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This paper reports the filamentation of hollow Gaussian beams of the first, second, third, and fourth orders during propagation in a cubic and quintic nonlinear medium. Due to spatial modulation instability, the hollow Gaussian beams split to form either co-centric circular filaments or ultrashort pulses. It is found that the properties of the nonlinear medium used for propagation have a strong influence on certain characteristics of the formed filaments, such as peak intensity and pulse width. This correlation between the system parameters of the medium and filament characteristics represents a method for calculating the system parameters of the medium. This investigation can be helpful in the development of a hollow Gaussian beam-based artificial intelligence system that can be used to measure the system parameters of the studied nonlinear medium.

In this work, by considering spatial uniform meshes and stencils having five adjacent discretization nodes, we furnish a numerical scheme to solve the time-fractional Black–Scholes (partial differential equation) PDE to price financial options under the generalized multiquadric radial basis function (RBF). The time-fractional derivative is estimated by an L1-scheme but the spatial variable is discretized using fourth-order RBF-FD methodology. As a matter of fact, the PDE problem is transformed in the form of a linear set of algebraic equations. To support analytical discussions, numerical tests are furnished and reveal the efficacy of the presented solver.

This paper proposes a computational solver via the localized radial basis function finite difference (RBF-FD) scheme and the use of graded meshes for solving the time-dependent Bates partial integro-differential equation (PIDE) arising in computational finance. In order to avoid facing a large system of discretization systems, we employ graded meshes along both of the spatial variables, which results in constructing a set of ordinary differential equations (ODEs) of lower sizes. Moreover, an explicit time integrator is used because it can bypass the need to solve the large discretized linear systems in each time level. The stability of the numerical method is discussed in detail based on the eigenvalues of the system matrix. Finally, numerical tests revealed the accuracy and reliability of the presented solver.

The Heston–Hull–White three-dimensional time-dependent partial differential equation (PDE) is one of the important models in mathematical finance, at which not only the volatility is modeled based on a stochastic process but also the rate of interest is assumed to follow a stochastic dynamic. Hence, an efficient method is derived in this paper based on the methodology of the localized radial basis function generated finite difference (RBF-FD) scheme. The proposed solver uses the RBF-FD approximations on graded meshes along all three spatial variables and a high order time-stepping scheme. Stability is also studied in detail to show under what conditions the proposed method is stable. Computational simulations are given to support the theoretical discussions.

Thermal treatment followed by subsequent cooling conditions (slow and rapid) can induce damage to the rock surface and internal structure, which may lead to the instability and failure of the rock. The extent of the damage is measured by the damage factor (DT), which can be quantified in a laboratory by evaluating the changes in porosity, elastic modulus, ultrasonic velocities, acoustic emission signals, etc. However, the execution process for quantifying the damage factor necessitates laborious procedures and sophisticated equipment, which are time-consuming, costly, and may require technical expertise. Therefore, it is essential to quantify the extent of damage to the rock via alternate computer simulations. In this research, a new predictive model is proposed to quantify the damage factor. Three predictive models for quantifying the damage factors were developed based on multilinear regression (MLR), artificial neural networks (ANNs), and the adoptive neural-fuzzy inference system (ANFIS). The temperature (T), porosity (ρ), density (D), and P-waves were used as input variables in the development of predictive models for the damage factor. The performance of each predictive model was evaluated by the coefficient of determination (R2), the A20 index, the mean absolute percentage error (MAPE), the root mean square error (RMSE), and the variance accounted for (VAF). The comparative analysis of predictive models revealed that ANN models used for predicting the rock damage factor based on porosity in slow conditions give an R2 of 0.99, A20 index of 0.99, RMSE of 0.01, MAPE of 0.14, and a VAF of 100%, while rapid cooling gives an R2 of 0.99, A20 index of 0.99, RMSE of 0.02, MAPE of 0.36%, and a VAF of 99.99%. It has been proposed that an ANN-based predictive model is the most efficient model for quantifying the rock damage factor based on porosity compared to other models. The findings of this study will facilitate the rapid quantification of damage factors induced by thermal treatment and cooling conditions for effective and successful engineering project execution in high-temperature rock mechanics environments.

In this article, the Laplace distribution is employed in lieu of the well-known normal distribution for finding better scalar values of risk. Explicit formulas for value-at-risk (VaR) and conditional value-at-risk (CVaR) are studied and used to manage the risk involved in a stock movement by using the GARCH model. Numerical simulations are given for a variety of stocks in equity markets to uphold the findings.

The target of this work is to present a multiplication-based iterative method for two Hermitian positive definite matrices to find the geometric mean. The method is constructed via the application of the matrix sign function. It is theoretically investigated that it has fourth order of convergence. The type of convergence is also discussed, which is global under an appropriate choice of the initial matrix. Numerical experiments are reported based on input matrices of different sizes as well as various stopping termination levels with comparisons to methods of the same nature and same number of matrix–matrix multiplications. The simulation results confirm the efficiency of the proposed scheme in contrast to its competitors of the same nature.

In this paper, we first derive a family of iterative schemes with fourth order. A weight function is used to maintain its optimality. Then, we transform it into methods with several self-accelerating parameters to reach the highest possible convergence rate 8. For this aim, we employ the property of the eigenvalues of the matrices and the technique with memory. Solving several nonlinear test equations shows that the proposed variants have a computational efficiency index of two (maximum amount possible) in practice.

Nowadays, the problem of solar thermal absorption plays a vital role in energy storage in power plants, but within this phenomenon solar systems have a big challenge in storing and regulating energies at extreme temperatures. The solar energy absorber based on hybrid nanofluids tends to store thermal energy, and the hybrid nanofluids involve the stable scattering of different nano dimension particles in the conventional solvent at a suitable proportion to gain the desired thermophysical constraints. The authors focus on the behavior of the inclined plate absorber panel as the basic solution of water is replaced by a hybrid nanofluid, including Cu (Copper) and Al2O3 (Aluminum Oxide), and water is utilized as a base surfactant in the current investigation. The inclined panel is integrated into a porous surface with the presence of solar radiations, Joule heating, and heat absorption. The fundamental equations of the flow and energy model are addressed with the similarity transformations. The homotopy analysis method (HAM) via Mathematica software is used to explore the solution to this problem. Furthermore, the important physical characteristics of the rate of heat transfer, omission and absorption of solar radiation, and its impact on the solar plant are observed.

The goal of this article is to investigate a new solver in the form of an iterative method to solve X+A∗X−1A=I as an important nonlinear matrix equation (NME), where A,X,I are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions.