Differential Equation Approach Research Articles

On the stochastic uncertainties of thermally and optically stimulated luminescence signals: A Monte Carlo approach

Phenomenological models are frequently used to analyze experimental signals in thermally and optically stimulated luminescence experiments. Typically, these models consist of systems of differential equations describing various electronic transitions. An alternative to the differential equation approach is the use of Monte Carlo (MC) methods, which also allow an estimation of the theoretical stochastic uncertainty of the intensity of the luminescence signal. By running and averaging several MC variants, these stochastic uncertainties are estimated in this paper for various luminescence models.In the case of first-order kinetics processes, the MC results compare well with previously published analytical results for the coefficient of variation (CV) in stochastic linear pure death processes. By contrast, no analytical results are available for the more general one trap one recombination center model (OTOR), and MC is the only method available for estimating the stochastic uncertainties.In this paper the CV coefficients are simulated for three commonly used experimental stimulation modes, namely thermally stimulated luminescence (TL), continuous-wave optically stimulated luminescence (CW-OSL) and linearly modulated OSL (LM-OSL). The results of the simulations show that CW-OSL signals have the smallest CV values among the three stimulation modes, and therefore these signals are least likely to exhibit stochastic variations. The stochastic uncertainties in these phenomenological models are discussed in the context of single grain luminescence experiments and nanodosimetric materials, in which one deals with small numbers of charge carriers.

Efficient valuation of a variable annuity contract with a surrender option

We study the partial differential equation (PDE) approach for efficient and accurate valuation of a variable annuity (VA) contract with a surrender option. Specifically, using the Laplace–Carson Transform (LCT), we derive an analytic pricing formula for a VA contract with a surrender option which is formulated as a PDE with an optimal surrender boundary. To demonstrate the efficiency and accuracy of our approach, we show that our pricing formula efficiently provides the exact value of a VA contract. Moreover, we compare the performance of three numerical methods for Laplace inversion to find the most efficient method. Among these methods, we found that the Gaver–Stehfest method is the most efficient pricing for a VA contract with a surrender option.

Time Dimensional Consistency Aware Analysis of Voltage Mode and Current Mode Active Fractional Circuits

In this research, the analysis of the active fractional circuits has been performed by using the fractional differential equation approach. Both voltage and current mode circuits have been taken into account. The fractional time component parameters have been included in the derivative terms within the fractional differential equations. This is because the consistency in time dimension between the fractional derivative and the conventional one which is also related to the physical measurability, is concerned. The fractional derivatives have been interpreted in Caputo sense. The resulting analytical solutions of the time dimensional consistency aware fractional differential equations have been determined. We have found that the dimensional consistency between both sides of the equations of the solutions which cannot be achieved in the previous works, can be obtained. By applying different source terms to the obtained analytical solutions, the response of both voltage and current mode circuits have been determined and the behaviours of the circuits have been analysed. The fractional time constant and pole locations in the F-plane of these circuits have been determined. Their dynamic behaviours, stabilities have been analysed. Moreover, the discussion on circuit realizations with fractional capacitor has also been made.

Non-stationary spatial regression for modelling monthly precipitation in Germany

It is widely accepted that spatial dependencies have to be acknowledged appropriately in data that are spatially aligned. However, most spatial models still assume that the dependence structure does not vary over space, i.e., it is stationary. While assuming stationarity considerably facilitates estimation, it is often too restrictive when describing atmospheric phenomena such as precipitation. Nonetheless, the applicability of non-stationary models is often hindered, as their use reveals to be cumbersome and improvements over stationary models can be hard to identify. The stochastic partial differential equation (SPDE) approach to spatial modelling allows for flexible specifications of non-stationary models. In particular, given the German orographic diversity, it makes sense to investigate potential non-stationarity in precipitation patterns. Taking such potential non-stationarities into account may, in particular, leads to improved smoothing of the observed precipitation pattern taken on a finite set of measurement stations, and therefore to improved inputs to hydrological models. We suggest an SPDE-based model where the mean and dependence structure are allowed to vary with elevation, as well as a more flexible non-parametric alternative based on multivariate B-Splines. As factors such as wind may cause higher dependence in a given direction, we include anisotropy in the model. Results show that, according to the widely applicable Bayesian information criterion, a non-stationary model provides a better fit to the data. Taking German monthly precipitation as a motivation, we set up a simulation study to explore the ability of the elevation and spline-based models to correctly identify non-stationarity.

