According to the World Health Organization (WHO), Chronic Heart Disease (CHD) is one of the greatest defies currently confronting humankind which is sweeping the whole globe, with an expanding trend in developing countries. In this paper, a mathematical model (MM) was proposed to study the connection between fish consumption and CHD mortality in Egypt, by considering a system of ordinary differential equations (ODEs) involving time-fractional derivative (FD). We considered here the study on Egypt for the ease of obtaining real data, but the method and approach adopted here is not limited to Egypt only and can be applied to any country in the world with the information of the real data related to the subject of the study. Additionally, the control function which represents the metabolic and the behavioural risk factors of CHD that help to reduce the number of mortality due to CHD is incorporated in the proposed MM. A fractional optimal control problem (FOCP) with a proposed control is formulated and studied theoretically using the Pontryagin maximum principle, to minimize the susceptible population and also to decrease the mortality rate of CHD. Moreover, firstly we discussed the positivity and boundedness of solutions; then, the model equilibria are determined and their local stability analysis was investigated; furthermore, we use the improved forward-backward sweep method (FBSM) based on the predictor-corrector method (PCM) in order to obtain the solution of proposed FOCP. In addition, some numerical simulations were performed to show the effect of the proposed optimal control (OC) besides the impact of fish consumption on the mortality of CHD.

The present study introduces a novel analysis of the effect of the glass phase on the energy storage and the pyroelectric properties of 0.65(Bi_0.5Na_0.5)TiO₃-0.25SrTiO₃–0.1BaTiO₃ (abbreviate NBT-ST-BT) lead-free ceramics. The glass phase was prepared by melt-quenching of (Ba_0.9B_0.1)TiO₃ (BBT) calcined powder. Different content of BBT glass phase was introduced into the ceramic matrix [(1-x)(NBT-ST-BT)—x(BBT)] (x = 0.0, 2.5, 5, 7.5 and 10%) solid solution. The crystal structure shows rhombohedral and orthorhombic coexistence phases, increasing the R-phase volume fraction by increasing BBT glass content. The grain size was suppressed to a sub-micrometer by increasing the BBT glass amount, denoting the enhanced dielectric breakdown strength (BDS). The most significant recoverable energy storage density (W_rec = 2.5 J cm⁻³) with the highest energy storage efficiency (η ∼ 87%) has been obtained at 200 kV cm⁻¹ of BBT 5%. The variation in W_rec of the optimum sample is less than 4% from 25 °C to 150 °C, indicating the high thermal stability of energy storage properties. The pyroelectric coefficient (PE) was estimated using an approximate numerical method of differentiating remnant polarization P_r concerning temperature. Adding the BBT glass phase enhanced the pyroelectric properties and figure of merit (FOM). The FOM increased from 7 × 10⁻¹⁰ to 8 × 10⁻¹⁰ C/cm². °C at T = 150 °C when glass content increased from 0.0 to 0.1. These results prove that the addition of the BBT glass phase resolves the difference between high energy storage properties and lower sintering temperatures of ceramic materials, enhancing the pyroelectric properties for practical applications.

In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM on systems of nonlinear time-fractional PDEs. Precisely, we consider some important fractional-order nonlinear systems, namely Broer–Kaup (BK) and Burgers, which have found major significance because they arise in many physical applications such as shock wave, wave processes, vorticity transport, dispersal in porous media, and hydrodynamic turbulence. The analysis of these methods is implemented on the BK, Burgers systems and solutions have been offered in a simple formula. We show our results in figures and tables to demonstrate the efficiency and reliability of the used methods. Furthermore, our outcome converges rapidly to the given exact solutions.