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In this article, thin film flow of non-Newtonian pseudo-plastic fluid is investigated on a vertical wall through homotopy-based scheme along with fractional calculus. Three cases were examined after considering (i) partial fractional differential equation (PFDE) by altering first-order derivative to fractional derivative in the interval (0, 1), (ii) PFDE by altering second-order derivative to fractional derivative in the interval (1, 2), and (iii) fully FDE by altering first-order derivative to fractional derivative in (0, 1) and second-order derivative to fractional derivative in (1, 2). Different physical quantities such as the velocity profile and volume flux were computed and analyzed. Validity of obtained results was checked by finding residuals. Moreover, consequence of different parameters on the velocity were also explored in fractional space.

The concept of fuzzy calculus in fluid modelling offers a feasible approach to address ambiguity and uncertainty in physical phenomena. This study aims to model and analyse thin film flow of Johnson Segalman nonofluid (JSNF) on a vertical belt in fuzzy environment for lifting and drainage settings. By incorporating Triangular fuzzy numbers (TFNs), a more accurate representation of the uncertain nature of JSNF flow is obtained which leads to a better understanding of fluid behaviour and its potential applications. The fluid problems are modelled with uncertainties and numerically solved through fuzzy extension of He-Laplace algorithm. The validity and convergence of the proposed methodology is checked by computing residual errors in each case. The obtained solutions provide fuzzy velocity profiles and volumetric flow rates in lift and drain cases. As the parameter $r - cut$ r − c u t approaches 1, the velocity profiles at the upper and lower bounds merge, indicating solution consistency. [ABSTRACT FROM AUTHOR]

This paper contains modeling of a fuzzy-fractional financial chaotic model based on triangular fuzzy numbers (TFNs) to predict the idea that long-term dependency and uncertainty both have an impact on the financial market. For solution purposes, the He–Mohand algorithm is proposed where homotopy perturbation is hybrid with Mohand transform in a fuzzy-Caputo sense. In analysis, solutions and corresponding errors at upper and lower bounds are estimated. The obtained numerical results are displayed in tables to show the reliability and efficiency of the proposed methodology. Upper bound errors range from 10 − 6 to 10 − 12 and lower bound errors from 10 − 6 to 10 − 11 . For graphical analysis, system profiles are illustrated as two-dimensional and three-dimensional plots at diverse values of fractional parameters and time to comprehend the physical behavior of the proposed fuzzy-fractional model. These plots demonstrate that the interest rate, price index, and investment demand decrease with the increase of the value of the r -cut at the lower bound. At the upper bound, this behavior is totally opposite. The chaotic behavior of the system at smaller values of saving rate, elasticity of demands, and per-investment cost is greater in contrast to their larger value. Analysis reveals that the proposed methodology (He–Mohand algorithm) provides a new way of understanding the complicated structure of financial systems and provides new insights into the dynamics of financial markets. This algorithm has potential applications in risk management, portfolio optimization, and trading strategies. [ABSTRACT FROM AUTHOR]

The current manuscript focuses on the solution and analysis of space and time fractional coupled Schrödinger system that belongs to a class of evolution equations. These systems encounter in different fields related to plasma waves, optics, and quantum physics. The fractional He-Laplace approach is proposed for the series form solutions of fractional systems. This approach contains hybrid of Laplace transform and homotopy perturbation along with Caputo fractional derivative. The current study provide new results on time and space fractional coupled Schrödinger systems which are not captured in existing literature. Reliability of proposed algorithm in both time and space fractional scenarios is observed through residual error concept throughout fractional domains. The effect of fractional parameters on wave profiles are analyzed numerically and graphically as 2D and 3D illustrations. Analysis reveals that proposed algorithm is suitable for non-linear time-space fractional systems encountering in different fields of sciences. [ABSTRACT FROM AUTHOR]