Regular and Jost states for the S-wave Manning–Rosen potential

A regular solution for the Manning–Rosen potential is constructed by adapting the differential equation approach to the problem. The Jost solution and the Jost function are found by exploiting the relation between regular and irregular solutions. The Jost function thus obtained is applied for the first time to find bound state energies and the scattering phase shifts for nuclear systems.

Mechanism-Enabled Population Balance Modeling of Particle Formation en Route to Particle Average Size and Size Distribution Understanding and Control.

The concept of Mechanism-Enabled Population Balance Modeling (ME-PBM) is reported, illustrated by its application to a prototype Ir(0)n nanoparticle formation reaction. ME-PBM is defined herein as the use of now available, experimentally established, disproof-based, deliberately minimalistic mechanisms of particle formation as the required input for more rigorous Population Balance models, critically including an experimentally established nucleation mechanism. ME-PBM achieves the long-sought goal of connecting such now available experimental minimum mechanisms to the understanding and rational control of particles size and size distributions. Twelve pseudoelementary step, particle-formation mechanisms are considered so that the approach to the ME-PBM is also extensively disproof-based. Resurrection of Smoluchowski's 1918 full Ordinary Differential Equation (ODE) approach to the PBM is another, critical aspect of our approach which, in turn, allows unbiased fitting of the information-laden particle-size distribution (PSD) including its shape. The results provide one solution to the "inverse problem" in which the PSD informs one as to the correct particle formation mechanism: A new, deliberately minimalistic 3-step particle-formation mechanism has been uncovered that is a single-additional-step expansion of the now broadly used Finke-Watzky (FW) 2-step mechanism, the new 3-step mechanism being: A → B (rate constant k1), A + B → C (rate constant k2), and A + C → 1.5C (rate constant k3), where A represents the monomeric nanoparticle precursor, B represents "small" nanoparticles, and C represents "larger" nanoparticles. The results strongly support three paradigm shifts for nucleation and growth of particles, the most critical paradigm shift being that the "burst" nucleation assumption in LaMer's 1950s model of particle formation is not required to produce narrow, near-monodisperse PSDs. Instead, narrow PSDs can be and are achieved despite continuous nucleation because smaller particles grow faster than larger ones, k2 > k3, thereby allowing the smaller particles to catch up in size to the more slowly growing larger particles.

Flexible modelling of spatial variation in agricultural field trials with the R package INLA

Key messageEstablished spatial models improve the analysis of agricultural field trials with or without genomic data and can be fitted with the open-source R package INLA.The objective of this paper was to fit different established spatial models for analysing agricultural field trials using the open-source R package INLA. Spatial variation is common in field trials, and accounting for it increases the accuracy of estimated genetic effects. However, this is still hindered by the lack of available software implementations. We compare some established spatial models and show possibilities for flexible modelling with respect to field trial design and joint modelling over multiple years and locations. We use a Bayesian framework and for statistical inference the integrated nested Laplace approximations (INLA) implemented in the R package INLA. The spatial models we use are the well-known independent row and column effects, separable first-order autoregressive (mathrm{AR1} otimes mathrm{AR1}) models and a Gaussian random field (Matérn) model that is approximated via the stochastic partial differential equation approach. The Matérn model can accommodate flexible field trial designs and yields interpretable parameters. We test the models in a simulation study imitating a wheat breeding programme with different levels of spatial variation, with and without genome-wide markers and with combining data over two locations, modelling spatial and genetic effects jointly. The results show comparable predictive performance for both the mathrm{AR1} otimes mathrm{AR1} and the Matérn models. We also present an example of fitting the models to a real wheat breeding data and simulated tree breeding data with the Nelder wheel design to show the flexibility of the Matérn model and the R package INLA.

