Phase-lag mixed integral equation of a generalized symmetric potential kernel and its physical meanings in (3+1)...

Currently, there has been no research conducted on the impact of time on the structure of the overall nucleus, nor has there been any investigation into the influence of time delay on the three-dimensional integral equation, considering the temporal changes. This research is being studied to understand its meaning. In order to analyze a phase-lag mixed integral equation (P-LMIE) in dimensions (3+1), in $L_2\left(\Omega \right)\times C\left[0,T\right],T<1\text{,}$ where $\Omega =\left\{\left(x,y,z\right)\in \Omega :\sqrt{x^2+y^2}\le a,z=0\right\}$ is the position domain of integration and T is the time. Some specific assumptions were established. The position kernel was imposed, according to Hooke's law, as a generalized potential function in $L_2\left(\Omega \right)\text{.}$ By applying the properties of fractional calculus, it is possible to get an integro-differential Fredholm-Volterra integral equation (Io-DF-VIE). The kernel employs the generalized Weber-Sonien integral formula by utilizing polar coordinates. Moreover, the separation approach is utilized to convert the MIE into m-harmonic Fredholm integral equations (FIEs) with kernels expressed in the Weber-Sonien integral forms and coefficients involving both temporal and fractional components. The degenerate method is employed to deduce the linear algebraic system (LAS). In addition, our endeavor yielded novel and distinct instances. In addition, Maple 2018 and mathematical programming are utilized to calculate numerical values for various coefficients related to the Weber-Sonien integral and its harmonic degree.