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Access status: Open Access ,
A finite difference method for nonlinear parabolic-elliptic systems of second-order partial differential equations
(2007) Malec, Marian; Sapa, Lucjan
This paper deals with a finite difference method for a wide class of weakly coupled nonlinear second-order partial differential systems with initial condition and weakly coupled nonlinear implicit boundary conditions. One part of each system is of the parabolic type (degenerated parabolic equations) and the other of the elliptic type (equations with a parameter) in a cube in $\mathbf{R}^{1+n}$. A suitable finite difference scheme is constructed. It is proved that the scheme has a unique solution, and the numerical method is consistent, convergent and stable. The error estimate is given. Moreover, by the method, the differential problem has at most one classical solution. The proof is based on the Banach fixed-point theorem, the maximum principle for difference functional systems of the parabolic type and some new difference inequalities. It is a new technique of studying the mixed-type systems. Examples of physical applications and numerical experiments are presented.
Access status: Open Access ,
Existence, uniqueness and estimates of classical solutions to some evolutionary system
(2015) Sapa, Lucjan
The theorem of the local existence, uniqueness and estimates of solutions in Hölder spaces for some nonlinear differential evolutionary system with initial conditions is formulated and proved. This system is composed of one partial hyperbolic second-order equation and an ordinary subsystem with a parameter. In the proof of the theorem we use the Banach fixed-point theorem, the Arzeli-Ascola lemma and the integral form of the differential problem.
Access status: Open Access ,
Estimates of solutions for parabolic differential and difference functional equations and applications
(2012) Sapa, Lucjan
Theorems on the estimates of solutions for nonlinear second-order partial differential functional equations of parabolic type with Dirichlet’s condition and for suitable implicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent and stable difference method is considered without the assumption of the global generalized Perron condition posed on the functional variable but with the local one only. It is a consequence of our estimates theorems. In particular, these results cover quasi-linear equations. However, such equations are also treated separately. The functional dependence is of the Volterra type.
Access status: Open Access ,
Quaternionic Quantum Mechanics: the Particles, Their q-Potentials and Mathematical Electron Model
(AGH University Press, 2026) Bożek, Bogusław; Danielewski, Marek; Sapa, Lucjan
In this work we show the quaternionic quantum descriptions of physical processes from the Planck to macro scale. The results presented here are based on the concepts of the Cauchy continuum and the elementary cell at the Planck scale. The structurally symmetric quaternion relations and the postulate of the quaternion velocity have been important in the present development. The momentum of the expansion and compression $u̇_{0}(t, x)$ is the consequence of the scalar term $\sigma_{0}(t, x)$ in the quaternionic deformation potential. The quaternionic $G_{0}(m)(σ_{0} + \hat{\textstyle \phi} )$, vectorial $G_{0}(m) \hat{\textstyle \phi}$ and scalar $G_{0}(m)\sigma_{0}$ propagators are used to generate the second order PDE systems for the proton, electron and neutron. A mathematical model of an electron is formulated. It is described by the hyperbolic-elliptic partial differential system of quaternion equations with the initial-boundary conditions. The boundary conditions are generated by the quaternion energy flux that is found with the use of the Gauss theorem, the Cauchy–Riemann derivative and other mathematical formulas. The rigorous assessment of the second order PDE systems allows the proposal of two second order PDE systems for the $u$ and $d$ quarks from the up and down groups. It was verified that both the proton and the neutron obey experimental findings and are formed by three quarks. The proton and neutron are formed by the $d$-$u$-$u$ and $d$-$d$-$u$ complexes, respectively. The u and d quarks do not comply with the Cauchy equation of motion. The inconsistencies of the quarks’ PDE with the quaternion forms of the Cauchy equation of motion account for their short lifetime and the observed Quarks Chains. That is, they explain the Wilczek phenomenological paradox: Quarks are Born Free, but everywhere they are in Chains.

Sapa, Lucjan

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