Band 25, Heft 5-6

The shock propagation in non-ideal self-gravitating gas with the azimuthal magnetic field and the monochromatic radiation effect have been investigated utilizing the Lie group theoretic method. The gas is assumed to be at rest having constant initial density and the magnetic fields in front of the shock front are assumed to be varying with time. For this problem, we obtained analytical and numerical solutions for the different classes of infinitesimal generators. The impacts of varying the strength of the magnetic field, gravitational parameter, parameter of non-idealness, and adiabatic exponent of the gas on the flow variable in the flow field region behind the shock, and on the moving shock with an exponential shock path have been studied in detail. The software package “Mathematica” has been used for numerical computation to obtain a similarity solution. It has been shown that the presence of magnetic field, non-idealness of the gas, and adiabatic exponent of the gas have decaying effect on the shock wave, but in the presence of the gravitation the shock strength is enhanced. The entire energy of the disturbance is not constant is not constant and varies with some power of the shock radius in the flow field behind the shock wave.

The standard Halanay inequality is generalized by considering fractional derivatives instead of first-order derivatives and also the nonlinear case. Moreover, we shall deal with a large class of kernels in distributed delays. It is then used to prove the Mittag-Leffler stability of a fractional nonlinear Hopfield neural network system with activation functions that are not necessarily Lipschitz continuous. The findings are supported by two explicit classes of admissible kernels and an example.

Aided by flux-approximation limits of solutions to the Suliciu relaxation system with one independent parameter, we identify the phenomena of concentration and cavitation. It turns out that, as flux perturbation decreases to a certain critical value, any Riemann solution containing two shocks and possibly one-contact-discontinuity of the perturbed Suliciu relaxation system converges to its own delta-shock solution. As the parameter goes to zero further, the limiting solution is just the delta-shock solution of the pressureless gas dynamics model; any Riemann solution containing two rarefaction waves and possibly one contact-discontinuity converges to a solution involving vacuum state of the pressureless gas dynamics model. In the end we show some numerical results to exhibit the formation process of delta shock wave and vacuum states as the flux perturbation decreases.

Lumpy skin disease, attributed to the lumpy skin disease virus, is a viral ailment that affects cattle. This study presents a novel operational matrix employing Bernstein wavelets to examine the nonlinear dynamics of the fractional-order model of lumpy skin disease. The precise mechanism of the Riemann–Liouville arbitrary integral operator applied to Bernstein wavelets is elucidated, employing an arbitrary-order derivative in the Liouville–Caputo approach. Further, the operational matrix is employed to convert the given differential equations into a system of algebraic equations. The Adams–Bashforth–Moulton (ABM) method is also investigated as a potential solution to address the problem. A comparative analysis was performed using the ABM scheme to evaluate the precision and appropriateness of the Bernstein wavelet approach. The proposed method was validated through a convergence analysis of the Bernstein wavelet. To verify the theoretical insights and demonstrate the efficiency of the methodology, numerical simulations were ultimately performed.