Some New and Sharp Inequalities of Composite Simpson’s Formula for Differentiable Functions with Applications

Composite integral formulas offer greater accuracy by dividing the interval into smaller subintervals, which better capture the local behavior of function. In the finite volume method for solving differential equations, composite formulas are mostly used on control volumes to achieve high-accuracy solutions. In this work, error estimates of the composite Simpson’s formula for differentiable convex functions are established. These error estimates can be applied to general subdivisions of the integration interval, provided the integrand satisfies a first-order differentiability condition. To this end, a novel and general integral identity for differentiable functions is established by considering general subdivisions of the integration interval. The new integral identity is proved in a manner that allows it to be transformed into different identities for different subdivisions of the integration interval. Then, under the convexity assumption on the integrand, sharp error bounds for the composite Simpson’s formula are proved. Moreover, the well-known Hölder’s inequality is applied to obtain sharper error bounds for differentiable convex functions, which represents a significant finding of this study. Finally, to support the theoretical part of this work, some numerical examples are tested and demonstrate the efficiency of the new bounds for different partitions of the integration interval.