The research paper aims to investigate the space-time fractional cubic-quartic non-linear Schrödinger equation in the appearance of the third, and fourth-order dispersion impacts without both group velocity dispersion, and disturbance with parabolic law media by utilizing the extended sinh-Gordon expansion method. This method is one of the strongest methods to find the exact solutions to the non-linear partial differential equations. In order to confirm the existing solutions, the constraint conditions are used. We successfully construct various exact solitary wave solutions to the governing equation, for example, singular, and dark-bright solutions. Moreover, the 2D, 3D, and contour surfaces of all obtained solutions are also plotted. The finding solutions have justified the efficiency of the proposed method.

Many dynamic systems can be modeled by fractional differential equations in which some external parameters occur under uncertainty. Although these parameters increase the complexity, they present more acceptable solutions. With the aid of Atangana-Baleanu-Caputo (ABC) fractional differential operator, an advanced numerical-analysis approach is considered and applied in this work to deal with different classes of fuzzy integrodifferential equations of fractional order fitted with uncertain constraints conditions. The fractional derivative of ABC is adopted under the generalized H-differentiability (g-HD) framework, which uses the Mittag-Leffler function as a nonlocal kernel to better describe the timescale of the fuzzy models. Towards this end, applications of reproducing kernel algorithm are extended to solve classes of linear and nonlinear fuzzy fractional ABC Volterra-Fredholm integrodifferential equations. Based on the characterization theorem, preconditions are established under the Lipschitz condition to characterize the fuzzy solution in a coupled equivalent system of crisp ABC integrodifferential equations. Parametric solutions of the ABC interval are provided in terms of rapidly convergent series in Sobolev spaces. Several examples of fuzzy ABC Volterra-Fredholm models are implemented in light of g-HD to demonstrate the feasibility and efficiency of the designed algorithm. Numerical and graphical representations of both classical Caputo and ABC fractional derivatives are presented to show the effect of the ABC derivative on the parametric solutions of the posed models. The achieved results reveal that the proposed method is systematic and suitable for dealing with the fuzzy fractional problems arising in physics, technology, and engineering in terms of the ABC fractional derivative.

In real-world inventory problems, the deterioration rate is usually treated as an uncontrolled factor. However, by adopting certain preservation techniques, such deterioration rate can be controlled up to a desired level. Moreover, in certain business affairs, the supplier also provides a permissible delay in payment to encourage the vendors to have further sales. In this paper, we develop an inventory model that considers the power pattern demand under trade credit, permissible completely backlogged shortages, and suitable investment in preservation technology. Additionally, to make the model more realistic, we apply the learning effect to the holding costs. We develop the mathematical model of the problem and its solving policy in both crisp and fuzzy environments. The main purpose of the model is to obtain the optimum preservation technology-based cost and cycle time that maximize the overall profit. Furthermore, to validate our findings, we consider numerical examples and subsequently demonstrate the concavity of the profit function via Mathematica 11.1.1 software. Finally, we study the sensitivity analysis and accordingly present several managerial insights for the benefit of inventory managers.

Most of the researchers developed their inventory models to forecast the optimal replenishment quantity and time in view of minimizing the total inventory cost by considering deterministic demand and the deterioration of the items. But, in real business these demands and deterioration are mostly fuzzy in nature due to many practical factors, such as increase or decrease in goodwill of the product, competition from the substitute products, scientific advancement in preserving facilities, change in environmental conditions and so on. So by following researcher's classical inventory model, retailer may order less or excess amount of items than the actual requirement. As a result, retailer may face loss in business or increase in cost. Moreover, in many cases, suppliers offer trade credit to increase their sales, and by availing the trade credit facility the retailer purchases a number of items more than the existing storage capacity (in own warehouse) in order to minimize the ordering cost and investment capital. To accommodate these excess amounts of items retailer may hire a warehouse on rent basis. In the light of these facts, we develop a cost optimization model for the inventory items having fuzzy demand and deterioration with two-warehouse facility under trade credit financing by considering triangular fuzzy numbers for the associated parameters. The Graded Mean Integration Representation defuzzification technique is used and numerical examples are provided to justify the validity of the proposed model. Finally, sensitivity analysis of major parameters has been incorporated to draw the managerial insight on optimal solution.