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This paper comprehensively studies effective numerical methods for solving the simplest chaotic circuit model. We introduce a novel scheme for the Atangana–Baleanu Caputo fractional derivative (ABC-FD), coupled with the Laplace decomposition method (LDM). Furthermore, we rigorously compare the performance of these proposed methods with the Runge–Kutta fourth-order method. Using two mathematical techniques, we have discovered effective and highly convergent solutions to the chaotic model. We gave different values to the parameters to plot the chaos and create a phase portrait of the system. Therefore, the provided methods can be applied to more sophisticated examinations of different models. This study advances numerical techniques for understanding chaotic dynamics in complex systems. By introducing a novel scheme for the Atangana–Baleanu Caputo fractional derivative and the Laplace decomposition method, we provide a robust framework for effectively solving the simplest chaotic circuit model. This framework enhances accuracy and efficiency in unraveling chaotic behaviors, contributing to a broader understanding of chaotic dynamics across scientific domains in the future.

In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a mean field game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader facing a “background noise” (or “mean field”). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He also has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of “extended MFG”, we hence provide generic results to address these “MFG of controls”, before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of “heterogenous preferences” (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can “learn” it day after day, observing others’ behaviors.

Game theory is a field of applied mathematics that studies strategic behavior of rational factors. In other words, game theory is a collection of analytical tools that can be used to make optimal choices in interactional and decision making problems. Optimization in mathematics and computer science is the choice of the best member of an existing collection for a specific purpose. Several optimization methods have been used in many problems to minimize costs or maximize profits. From a particular point of view, it can be said that the game theory is in fact a kind of optimization. In this paper, a combined use of game theory and optimization algorithms has been reviewed and a new categorization is presented for researches which have been conducted in this area. In some of these combinations, game theory has been used to improve the performance of optimization algorithms, and in some others, optimizations methods help to solve game theory problems. Game theory and optimization algorithms are also used together to solve some other problems.

When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple “flattening of the curve”. Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.

A data-driven approach called CaNN (Calibration Neural Network) is proposed to calibrate financial asset price models using an Artificial Neural Network (ANN). Determining optimal values of the model parameters is formulated as training hidden neurons within a machine learning framework, based on available financial option prices. The framework consists of two parts: a forward pass in which we train the weights of the ANN off-line, valuing options under many different asset model parameter settings; and a backward pass, in which we evaluate the trained ANN-solver on-line, aiming to find the weights of the neurons in the input layer. The rapid on-line learning of implied volatility by ANNs, in combination with the use of an adapted parallel global optimization method, tackles the computation bottleneck and provides a fast and reliable technique for calibrating model parameters while avoiding, as much as possible, getting stuck in local minima. Numerical experiments confirm that this machine-learning framework can be employed to calibrate parameters of high-dimensional stochastic volatility models efficiently and accurately.

Nowadays, engineers and biochemical industries have benefited greatly from optimal control analysis and its computational methods. Furthermore, the optimal control theory is a powerful instrument in infectious disease modeling and control of vibration in civil engineering structures under random loadings. In this paper, a new solution representation and optimal control of second-order Hilfer fractional stochastic integro-differential systems (HFSIDSs) with non-instantaneous impulsive (NI) are studied. Existence and uniqueness of solutions are proved in the finite-dimensional space by using Schaefer’s type fixed-point theorem with low conservative conditions on nonlinear part. Further, Lagrange problem is considered to establish optimal control results for HFSIDSs with NI. Finally, a pharmacotherapy type Hilfer fractional model is discussed in the example section.

Coupling of mechanical, thermal and magnetic fields attracts the scientific community due to its numerous applications in geophysics, engineering, structures, aeronautics etc. To study the magneto-thermo-mechanical-interactions caused by laser heat input inside an infinite half-space structure, current investigation address a new generalized thermoelastic model incorporating nonlocal Moore–Gibson–Thompson approach with memory-dependent derivatives. A heat transfer equation half-space media is being pronounced with the magnetic field. A heat transfer equation based on memory-dependent derivatives is formulated by Eringen’s assumptions of nonlocal impact. The closed form solutions for the half-space system are determined in the Laplace transform domain. The distributions of physical fields such as temperature, displacement, thermal stress and stain are obtained in physical domain by adopting an approximation algorithm. With the help of computational outcomes and the graphical figures, the effects of effective parameters such as non-singular kernel functions, time delay and nonlocal quantum are revealed on the variations of the field quantities. Further, in order to exhibit the attractiveness of the nonlocal MGT model, a comparison of the current thermal conductivity model with previously established nonlocal classical and nonlocal generalized thermal conductivity models is made through the graphical results.

The integration of machine learning (Keplerian paradigm) and more general artificial intelligence technologies with physical modeling based on first principles (Newtonian paradigm) will impact scientific computing in engineering in fundamental ways. Such hybrid models combine first principle-based models with data-based models into a joint architecture. This paper will give some background, explain trends and showcase recent achievements from an applied mathematics and industrial perspective. Examples include characterization of superconducting accelerator magnets by blending data with physics, data-driven magnetostatic field simulation without an explicit model of the constitutive law, and Bayesian free-shape optimization of a trace pair with bend on a printed circuit board.

Gas flow through layers of porous materials plays a crucial role in technical applications, geology, petrochemistry, and space sciences (e.g., fuel cells, catalysis, shale gas production, and outgassing of volatiles from comets). In many applications the Knudsen regime is predominant, where the pore size is small compared to the mean free path between intermolecular collisions. In this context common parameters to describe the gas percolation through layers of porous media are the probability of gas molecule transmission and the Knudsen diffusion coefficient of the medium. We show how probabilistic considerations on layer partitions lead to the analytical description of the permeability of a porous medium to gas flow as a function of layer thickness. The derivations are made on the preconditions that the molecule reflection at pore surfaces is diffuse and that the pore structure is homogenous on a scale much larger than the pore size. By applying a bi-hemispherical Maxwell distribution, relations between the layer transmission probability, the half-transmission thickness, and the Knudsen diffusion coefficient are obtained. For packings of spheres, expressions of these parameters in terms of porosity and grain size are derived and compared with former standard models. A verification of the derived equations is given by means of numerical simulations, also providing evidence that our analytical model for sphere packing is more accurate than the former classical models.

The free vibration behavior of a new advanced functionally graded (FG) nanobeam is presented in this work using the recently proposed nonlocal higher-order shear deformation theory. In the present theory, the stress tensor can satisfy the parabolic variation of the shear stress distribution throughout the thickness direction and also fulfill the requirement that the shear stress on the top and bottom surfaces of the FG nanobeam is zero. Two common types of FG structures, namely, FG hardcore and FG softcore, are considered here for analysis with three schemes. The material properties of the FG nanobeam are assumed to vary continuously in both the longitudinal and transversal directions according to a combined simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of the FG nanobeam with simply supported boundary conditions are derived using the proposed higher-order shear deformation plate theory. The nonlocal strain gradient theory is employed to capture the microstructure-dependent effect. The influence of the structural geometry, the gradient index, and the nonlocal and length scale parameters on the vibration frequency is investigated. Finally, many new results are also reported in the current study, which will serve as a benchmark for future research.