method,when applied to boundary value problems.Therefore,we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum,steady-state heat transfer through a cooling web,and the structural analysis of a slender tower based on second-order beam theory.Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.展开更多
文摘In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouville sense. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical results are introduced to satisfy the accuracy and the applicability of the proposed method.
基金The research is supported by Zhejiang Provincial Natural Science Foundation of China
文摘We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness wФ^r（f, t） where Ф is an admissible step-weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.
基金Fruitful discussions with Gerhard Hofinger,from Feb 2007 until Dec 2010 research assistant at the Institute for Mechanics of Materials and Structures,Vienna University of Technology,are gratefully acknowledged.
文摘Many problems in engineering sciences can be described by linear,inhomogeneous,m-th order ordinary differential equations(ODEs)with variable coefficients.For this wide class of problems,we here present a new,simple,flexible,and robust solution method,based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals.The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus.Based on cubic approximation polynomials,the presented method can be expected to perform(i)similar to the Runge-Kutta method,when applied to stiff initial value problems,and(ii)significantly better than the finite difference method,when applied to boundary value problems.Therefore,we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum,steady-state heat transfer through a cooling web,and the structural analysis of a slender tower based on second-order beam theory.Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.
