The comprehension of inequalities in preinvexity is very important for studying fractional calculus and its effectiveness in many applied sciences. In this article, we develop and study of fractional integral inequalities whose second derivatives are preinvex functions. We investigate and prove new lemma for twice differentiable functions involving Riemann-Liouville(R-L) fractional integral operator. On the basis of this newly developed lemma, we make some new results regarding of this identity. These new results yield us some generalizations of the prior results. This study builds upon on a novel new auxiliary result which enables us to develop new variants of Ostrowski type inequalities for twice differentiable preinvex mappings. As an application, several estimates concerning Bessel functions of real numbers are also illustrated.

We investigate and prove a new lemma for twice differentiable functions with the fractional integral operator $AB$. Based on this newly developed lemma, we derive some new results about this identity. These new findings provide some generalizations of previous findings. This research builds on a novel new auxiliary result that allows us to create new variants of Ostrowski type inequalities for twice differentiable convex mappings. Some of the newly presented results' special cases are also discussed. As applications, several estimates involving special means of real numbers and Bessel functions are depicted.

This manuscript involves the new exact solitary wave solutions of fractional reaction-diffusion model using the exp $(-\phi \left(\eta \right))$-expansion method. The spatial model of fractional form is applied in modeling super-diffusive systems in the field of engineering, biology, physics (neutron diffusion theory), ecology, finance, and chemistry. The findings of miscellaneous studies showed that presented method is efficient for exploring new exact solutions to solve the complexities arising in mathematical physics and applied sciences. The new solutions which are obtained in the form of the rational, exponential, hyperbolic and trigonometric functions have a wide range in physics and engineering fields. Several results would be obtained under various parameters which shows good agreement with the previous published results of different papers. The proposed method can be extended to solve further problems arising in the engineering fields. My main contribution is programming and comparisons.