The analytical solution of the Black-Scholes equation can lead to the attainment of the price of an option in an idealized fiscal market. However, this is not practically beneficial enough. This happens due to the constricting assumptions based on which the Black-Scholes model is derived. In the real financial market, one can question the constant nature of the coefficients of the Black-Scholes equation. In this paper, the solution of the Black-Scholes equation with constant parameters is reviewed. Next, the Black-Scholes equation solution taking time-dependent volatility and time-dependent risk-free interest rate into consideration is studied. Finally, a fully nonlinear model of Black-Scholes equation (Barls and Soner’s model) is considered. Because of the nonlinear nature of the model, there is not an analytical solution and numerical method is mandatory to price the derivative. Saul’yev finite difference scheme is proposed which not only is explicit, but also is unconditionally stable.

This article develops duality principles applicable to some originally non-convex primal variational formulations. More specifically, in a first step, we develop applications to a full complex Ginzburg-Landau system in superconductivity, including a magnetic field and respective magnetic potential. The results are obtained through basic tools of functional analysis, calculus of variations, duality and optimization theory in infinite dimensional spaces. It is worth emphasizing we have obtained convex dual variational formulations which may be applied to a large class of similar models in the calculus of variations. In the subsequent sections we also present a procedure for improving the convexity conditions of an originally non-convex primal formulation which is also applied to a Ginzburg-Landau type equation. Finally, in the last sections, we develop duality principles and related numerical examples for models in phase transition.

This paper investigates the existence, uniqueness, and stability of weak \(T\)-periodic solutions for nonlinear cooperative parabolic systems with Neumann boundary conditions a class of problems central to understanding recurrent phenomena in nature. By combining monotone iteration techniques with the method of sub- and super-solutions in ordered Banach spaces, we develop a robust framework that yields the existence of extremal periodic solutions between ordered barriers under general Carathéodory conditions and a cooperativity assumption on the nonlinearities. Uniqueness is further established under a mild Lipschitz condition. The power and applicability of the abstract results are demonstrated through their application to a time-periodic model of water-solute transport in porous media, offering new insights into the dynamics of solutes under environmental influence. Thus, the results of this study contribute to the theoretical advancement of periodic parabolic systems, mathematical modeling, and numerical simulation of periodic phenomena in hydrological and environmental sciences using the IMEX scheme, where seasonal phenomena such as annual recharge cycles, climatic fluctuations, and seasonal agricultural activities are particularly influenced by periodic conditions.

In this paper, we introduce the function class \(G_p^\eta(\mathbb{Z})\) alongside a more general class \(G_p(\mathbb{Z})\) to serve as parameter functions for generalized Morrey sequence spaces and generalized mixed Morrey double-sequence spaces. Using these classes, we establish necessary and sufficient conditions for the boundedness of the discrete Hardy–Littlewood maximal operator and its higher-order commutator on generalized Morrey sequence spaces and generalized mixed Morrey double-sequence spaces over \(\mathbb{Z}\). The boundedness of the maximal operator is characterized in terms of relations between the parameter functions. To obtain the sufficiency and necessity conditions, we utilize the properties of the \(G_p(\mathbb{Z})\) class. For the necessity conditions, we compute the norms of the characteristic functions of a discrete interval in generalized Morrey sequence spaces and of a rectangle in \(\mathbb{Z}\times\mathbb{Z}\) in generalized mixed Morrey double-sequence spaces. Furthermore, we characterize the boundedness of the higher-order commutator via the \(\operatorname{BMO}\) space. Specifically, the proof of sufficiency relies on the Fefferman–Stein inequality within these spaces, whereas the necessity proof adapts the techniques developed for the maximal operator itself.

In this paper, we study the extension of the Riemann zeta function to finite-dimensional commutative algebras and present examples of such extensions for several hypercomplex systems. We also investigate the zeros of the quaternionic zeta function and prove that its nontrivial zeros are spherical.