Enhancement of directivity of dipole antenna by a complex conjugate cylinder

ABSTRACT The directional properties of a dipole antenna are studied by placing it in the vicinity of a complex conjugate cylinder. An exact formal solution for the fields is obtained via the differential equation approach. The far field approximation is obtained by asymptotic analysis. The variation of radiation patterns with the imaginary part of permittivity and/or permeability is observed and the results are compared with the lossless dielectric. It is observed that the directivity of the antenna is enhanced in the presence of the cylinder.

Mean-variance asset-liability management with affine diffusion factor process and a reinsurance option

ABSTRACTThis paper considers an optimal asset-liability management (ALM) problem for an insurer under the mean-variance criterion. It is assumed that the value of liabilities is described by a geometric Brownian motion (GBM). The insurer's surplus process is modeled by a general jump process generated by a marked point process. The financial market consists of one risk-free asset and n risky assets with the risk premium relying on an affine diffusion factor process. By transferring a proportion of insurance risk to a reinsurer and investing the surplus into the financial market, the insurer aims to maximize the expected terminal net wealth and, at the same time, minimize the corresponding variance of the terminal net wealth. By using a backward stochastic differential equation (BSDE) approach, closed-form expressions for both the efficient strategy and efficient frontier are derived. To illustrate the main results, we study an example with the Heston stochastic volatility (SV) model and numerically analyze the economic behavior of the efficient frontier. Finally, a generalization of the Mutual Fund Theorem is obtained.

A finite difference discretization method for heat and mass transfer with Robin boundary conditions on irregular domains

This paper proposes a finite difference discretization method for simulations of heat and mass transfer with Robin boundary conditions on irregular domains. The level set method is utilized to implicitly capture the irregular evolving interface, and the ghost fluid method to address variable discontinuities on the interface. Special care has been devoted to providing ghost values that are restricted by the Robin boundary conditions. Specifically, it is done in two steps: 1) calculate the normal derivative in cells adjacent to the interface by reconstructing a linear polynomial system; 2) successively extrapolate the normal derivative and the ghost value in the normal direction using a linear partial differential equation approach. This method produces second-order accurate solutions for both the Poisson and heat equations with Robin boundary conditions, and first-order accurate solutions for the Stefan problems. The solution gradients are of first-order accuracy, as expected. It is easy to implement in three-dimensional configurations, and can be straightforwardly generalized into higher-order variants. The method thus represents a promising tool for practical heat and mass transfer problems involving Robin boundary conditions.

Off-shell Jost function for the Hulthén potential in all partial waves

A new expression for the Hulthén off-shell Jost function in all partial waves is constructed in its maximal reduced form. As a basic requirement the on-shell solutions are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics. Utilizing the properties of the Jost function, the binding energies and phase shifts for N-N and n-d systems are computed and found excellent agreement with standard data.

Optimal Release Time Determination Using FMOCCP Involving Randomized Cost Budget for FSDE-Based Software Reliability Growth Model

This paper presents a software reliability growth model (SRGM) based on a fuzzy stochastic differential equation (FSDE) approach. The software shall comprise of dependent and independent main and sub-components owing to a wide range of functionalities in the system. In order to obtain the optimal date of product release, a two-phase fuzzy goal programing (FGP) method has been applied for solving a fuzzy multi-objective chance-constrained programming (FMOCCP) model. While framing the cost function, the costs associated with changes in software specifications have been considered along with other cost factors. In most cases, the changes in software specifications during the software development process are dynamic in nature necessitating the cost budget to be considered as a random variable while formulating the fuzzy chance-constrained model. The SRGM proposed has been validated on real-time data sets and numerical illustrations have been provided to instantiate the optimal time of product release.

