Explicit Solutions Research Articles

ALM for insurers with multiple underwriting lines and portfolio constraints: a Lagrangian duality approach

We investigate how an insurance firm can optimally allocate its assets to back up liabilities from multiple non-life business lines. The insurance risks are modeled by a multidimensional jump-diffusion process that accounts for simultaneous claims in different insurance lines with policy limits. We use Lagrangian convex duality techniques to derive optimal investment-underwriting strategies that maximize the expected utility from dividends and final wealth over a finite horizon. We examine how risk aversion, prudence, portfolio constraints, and multivariate insurance risk affect the firm’s earnings retention. We obtain explicit solutions for optimal strategies under constant relative risk aversion preferences. Finally, we illustrate our results with numerical examples and show the impact of co-integration for asset-liability management with multiple sources of insurance risk.

Simulations of exact explicit solutions of simplified modified form of Camassa–Holm equation

Simulations of exact explicit solutions of simplified modified form of Camassa–Holm equation

Representation of solutions to fuzzy linear fractional differential equations with a piecewise constant argument

Abstract This study marks the first exploration of fuzzy linear fractional differential equations with a piecewise constant argument (FLFDEs-PCA), incorporating the concept of Caputo's type gH-differentiability with the order $\alpha \in (0,1]$. Such problems are noteworthy as they represent hybrid systems, blending the characteristics of continuous and discrete dynamical systems and integrating aspects from both differential and difference equations. The primary objective of this research is to establish a standardized framework for deriving explicit solution formulas for FLFDEs-PCA under various scenarios. Additionally, illustrative examples are provided to demonstrate the practical implications of our theoretical findings.

A simple, accurate, and explicit form of the Green–Ampt model to estimate infiltration, sorptivity, and hydraulic conductivity

AbstractFinding an explicit solution to the widely used Green–Ampt (G–A) one‐dimensional infiltration model has been subject of efforts for more than half a century. We derived an explicit semiempirical approach that combines accuracy with simplicity, a concept that has been generally neglected in previous studies. The equation is , with F (L), Ks (L/T), S (L/T0.5) and t (T) being cumulative infiltration, saturated hydraulic conductivity, sorptivity, and time, respectively. Relative errors (ɛ) by the application of this equation generally do not exceed ±0.3% in most applied infiltration problems faced by water resources engineering today. We show both numerically and mathematically that │ɛ│> 0.3% could only occur if Kst/F > 0.904, a criterion that could apply to sand and loamy sand soils (i.e., coarse texture) and if they experience infiltration rates for over 6 h and 19 h, respectively. Hence, we also derived a simple linear adjustment in the model as Fadj ≅ 0.9796 F + 0.335 S2/Ks to address these longer infiltration rates, and to assure that ɛ remains within the expected ±0.3% range of uncertainty. A linearized regression technique was also developed to accurately estimate S and Ks when the G–A model is used. We numerically demonstrated that our fitting method could be used even when the G–A approach is less valid (diffusive soils), provided that the actual value of the capillary length (λ) is initially known. An added benefit of our approach is that by setting λ equal to 1/3 and 2/3, it can significantly limit the range of initializing, unknown, a priori values of S and Ks, as these two parameters are estimated through the inverse solution of implicit infiltration models. Due to the model's simplicity and accuracy, our solution should find application among hydrologists, natural resource scientists, and engineers who wish to easily derive accurate estimations from the G–A infiltration approach and/or estimate sorptivity and hydraulic conductivity without encountering divergence problems.

Progressive damage analysis of embedment of metal washer into timber

ABSTRACT Metal washers are widely used in joints of timber structures. Thick washers embed fully into timber, whereas thin washers partially embed into timber due to their bending deformation. For thick washers, physical models have been proposed to estimate the stiffness and strength of timber. However, for thin washers, these estimation methods are insufficient. The objective of this study is to propose a finite-element analysis (FEA)-based approach to predict the stiffness and strength of wood during washer embedding, as well as forecast plasticity and damage behaviours of timber, regardless of the washer size. The FEA used a constitutive model with plastic and damage criteria for timber. In the FEA, an explicit dynamic solution method was used to prevent solution divergence. The FEA reproduced the embedment load–displacement relation and damage behaviours of three species. In addition, the FEA predicted the initial stiffness, second stiffness, and yield load with error margins of 12%, 16%, and 7%, respectively. The influences of the loading speed and mesh fineness on the FEA were investigated numerically to validate the results. In addition, parametric studies on contact stiffness and damage constants were conducted to understand the influences of material constants on the embedment behaviours of FE models.