We develop fixed point and periodic point results for self-mappings on \(n\)-G-metric spaces. The definition is stated in a form compatible with the classical case \(n=3\), while the paper specializes to the genuinely higher-order case \(n\geq4\). The first result proves a sharp bound: a mapping satisfying an \(n\)-tuple contraction on pairwise distinct points has at most \(n-1\) periodic points, and hence at most \(n-1\) fixed points. This formulation makes clear the role of short periodic orbits, which are the natural obstruction to applying a distinct-tuple contraction along Picard blocks. Under \(d_G\)-completeness, orbital continuity and an \(n\)-orbit-separation condition, every Picard orbit either reaches a fixed point or converges to one. We also state direct \(n\)-ary Kannan- and Reich-type analogues and separate them from the standard consequences obtained from the associated metric. Finally, the total pairwise polygonal-length functional on normed spaces provides a concrete geometric model of the theory. To make the model computationally explicit, we add a Picard algorithm in \(n\)-G-metric spaces, pseudocode, convergence and error tables for several values of \(n\), \(\lambda\) and \(\rho\), and both linear and nonlinear discrete polygonal systems illustrating the comparison between the associated metric \(d_G\) and the pairwise polygonal-length dynamics.

An FR-PCM composite was produced for the development of an efficient material for thermal energy storage and fire protection through the incorporation of paraffin wax into a cassava starch-oil palm empty fruit bunch lignin-nano-bentonite-boric acid-based biopolymer-mineral matrix. Characterization of the material was done through FTIR, TGA/DTG, scanning electron microscopy (SEM), energy-dispersive spectroscopy (EDS), XRD, Brunauer–Emmett–Teller (BET) analysis, leakage test, thermal cycling, cone calorimetry, smoke-toxicity screening, mechanical properties assessment, and water uptake analysis. FTIR analysis revealed the presence of the C–H bands associated with paraffin wax as well as hydrogen bonding and borate interactions within the biopolymer-mineral matrix. TGA/DTG revealed the increase in the onset of decomposition from about 180 \(^\circ\)C for pure paraffin to about 245 \(^\circ\)C and the increase in residue at 600 \(^\circ\)C from about 15% for pure paraffin to about 30%. Cone-calorimetry smoke/toxicity screening test demonstrated the enhancement in fire performance attributes, such as the increase in time to ignition from 35 s for pure paraffin to 85 s, decrease in peak heat release rate from 900 to 380 kW m\(^{-2}\) and decrease in fire growth index from 2600 to 900 W s\(^{-1}\). Laboratory flame-spread testing revealed reduction in flame spread rate from 3.50 to 1.36 cm min\(^{-1}\) and increase in ignition delay to 97 s. SEM, EDS, XRD and BET analysis revealed the successful incorporation of biopolymer and mineral materials, maintenance of paraffin crystalline nature, control of mesoporosity and formation of cohesive protective char layer. Leakage and thermal cycling revealed the ability of the matrix to preserve 92% shape retention at 70 \(^\circ\)C and more than 90% latent heat retention after 100 thermal cycles. Mechanical and water-uptake properties assessment revealed improvement in the material structural characteristics together with introduction of mild moisture sensitivity. Overall, the FR-PCM composite provides a practical balance between latent heat storage, thermal transport, shape stability, and fire safety for energy-efficient building applications.

In selecting the suitable UAV for agricultural spraying, it is necessary to consider simultaneously the payload capacity, field productivity, endurance, spraying capacity, energy capacity, range of operation, weight, and charging requirement. In this paper, the authors have presented a multi-attribute decision analysis technique to rank alternative spraying UAVs based on a common set of criteria. The AHP method was used to evaluate the weights of criteria through reciprocal pairwise comparison, and then, the criterion weights are utilized to calculate the ranking orders of three spraying UAV alternatives using TOPSIS, MABAC, and ARAS methods. From AHP result, it can be seen that the weights of payload capacity, work efficiency, endurance time, and spraying flow performance are higher than other criteria. As a consequence, the results obtained from the above three ranking methods have been completely coincided and showed that the order of UAV alternatives are D-T5 > D-T4 > D-T2A. It shows that the decision is not affected by the choice of computational logic. In addition, sensitivity analysis on payload capacity and work efficiency criteria was performed, and there was no change in ranking order in considered intervals. Hence, the results revealed that D-T5 is superior alternative compared with other two alternatives in terms of capacity, efficiency, spray capability, and operational range while D-T4 and D-T2A are ranked in second and third positions, respectively. The study offers a transparent and technically consistent decision procedure for UAV selection in precision agriculture and related equipment-selection problems.