Uncertainty Quantification of Analytical Solutions for the Nonlinear Space and Time Fractional Order ϕ 4 $$ {\phi}^4 $$ Equation Under Fuzziness

Abstract

In this study, the nonlinear time fractional ${\varphi }^4$ equation is taken and made into a space and time nonlinear fractional ${\varphi }^4$ equation. After that, fuzzy theory is utilized to study our simple modified space and time nonlinear fractional ${\varphi }^4$ equation under fuzziness. Our aim is to extract analytical solutions for the proposed model under fuzziness. A fuzzified fractional traveling wave transformation and conformable time fractional derivative are applied to reduce the governing fuzzy space and time fractional nonlinear ${\varphi }^4$ equation into a fuzzy ordinary differential equation (FODE), which is solved using the extended hyperbolic function method (EHFM). By considering different cases and fuzzy lower and upper bounds, multiple types of soliton solutions are derived, including bright solitons, periodic solutions, and exponential-type wave solutions. These solutions describe various nonlinear wave structures that can propagate in dispersive media. To illustrate the physical behavior of the fuzzy space and time fractional nonlinear ${\varphi }^4$ model, two-dimensional, three-dimensional, and contour plots are presented using Mathematica software to show the effects of fractional derivatives and fuzzy parameters on soliton amplitude, localization, and propagation. The final three graphs in every figure further quantify uncertainty by displaying the divergence between lower and upper fuzzy solutions, highlighting enhanced sensitivity near the soliton core.