Dynamically driven correlations in elastic net models reveal sequence of events and causality in proteins.

An explicit analytic solution is given for the Langevin equation applied to the Gaussian Network Model of a protein subjected to both a random and a deterministic periodic force. Synchronous and asynchronous components of time correlation functions are derived and an expression for phase differences in the time correlations of residue pairs is obtained. The synchronous component enables the determination of dynamic communities within the protein structure. The asynchronous component reveals causality, where the time correlation function between residues i and j differs depending on whether i is observed before j or vice versa, resulting in directional information flow. Driver and driven residues in the allosteric process of cyclophilin A and human NAD-dependent isocitrate dehydrogenase are determined by a perturbation-scanning technique. Factors affecting phase differences between fluctuations of residues, such as network topology, connectivity, and residue centrality, are identified. Within the constraints of the isotropic Gaussian Network Model, our results show that asynchronicity increases with viscosity and distance between residues, decreases with increasing connectivity, and decreases with increasing levels of eigenvector centrality.

EUROPEAN OPTION PRICING IN THE GENERALIZED MIXED WEIGHTED FRACTIONAL BROWNIAN MOTION

In order to describe the self-similarity and long-range dependence of financial asset prices, this paper adopts a new fractional-type process, i.e, the generalized mixed weighted fractional Brownian motion to describe the dynamic change process of risky asset prices. A European option pricing model driven by the generalized mixed weighted fractional Brownian motion is constructed, and explicit solutions to the pricing formulas of European call options and European put options are derived by using the arbitrage-free pricing theory. Finally, through numerical simulation, the influence of the parameter on the option price is analyzed.

Correlated Equilibria for Mean Field Games with Progressive Strategies

In a discrete space and time framework, we study the mean field game limit for a class of symmetric N-player games based on the notion of correlated equilibrium. We give a definition of correlated solution that allows us to construct approximate N-player correlated equilibria that are robust with respect to progressive deviations. We illustrate our definition by way of an example with explicit solutions. Funding: O. Bonesini acknowledges financial support from Engineering and Physical Sciences Research Council [Grant EP/T032146/1]. M. Fischer acknowledges partial support through the University of Padua [Research Project BIRD229791 “Stochastic mean field control and the Schrödinger problem”].

Robust time-consistent strategies of DC pension plans with the return of premiums clauses under inflation

. This article investigates the robust time-consistent investment strategy for a DC pension plan with model uncertainty under the mean-variance optimization objective. For the avoidance of reductions in investment returns due to inflation risk and premature death of members, an inflation-indexed bond is introduced into the financial market and a return of premium clause is incorporated into the model. By establishing extended Hamilton-Jacobi-Bellman (HJB) equations, the explicit solutions of both the robust precommitment strategy and the robust time-consistent strategy are presented by virtue of the robust optimal control theory and the stochastic dynamic programming approach. Furthermore, two degradation cases are derived in detail. Finally, a numerical example demonstrates the analysis of the obtained results.

Dynamics of Cosmological Scalar Fields Revisited

This paper reviews the dynamics of a single isotropic and homogeneous scalar field φ(t) in the context of cosmological models. A non-standard approach to the solution of the Einstein–Klein–Gordon equations is described which uses the scalar field as the evolution parameter for cosmic dynamics. General conclusions about the qualitative behaviour of the solutions can be drawn, and examples of how to obtain explicit solutions for some cosmological models of interest are given. For arbitrary potentials, analytical results can be obtained from the slow-roll approximation by using a series expansion for the Hubble parameter H[φ], from which a quantitative estimate for the number of e-folds of expansion is obtained. This approach is illustrated with the examples of quadratic potentials and hilltop models, with special consideration of Higgs-type potentials. The GUT-scale is shown to come out of such a model quite naturally. Finally, it is discussed how to find scalar potentials giving rise to a predetermined scalar-field behaviour and the associated evolution of the scale factor.

A population balance model for the kinetics of covalent organic framework synthesis.