A quasi-Lagrangian moving mesh discontinuous Galerkin method for hyperbolic conservation laws

A moving mesh discontinuous Galerkin method is presented for the numerical solution of hyperbolic conservation laws. The method is a combination of the discontinuous Galerkin method and the mesh movement strategy which is based on the moving mesh partial differential equation approach and moves the mesh continuously in time and orderly in space. It discretizes hyperbolic conservation laws on moving meshes in the quasi-Lagrangian fashion with which the mesh movement is treated continuously and no interpolation is needed for physical variables from the old mesh to the new one. Two convection terms are induced by the mesh movement and their discretization is incorporated naturally in the DG formulation. Numerical results for a selection of one- and two-dimensional scalar and system conservation laws are presented. It is shown that the moving mesh DG method achieves the second and third order of convergence for P1 and P2 elements, respectively, for problems with smooth solutions and is able to capture shocks and concentrate mesh points in non-smooth regions. Its advantage over uniform meshes and its insensitiveness to mesh smoothness are also demonstrated.

Multilayer Airfoil Ice Accretion Simulations Using a Level-Set Method with B-Spline Representation

This paper presents a partial differential equation approach to model the icing front evolution on airfoils via the level-set equation, offering a high-fidelity alternative to common algebraic approaches. Furthermore, the automation of the multilayer icing process is achieved by using B-spline curves to improve the geometry representation, which is usually limited to linear segments. The B-splines are generated using a curve approximation algorithm based on an error-controlled knot insertion together with a least-square minimization process, allowing an adequate shape representation of the ice irregular features. The evolution of the level-set interface depends on the so-called icing velocity, which is computed locally in the field from the propagated surface ice mass accretion rate solution. Additionally, an explicit swept volume tracking method controls actively the global conservation of the ice volume. A verification phase is performed on analytical cases to highlight the characteristics of the methods and discretization schemes. Multilayer icing results are then presented on two cases and compared with experimental data providing a reasonable global mass error level in each case. Overall, the method shows great potential to model the ice geometry evolution in a globally conservative manner while ensuring a smooth geometry representation that improves multilayer automation.

Proving the existence of bound states for attractive potentials in one-dimension and two-dimensions without calculus

Schrödinger developed an operator method for solving quantum mechanics. While this technique is overshadowed by his more familiar differential equation approach, it has found wide application as an illustration of supersymmetric quantum mechanics. One reason for reticence in its usage for conventional quantum instruction is that the approach for simple problems like the particle-in-a-box is much more complicated than the differential equation approach, making it appear to be less useful for pedagogy. We argue that the approach is still quite attractive because it employs only algebraic methods, and thereby has a much lower level of math background needed to use it. We show how Schrödinger’s operator method can be streamlined for these particle-in-a-box problems greatly reducing the complexity of the solution and making it much more accessible. As an application, we illustrate how this approach can be used to prove an important result, the existence of bound states for one- and two-dimensional attractive potentials, using only algebraic methods. The approach developed here can be employed in undergraduate classes and possibly even high school classes because it employs only algebra and requires essentially no calculus.

The Time Dimensional Measurability Aware FDE Based Analysis of Active Circuit in The Fractional Domain

In this research, the analysis of the active circuit in the fractional domain has been performed by using the fractional differential equation approach where themeasurability in time dimension of the derivative term has also been concerned unlike the previous work. The OTA-C filter has been adopted as the candidate active circuit due to its compactness and renown. The derivative term of the fractional differential equation which includes the fractional time component parameters for obtaining such time dimensional measurability, has been interpreted in Caputo sense and the analytical solution of such equation has been determined with the aid of Laplace transformation. With the obtained solution, the time dimensional measurability aware fractional derivative based circuit responses to various inputs have been determined, the circuit fractional time constant and other crucial time parameters has been determined and thetemporal behaviour of the circuit has been analysed in the fractional domain. The loci of the pole on W-plane has been determined and the stability analysis has also been presented. Moreover, we also mathematically proof that the OTA-C filter in the fractional domain can be electronically realized with time dimensionalmeasurability awareness by using the state of the art fractional capacitor.

