Avit, Özkan; Anac, Halil

The present study used the novel conformable fractional methods to examine the nonlinear Kuramoto-Sivashinsky equations of the conformable fractional derivative. The Kuramoto-Sivashinsky equation is a mathematical model employed to elucidate a diverse range of chemical and physical phenomena, encompassing chemical reaction-diffusion processes, plasma instabilities, viscous flow issues, flame front propagation, and magnetized plasmas. The present inquiry provides an examination of convergence and error for the future scheme. The conformable q-homotopy analysis transform method (Cq-HATM) provides h-curves that illustrate the convergence range of the series solution achieved. In order to validate the effectiveness and suitability of the Cq-HATM, we examine separate instances. In this study, we introduce an application that demonstrates the potential benefits and effectiveness of the proposed approach. Furthermore, an error analysis is performed in order to verify the accuracy of the scheme. Numerical simulations are conducted to validate the precision of the forthcoming approach. The findings obtained from the numerical and graphical analyses are presented in this study. The method provided in this study showcases a notable degree of computational precision and ease in examining and resolving intricate phenomena related to conformable fractional nonlinear partial differential equations within the domains of science and technology.

Merdan, Mehmet; Atasoy, Nihal

In this study, random ordinary differential equations obtained by randomly choosing the coefficients or initial conditions of the ordinary differential equations will be analyzed by the Residual Power Series Method. The initial conditions or coefficients of the equations will be converted to random variables with Normal, Gamma and exponential distribution. Probability characteristics such as expected value, variance and confidence interval of the obtained random ordinary differential equations will be calculated. Obtained results will be drawn with the help of MATLAB (2013a) package program and random results will be interpreted.