The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Tra...The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.展开更多
A new formula with derivatives for numerical integration was presented. Based on this formula and the Richardson extrapolafion process, a numerical integration method was established. It can converge faster than the R...A new formula with derivatives for numerical integration was presented. Based on this formula and the Richardson extrapolafion process, a numerical integration method was established. It can converge faster than the Romberg＇s. With the same accuracy, the computation of the new numerical integration with derivatives is only half of that of Romberg＇s numerical integration.展开更多
文摘The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator.This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the(n+1)-composite Trapezoidal rule.In particular,the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule,which has accordingly been used to derive a more accurate fractional-order formula called the(n+1)-composite Trapezoidal rule.Additionally,in order to increase the accuracy of the proposed approximations by reducing the true errors,we incorporate the so-called Romberg integration,which is an extrapolation formula of the Trapezoidal rule for integration,into our proposed approaches.Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.
文摘A new formula with derivatives for numerical integration was presented. Based on this formula and the Richardson extrapolafion process, a numerical integration method was established. It can converge faster than the Romberg＇s. With the same accuracy, the computation of the new numerical integration with derivatives is only half of that of Romberg＇s numerical integration.
