In this paper, we propose a new type of FUZZY FRACTIONAL differential equations under FUZZY Kolwankar- Gangal LOCAL FRACTIONAL derivatives (for short, FUZZY KG-LFD) using FUZZY Riemann-Liouville differentiability. Then, we prove some basic results in this area like, the relation between different types of FUZZY KG-LFD and their r-cut representations, composition of the FUZZY KG-LFD and the FUZZY LOCAL FRACTIONAL integral. Also, an application is provided in details such that the explicit solutions are expressed through Mittag-Leffler function.

In the last decade, the theory of LOCAL FRACTIONAL calculus has been successfully used to describe and solve fundamental science and engineering problems. In this article, the LOCAL FRACTIONAL Yang-Laplace variational iteration method has been used for solving the LOCAL FRACTIONAL partial differential equation on a cantor set. The non-differentiable exact and approximate solutions are obtained for kind of LOCAL FRACTIONAL linear and nonlinear equations. It is shown that the used method is an efficient and easy method to implement for linear and nonlinear problems arising in science and engineering. In this article, we emphasize on the LFYLVM method which is a combination form of LOCAL FRACTIONAL variational iteration method and Yang-Laplace transform. Most of the obtained solutions from this method are in series form that converge rapidly to exact or approximate solutions. Illustrative examples demonstrate that the method is able to reduce the volume of computation compared to the existing classical methods.

Our aim in this paper is to study the FUZZY stability of a FRACTIONAL Lorenz system in the sense of the Caputo-Fabrizio derivative and a FRACTIONAL financial crisis in the sense of }−Hilfer derivative. Defining a new type of FUZZY control function that has a dynamic situation helps us to investigate new stability results for these mathematical models.

FUZZY FRACTIONAL heat equations (FFHEs) are utilized to an-alyze the behaviour of the certain phenomena in various mathematical and scienti, c models. The main goal of this paper is to construct the solution of FUZZY FRACTIONAL heat equations by taking a reliable recipe of Sumudu transformation method and homotopy analysis method into account. These method allow us to remove the di, culties and restric-tions confronted in other methods. The feasibility of this method is con, rmed by given numerical examples. The result presented that the proposed method is suitable, powerful and reliable for obtaining the solution of FUZZY FRACTIONAL problems with FFHEs.

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.

In this work, we extend the existing LOCAL FRACTIONAL Sumudu decomposition method to solve the nonlinear LOCAL FRACTIONAL partial differential equations. Then, we apply this new algorithm to resolve the nonlinear LOCAL FRACTIONAL gas dynamics equation and nonlinear LOCAL FRACTIONAL Klein-Gordon equation, so we get the desired non-differentiable exact solutions. The steps to solve the examples and the results obtained, showed the flexibility of applying this algorithm, and therefore, it can be applied to similar examples.