Ostrowski type inequalities for fractional calculus

The theory of integral equations is one of the most useful mathematical tools in both pure and applied mathematics. Many initial and boundary value problems associated with ordinary differential equation (ODE) and partial differential equation (PDE) can be transformed into problems of solving some approximate integral equations. However, some initial and boundary value domains are fractal curves, which are everywhere continuous but nowhere differentiable. As a result, we cannot employ the classical calculus, which requires that the defined functions should be differentiable. The theory of local fractional integrals and derivatives are one of useful tools to handle the fractal and continuously non-differentiable functions. It is worth to note the applications of fractional calculus in several scientific areas including special functions, control theory, chemical physics, stochastic processes, anomalous diffusion, etc. On the other hand, the theory of Conformable Fractional Calculus which is used to calculate the integral and derivative values in between 0 and 1 has become the important place in mathematics. Some researchers established fundamental inequalities such as Hermite-Hadamard, Ostrowski, Grüss and Ostrowski-Grüss. In this thesis, some basic methods based on local fractal calculus and conformable fractional calculus are used to obtain the new inequalities will be first established. For this, an integral identity for twice differentiable functions on local fractal spaces will be established. Afterwards, Ostrowski type inequalities for mappings whose second local fractional derivatives in modulus are bounded will be presented by using this equality. Furthermore, with the help of the results obtained in this thesis, some applications for numerical integration and special means will be given. Second aim of this study is to establish Ostrowski type results for conformable fractional integrals. Two generalized Ostrowski type estimations for mappings whose second conformable fractional derivatives are bounded will be derived, and special cases of these comprehensive inequalities will be observed. Moreover, by using these inequalities, some applications in numerical integrations will be examined.