High-order Sobolev-type equations are mathematical models used in many applied problems. As is known, in many cases, it is difficult to obtain analytical solutions to high-order equations; therefore, they are mainly solved by numerical methods. At present, the method of straight lines is often used to solve non-stationary problems of mathematical physics; in this method, discretization is first realized only in spatial variables, and the resulting system of ordinary differential equations of high dimension is solved by finite difference methods or finite elements of higher accuracy. In this study, for a system of ordinary differential equations of the fourth order, new multi-parameter difference schemes of higher accuracy based on the finite element method are constructed. The presence of parameters in the scheme makes it possible to regularize the schemes in order to optimize the implementation algorithm and the accuracy of the scheme. The stability and convergence of the constructed difference schemes are also proved, and accuracy estimates are obtained on their basis. An algorithm for the implementation of the constructed difference schemes is presented. The results obtained can be further applied in the numerical solution to initial-boundary value problems for the equations of dynamics of a compressible stratified rotating fluid, magnetic gas dynamics, ion-acoustic waves in a magnetized plasma, spin waves in magnets, cold plasma in an external magnetic field, etc.

When an irrigation canal is put on restriction, what is fair depends on the water each user receives, not on what was allocated: transport delay and the stabilising low-pass filter re-shape every order on the way to the offtake. We pose the problem as lexicographic max-min optimisation of the fraction of each user’s order delivered inside an agreed window, after the filter. The orders are the decision variables, re-shaped before the filter, leaving the controller and filter untouched. The delivered fractions are affine in the orders, so this is a finite sequence of linear programmes. Relaxing the control law alone yields a certified upper bound; an infeasibility certificate reports whose order cannot be met and what must give. On a published eight-pool test canal the criterion raises the worst-off delivered fraction by 0.074 to 0.180 over leaving orders unchanged; the bound is attained at nine of eleven operable supply levels, where the control layer loses nothing. A 50% larger volume budget leaves the lexicographic answer unchanged. The filter’s order decides whether a feasible schedule exists, its cut-off only what is delivered; neither moves the comparison between criteria much. On this canal filter design precedes the choice of fairness criterion.

On lossy irrigation trees, the nominal max–min guarantee has a closed form for deterministic data. With random demands and source capacity, the joint chance-constrained guarantee is the largest service level all users and periods meet jointly with a prescribed probability. A recursion on a loss-rescaled lattice brackets the probability that all capacities suffice; the one-period bracket on the guarantee has a leading-order relative width at most twice the binding-probability-weighted relative lattice step. In one-factor models, certified integration brackets the guarantee with bounded rounding errors and, under stated conditions, converges; in a generalized factor model, the guarantee does not decrease as period-factor correlations increase. Under stated conditions, its capacity elasticities equal binding probabilities. In the Gone Abat Jap network (rule-based capacities, assumed source parameters), all brackets at probability 0.95 lie below the nominal guarantee 0.85. If dry years also lower the source, the guarantee lies in [0.52755, 0.53282], whose lower end is 6.5–6.7% above estimated safe individual-constraint designs; the naive design misses the target. The source, the nominal bottleneck, binds at the guarantee with estimated conditional probability 0.926 (0.040 if deterministic).

This study presents numerical modeling and simulation of a two-temperature plasma system using the Python programming environment. This paper presents numerical modeling and simulation of a two-temperature plasma system using the Python programming environment. The model is based on a connected system of nonlinear equations of thermal conductivity describing the processes of energy exchange between electronic and ionic components. To obtain numerical solutions, an implicit finite-difference Crank–Nicholson scheme was implemented, providing second-order accuracy in time and space. The computational process was carried out using the NumPy, SciPy and SymPy libraries, and visualization of the results and user interaction were implemented using the Mathplotlib and Tkinter tools. The developed program allows you to interactively enter parameters, calculate temperature evolution and visualize the results in real time. Numerical experiments were carried out for the values of parameters corresponding to the natural and climatic conditions of the Republic of Karakalpakstan, where the density of the solar energy flux varies in the range of 1000-2800 W/m2. The results show the changes over time and space in electron and ion temperatures, confirming the Crank–Nicholson approach and the reliability of the developed software for modeling energy transfer in low-pressure plasma conditions.