Here we consider the ordinary and fractional approximation of functions by sublinear positive operators with applications to generalized convolution type expressed integrals such as Choquet Shilkret ones. The is under differentiability Caputo, Canavati Iterated-Caputo types. We produce Jackson inequalities basic initial conditions. So our way quantitative producing their right hand sides involv...

The advancements of integral inequalities with the help fractional operators have recently been focus attention in theory inequalities. In this study, we first review some fundamental concepts, and then using \(k\)-conformable integrals, establish a new identity for differentiable functions. Then, considering as an auxiliary result, several Ostrowski-type are presented functions whose modulus d...

In this paper, we study the existence of positive solutions for a system fractional differential equations with ?-Laplacian operators, Riemann–Liouville derivatives diverse orders and general nonlinearities which depend on several integrals differing orders, supplemented nonlocal coupled boundary conditions containing Riemann–Stieltjes varied derivatives. The from are continuous nonnegative fun...

In this paper we consider the gravitational field of fractal distribution of particles. To describe fractal distribution, we use the fractional integrals. The fractional integrals are considered as approximations of integrals on fractals. Using the fractional generalization of the Gauss’s law, we consider the simple examples of the fields of homogeneous fractal distribution. The examples of gra...

In this paper, we deal with the convolution series that are a far reaching generalization of conventional power and fractional exponents including Mittag-Leffler type functions. Special attention is given to most interesting case generated by Sonine kernels. first formulate prove second fundamental theorem for general integrals $n$-fold sequential derivatives both Riemann-Liouville Caputo types...

Fractional directional integrals are the extensions of the Riemann-Liouville fractional integrals from oneto multi-dimensional spaces and play an important role in extending the fractional differentiation to diverse applications. In numerical evaluation of these integrals, the weakly singular kernels often fail the conventional quadrature rules such as Newton-Cotes and Gauss-Legendre rules. It ...

*Correspondence: [email protected] Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, 32023, Taiwan, R.O.C. Abstract In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields of science and engineering. Many authors...

We develop spectral collocation methods for fractional differential equations with variable order with two end-point singularities. Specifically, we derive three-term recurrence relations for both integrals and derivatives of the weighted Jacobi polynomials of the form 1 x μ1 1 Æ x 2P a,b j x a, b, μ1, μ2 Æ1 , which leads to the desired differentiation matrices. We apply the new differentiation...