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This paper analyses a Kirchhoff type quasilinear space-time fractional integro-differential equation with memory $(\mathcal{K}^{s}_{\alpha})$. Various a priori bounds are derived in different norms on the solution of the considered equation. Utilizing these a priori bounds, existence and uniqueness of the weak solution to the proposed model are proved. Furthermore, regularity results on the solution of $(\mathcal{K}^{s}_{\alpha})$ are established. The contribution made in this work provides a framework for further investigation of such types of partial integro-differential equation $(\mathcal{K}^{s}_{\alpha})$.

Background: The phase-shift analysis for proton-proton scattering has been studied by various research groups using the realistic potentials to be comprised of various internal interactions based on an exchange of pions and mesons, involving a large number of parameters. Purpose: The goal of the research is to construct inverse potentials for various l-channels of proton-proton (pp) elastic scattering using the 3-parameter Morse function in combination with atomic Hulthen by utilizing the phase function method and variational optimization technique. Methodology: The implementation of variational optimization begins with randomly assigning initial values to the Morse model parameters. Utilizing the Morse + Hulthen potential as input, the phase equations for various l-channels are numerically solved using the RK-5 method for obtaining the simulated Scattering Phase Shift (SPS). Mean Squared error between simulated and expected SPS has been chosen as the cost function. Variational optimization proceeds iteratively by adjusting potential parameters and re-evaluating the cost function until convergence is achieved. Results: All the obtained scattering phase shifts for various l-channels have been found to converge to a mean squared error <= 0.3. The computed cross-sections matched the experimental ones to less than 1% for energies up to 25 MeV. The scattering parameters are also found to closely match the experimental data. Conclusion: The inverse potentials constructed for various l-channels using Morse + atomic Hulthen are on par with the currently available high-precision realistic potentials., Comment: 13 pages, 5 figures, 5 Tables

An algorithm$^{\ref{Fig1}}$ has been developed with the purpose of obtaining inverse potentials, where the Riccati-type non-linear differential equation, also called phase equation, has been kept in tandem with the Variational Monte Carlo method. The optimization of Gaussian function parameters is achieved such that the experimental phase shifts are reproduced. The obtained SPS for various $\ ell$ channels has been compared with experimental ones with mean absolute percentage error (MAPE) as a measure. The model parameters have been optimised by suitable optimisation technique by looking for minimum value of MAPE. The results for $\ell$=0$^+$, 2$^+$ and 4$^+$ partial waves have been obtained, to match with experimental SPS, with MAPE values of $2.9$, $4.6$ and $6.2$ respectively for data up to $23$ MeV, while for higher states 6$^+$, 8$^+$ and 10$^+$ has MAPE of $3.2, 4.5$ and $5.9$ respectively for data from $53-120$ MeV. On extrapolation for data in range E$_{\ell ab.}$ = $23-120$ MeV, using the optimised parameters, the SPS are found to be in close agreement with experimental ones for the first three channels., Comment: 18 pages, 7 figures, 25 references

In this article, we propose a numerical approach to solve quantum mechanical scattering problems, using phase function method, by considering neutron-proton interaction as an example. The nonlinear phase equation, obtained from the time-independent Schrodinger equation, is solved using the Runge-Kutta method for obtaining S-wave scattering phase shifts for neutron-proton interaction modeled using Yukawa and Malfliet-Tjon potentials. While scattering phase shifts of S-states using Yukawa match with experimental data for only lower energies of 50 MeV, Malfliet-Tjon potential with repulsive term gives very good accuracy for all available energies up to 350 MeV. Utilizing these S-wave scattering phase shifts, low energy scattering parameters, and total S-wave cross section have been calculated and found to be consistent with experimental results. This simulation methodology can be easily extended to study scattering phenomenon using phase wave analysis approach in the realms of atomic, molecular, and nuclear physics., Comment: 19 pages, 6 figures, 2 Tables