New Method of State-Space Formulation for Degenerate Circuit and Coupling Circuit

The state-space formulation overcomes many limitations of traditional differential equation approach and is utilized as alternative to many traditional approaches in the modern electrical field. This paper proposes a new method of finding the state equation for degenerate circuit and coupling circuit that have not been systematically solved now. This paper also introduces some sound improvements to solve complicated dependent-source circuits. Four comparative examples are demonstrated to show the significant merits that our method owns over the traditional approaches.

Prescribing Response Strategies to Manage Customer Opinions: A Stochastic Differential Equation Approach

Today, the reputation of a firm is profoundly influenced by user opinions expressed in online consumer reviews. Managing these opinions is, therefore, critical for the success of firms. We study the problem of devising an appropriate opinion management strategy (or response strategy) for a firm to respond to online customer reviews. To unravel the underlying mechanics of the problem, we develop a stochastic differential equation model that describes the evolution of review ratings over time for a given response strategy employed by the firm. This model is validated using data on online customer reviews and firm responses from two of the world’s largest online travel agents. When pitted against popular benchmark models, such as autoregressive moving average, generalized autoregressive conditional heteroscedasticity, moving average, exponential smoothing, and naive method, our approach not only achieves comparable (often better) predictive performance, it is also able to incorporate the response strategy into the data-generation process underlying the review ratings. Our approach, therefore, is not just predictive, but, more importantly, one that can be used in a prescriptive sense, namely to prescribe a response strategy that controls review ratings in a desired manner. We operationalize the theoretical response strategy in our stochastic model to an operational prescription that a firm can implement and show the applicability of our approach for different business objectives, such as mean control, mean-variance control, and service-level control. Finally, we demonstrate the flexibility of the stochastic differential equation model by extending it to encompass multiple state variables. The online appendices are available at https://doi.org/10.1287/isre.2018.0805 .

Fractional differential equation approach for convex optimization with convergence rate analysis

A fractional differential equation (FDE) based algorithm for convex optimization is presented in this paper, which generalizes ordinary differential equation (ODE) based algorithm by providing an additional tunable parameter $$\alpha \in (0,1]$$. The convergence of the algorithm is analyzed. For the strongly convex case, the algorithm achieves at least the Mittag-Leffler convergence, while for the general case, the algorithm achieves at least an $$O(1/t^\alpha )$$ convergence rate. Numerical simulations show that the FDE based algorithm may have faster or slower convergence speed than the ODE based one, depending on specific problems.

On the theoretical basis of memory-free approaches for fractional differential equations

PurposeA nonclassical method, usually called memory-free approach, has shown promising potential to release arithmetic complexity and meets high memory-storage requirements in solving fractional differential equations. Though many successful applications indicate the validity and effectiveness of memory-free methods, it has been much less understood in the rigorous theoretical basis. This study aims to focus on the theoretical basis of the memory-free Yuan–Agrawal (YA) method [Journal of Vibration and Acoustics 124 (2002), pp. 321-324].Design/methodology/approachMathematically, the YA method is based on the validity of two fundamental procedures. The first is to reverse the integration order of an improper quadrature deduced from the Caputo-type fractional derivative. And, the second concerns the passage to the limit under the integral sign of the improper quadrature.FindingsThough it suffices to verify the integration order reversibility, the uniform convergence of the improper integral is proved to be false. Alternatively, this paper proves that the integration order can still be reversed, as the target solution can be expanded as Taylor series on [0, ∞). Once the integration order is reversed, the paper presents a sufficient condition for the passage to the limit under the integral sign such that the target solution is continuous on [0, ∞). Both positive and counter examples are presented to illustrate and validate the theoretical analysis results.Originality/valueThis study presents some useful results for the real performance for the YA and some similar memory-free approaches. In addition, it opens a theoretical question on sufficient and necessary conditions, if any, for the validity of memory-free approaches.