In this paper, we investigate the solutions of a class of-Hilfer fractional differential equations with the initial values in the sense of-fractional integral by using the successive approximation techniques. Next, the continuous dependence of a solution for the given Cauchy-type problem is presented.

In this paper, we use the $(k, \psi)$-Hilfer fractional integral of functions with respect to another function to generalize Chebyshev-type fractional integral inequalities. Some inequalities involving $(k, \psi)$-Hilfer fractional integrals are also to be proved.

In present work, we discuss local existance and uniqueness of solution to the $\psi-$Hilfer fractional differential problem. By using the Picard successive approximations, we construct a computable iterative scheme uniformly approximating solution. Two illustrative examples are given to support our findings.

In this paper, we apply Newton's method to solve a class of integro-differential equations of the Volterra-Fredholm type with nonlocal characteristics, involving almost sectorial operators and Hilfer fractional derivatives. Since these equations play a key role in mathematics, engineering, biology, physics, chemistry, control theory and  economy, finding an appropriate solution is important. By Newton's method, we linearize a nonlinear Volterra-Fredholm integro-differential equation which is equivalent to a category of Volterra-Fredholm integro-differential equations with nonlocal properties, incorporating almost sectorial operators and Hilfer fractional derivatives and nearly sectorial operators. This technique has been shown to be an effective tool for the numerical solution of initial value problems in nonlinear integro-differential equations. The convergence analysis is also investigated.

In this paper, we introduce an extension of the Hilfer fractional derivative, the "Hilfer fractional quantum derivative", and establish some of its properties. Then, we introduce and discuss initial and boundary value problems involving the Hilfer fractional quantum derivative. The existence of a unique solution of the considered problems is established via Banach's contraction mapping principle. Examples illustrating the obtained results are also presented.

This paper deals with the existence and uniqueness results for a class of impulsive boundary value problem for implicit nonlinear fractional differential equations and k-Generalized-Hilfer fractional derivative involving both retarded and advanced arguments. Our results are based on the Banach contraction principle and Schauder's fixed point theorem. Suitable illustrative examples are provided.

Acne is a very common skin disease. Scars are seen in 95% of patients with acne. Although numerous treatments have been recommended to cure acne scars, researchers are still searching for a single modality to treat the complication due to its variety in shape and depth. In other words, a combination of available methods is required to reach satisfactory results. We compared the effects of fractional carbon dioxide (CO2) laser alone and in combination with subcision.Methods: This clinical trial study was performed in Skin Diseases and Leishmaniasis Research Center (Isfahan, Iran) during 2011-12. Eligible patients with atrophic acne scars were treated with fractional CO2 laser alone (five sessions with three-week interval) on right side of the face and fractional CO2 laser plus subcision (one session using both followed by four sessions of fractional CO2 laser, all with three-week intervals) on the left side. The subjects were visited one, two, and six months after the treatment. Patient satisfaction rate was analyzed by SPSS20.Findings: The average of recovery rate was 54.7% using the combination method and 43.0% using laser alone (P<0.001). The mean patient satisfaction rate according to visual analogue scale (VAS) score was significantly higher with the combination method than with laser alone (6.6 vs. 5.2; P<0.001). Bruising was only seen with the combination method and lasted for one week in 57.0% of patients and for two weeks in 43.0%. Erythema was a side effect of both methods. Post-inflammatory pigmentation and hyperpigmentation were only caused by the combination method. None of the patients had persistent side effects after six months.Conclusion: Using a combination of subcision and laser had suitable results regarding scar recovery and satisfaction rate. However, bruising, post-inflammatory pigmentation, and hyperpigmentation were the side effects of this method. In general, subcision plus laser can be beneficial in the treatment of atrophic acne scars.

Analysis of legal and economical aspects of currency derivatives and theirs effects. Derivatives are one of the financial instruments of capital market. Considering the nature of the basic asset, derivatives are categorized in three different types including: currency derivatives, securities derivatives, and commodity derivatives. Currency derivatives are the financial instruments of the money market. Money market participants are utilizing these derivatives with different purposes. Currency derivatives divided in to three main types as follows: currency options, currency swap and currency futures. One of the functions of the currency derivatives is to manage the risks of the currency price fluctuations. Although Iran economy always affected by the risk of the currency price, fluctuations which have had main effects to the economy activities during different periods and considering that the currency derivatives are useful instruments for risk management, but the legal nature, requirements and effects of this derivatives are not considered in Iranian law and therefor required mechanisms for establishment of integrated market of currency derivatives have not been formed in iran. This article argues about some legal issues related to the currency derivatives including their legal nature, basis and effects.