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We consider the Cauchy problem associated with a general parabolic partial differential equation in d dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as rigorous short-time error estimates. Using a boot-strapping technique, we also provide convergence results for arbitrarily large time intervals.

The charging time of Electric Vehicles (EVs) is an important parameter that greatly affects their operation. Its evaluation helps to estimate the distance traveled. The paper derives an analytical relationship that helps estimate the charging time of the remaining capacity of the EV battery. The relationship is based on a quadratic approximation of charging times for several categories of batteries with different capacities and charging stations with different power levels.

This study proposes a semi-analytic approximation to the laminar boundary layer growth in a polarized pressure field with temperature gradient represented by the joint Blasius-energy equation. We illuminate that is a probability density function (PDF) approximated by an amended Gaussian PDF with zero mean and standard deviation . This implies a diffusive structure for the molecular momentum conversion as well as the energy flux in the boundary layer. A new limit for the boundary layer edge is also presented. Results suggest an augmented boundary layer when compared to accepted values in the literature. We also reproduce the inverse proportionality of the free stream velocity to the diffusion of both momentum and energy.

The influence of landscape structure on epidemic invasion of agricultural crops is often underestimated in the construction and analysis of epidemiological models. Computer simulations of individual-based models (IBMs) are widely used to characterize disease spread under different management scenarios but can be slow in exploring large numbers of different landscape configurations. Here, we address the problem of finding an analytical measure of the impact of the spatial structure of a crop landscape on the invasion and spread of plant pathogens. We explore the potential of using an analytical approximation for the rate, r, at which susceptible crop fields become infected at the start of an epidemic to predict the effect that the spatial structure of a host landscape will have on an epidemic. We demonstrate the validity of this approach using two models: (i) a general IBM of the invasion and spread of a pathogen through an abstract host landscape; and (ii) an IBM of a real-life example for a virus disease spreading through a cassava landscape. Finally, we demonstrate that the analytical approach based on an estimate of the rate, r, can be used to identify spatial structures that effect deceleration of an invading pathogen.

The current paper addresses the lack of explicit analytical solutions for stress evaluations in variable-thickness flywheels by proposing an approximate formulation for conical profiles, where thickness varies linearly along the radius. The main objective was to develop a compact and practical expression to estimate radial and tangential stresses without relying on finite element analysis. Starting from a stress function, the model was simplified under the assumption of a small-thickness gradient, allowing the derivation of a closed-form solution. The resulting expression explicitly relates stresses to geometric and material parameters. To validate the approximation, stress distributions were computed for various outer-to-inner thickness ratios and compared with results obtained through FEA. The comparison, evaluated using the coefficient of determination, mean absolute percentage error, root mean squared error, normalized root mean squared error, and stress ratios, demonstrated strong agreement, especially for moderate-thickness ratios (1≦to/ti≦4.5). The method was more accurate for radial stress than tangential stress, particularly at higher gradients. The results confirmed that the proposed analytical approach provides a reliable and efficient alternative to numerical methods in the design and optimization of conical flywheels, offering practical value for early-stage engineering analysis and reducing reliance on time-intensive simulations.

We introduce a modern methodology for constructing global analytical approximations of special functions over their entire domains. By integrating the traditional method of matching asymptotic expansions—enhanced with Padé approximants—with differential evolution optimization, a modern machine learning technique, we achieve high-accuracy approximations using elegantly simple expressions. This method transforms non-elementary functions, which lack closed-form expressions and are often defined by integrals or infinite series, into simple analytical forms. This transformation enables deeper qualitative analysis and offers an efficient alternative to existing computational techniques. We demonstrate the effectiveness of our method by deriving an analytical expression for the Fermi gas pressure that has not been previously reported. Additionally, we apply our approach to the one-loop correction in thermal field theory, the synchrotron functions, common Fermi–Dirac integrals, and the error function, showcasing superior range and accuracy over prior studies.

In this paper, we investigate the pricing problem of barrier options under the Heston model. We innovatively develop a two-step solution process and present an analytical approximation formula of high efficiency and accuracy. In specific, upon assuming that all the future information of the volatility is known at the current time, the Heston model becomes a time-dependent Black-Scholes model, under which an analytical approximation for barrier option price is presented. The target barrier option price is essentially the expectation of the obtained conditional price with respect to the volatility, working out of which leads to an approximation involving a Fourier cosines series. Finally, the results of numerical experiments demonstrate that our formula has the potential to be applied in practice.

The aim of this paper is to derive an approximation formula for a single barrier option under local volatility models, stochastic volatility models, and their hybrids, which are widely used in practice. The basic idea of our approximation is to mimic a target underlying asset process by a polynomial of the Wiener process. We then translate the problem of solving first hit probability of the asset process into that of a Wiener process whose distribution of passage time is known. Finally, utilizing the Girsanov’s theorem and the reflection principle, we show that single barrier option prices can be approximated in a closed-form. Furthermore, ample numerical examples will show the accuracy of our approximation is high enough for practical applications.

Epidemiological modelling plays an important role in global food security by informing strategies for the control and management of invasion and spread of crop diseases. However, the underlying data on spatial locations of host crops that are susceptible to a pathogen are often incomplete and inaccurate, thus reducing the accuracy of model predictions. Obtaining and refining datasets that fully represent a host landscape across territories can be a major challenge when predicting disease outbreaks. Therefore, it would be an advantage to prioritize areas in which data refinement efforts should be directed to improve the accuracy of epidemic prediction. In this paper, we present an analytical method to identify areas where potential errors in mapped host data would have the largest impact on modelled pathogen invasion and short-term spread. The method is based on an analytical approximation for the rate at which susceptible host crops become infected at the start of an epidemic. We show how implementing spatial prioritization for data refinement in a cassava-growing region in sub-Saharan Africa could be an effective means for improving accuracy when modelling the dispersal and spread of the crop pathogen cassava brown streak virus.

The practical use of computational thermo-fluid dynamics (CFD) for food thermal process calculations still appears very premature due to both the high costs and the inhomogeneity and anisotropy of foods. Therefore, the traditional formula method with both Ball and Stumbo’s tables is still widely used due to its accuracy and safety. In both cases, the calculations require consulting and interpolating data from the respective tables, making the procedure slow and prone to human errors. The computerization of Ball’s tables to speed up and automate the calculations with a new mathematical approach based on the substitution of the integral exponential function and the initial cooling hyperbola has already been developed. The high accuracy obtained, superior to the direct regression of the table data, suggested adopting it also in the computerization of Stumbo’s tables. However, the latter are 14 times larger than those of Ball due to the extension of the thermo-bacteriological parameter z up to over 100 °C and the variability of the cooling lag factor Jcc. Therefore, the mathematical modelling was modified using an additional function, dependent on z and Jcc. The results obtained with the mathematical modelling showed a mean relative error and the standard deviation with respect to the Stumbo’s tables equal to MRE ± SD = 0.62% ± 1.29%. Further validation was obtained by calculating the thermal process time for different lethalities and thermo-bacteriological parameters with MRE ± SD compared to the Stumbo tables equal to 1.04% ± 0.82%.