This study presents a population balance model for the kinetics of nucleation and growth in covalent organic framework (COF) synthesis. The model incorporates second-order nucleation and first-order growth rates, consistent with proposals in the literature. Despite having non-linear terms, an implicit analytic solution is derived and then converted to explicit solutions for the monomer concentration and size distribution of COF flakes as a function of time. For experimental definitions of the induction time and the initial growth rate based on yield (y) vs time (t) curves, the model predicts power-law relationships: tind=0.409kN-1/3kG-2/3cA0-1 and dy/dtmax=0.965kN1/3kG2/3cA0, respectively. We discuss the implications for the interpretation of Arrhenius plots. We also discuss key discrepancies with experiments, including the predicted attainment of 100% yield instead of 30%-40% as observed and the value of the yield at the inflection point in the yield vs time curve. We suggest extensions to the model, including nucleation and growth kinetics with equilibrium solubility limitations and two-dimensional nucleation for the formation of multilayer COF particles.

Dynamic modelling of supplemental cooling systems with vapor compression refrigeration units in civil aircrafts

Newly developed airliners have adopted supplemental cooling systems (SCS) to provide for non-cabin cooling requirements. In the present study, dynamic component models for compressor and phase change heat exchangers were developed with dynamic decoupling and explicit solution methods, and pressure zone separation and solution methods are developed based to obtain dynamic pressures as key values. A dynamic VCRU subsystem model is established, and the result under single-unit startup condition shows pressure deviations less than 0.16 %, with a simulation time ratio of 5.65 %, exhibiting good overall performance at both precision and speed. A dynamic SCS system model is developed, and the simulation of a 9000 s flight took 1973.3 s to complete. Result in the cruise phase shows the condensation and evaporation temperatures are in the ranges of 28.39–31.33 °C and −17.93 to −10.71 °C, respectively, meeting their respective designed ranges of 30 ± 5 °C and −15 ± 5 °C. These results reflects good consistency of the model with actual system characteristics. Control optimization is conducted based on the developed model to eliminate temperature fluctuation. The optimization reduced the maximum temperature deviation from 0.4004 K to 0.0019 K after system starting process, proving the feasibility of control method optimization based on the developed dynamic simulation model.

Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results.

Novel solution strategies for multiparametric nonlinear optimization problems with convex objective function and linear constraints

AbstractThis paper expands the multiparametric quadratic programming (mp-QP) framework presented in Narciso et al. (Comput Chem Eng 164:107882, 2022. https://doi.org/10.1016/j.compchemeng.2022.107882) to the more general multiparametric nonlinear programming (mp-NLP) case. First, the vector of parameters in mp-NLP problems is recast so that a unique transformed parameter is implicitly assigned to each of the inequality constraints. Maps of critical regions in this transformed space of parameters feature a set of 1-dimensional parametric edges (two per inequality constraint), which then greatly facilitate solution calculation. In the mp-NLP case, however, parametric edges define nonlinear semi-infinite lines; this requires an adaptation to the mp-QP algorithm (deals with linear parametric edges only), to enable a suitable calculation path to the more general nonlinear case. Three routes are proposed to mp-NLPs: the first route delivers solutions in compact form (same format as in mp-QP) using a single reference point per edge; the second route delivers explicit solutions using a hybrid approach for critical region construction, where all active sets not detected in the parameters space are excluded from the solution (equivalent to first route concerning accuracy); the third route builds on the initial explicit solution and further partitions the parameters space until all solution fragments satisfy an error check. Five algorithms were coded for these routes, and tested in a large range of mp-NLP problems. These strategies enable significant improvements in terms of solution accuracy, algorithm efficiency, and interpretability when compared to the state-of-the-art mp-NLP algorithms.

Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equations

This paper aims to develop a direct approach, namely, the Cauchy matrix approach, to non-isospectral integrable systems. In the Cauchy matrix approach, the Sylvester equation plays a central role, which defines a dressed Cauchy matrix to provide τ functions for the investigated equations. In this paper, using the Cauchy matrix approach, we derive three non-isospectral nonlinear Schrödinger equations and their explicit solutions. These equations are generically related to the time-dependent spectral parameter in the Zakharov–Shabat–Ablowitz–Kaup–Newell–Segur spectral problem. Their solutions are obtained from the solutions of unreduced non-isospectral nonlinear Schrödinger equations through complex reduction. These solutions are analyzed and illustrated to show the non-isospectral effects in dynamics of solitons.

Relative controllability for conformable impulsive delay differential equations

Abstract In this paper, we mainly study a class of conformable impulsive delay differential equations (CIDDEs). We first define a conformable impulsive delayed matrix function, and construct an explicit solution for linear CIDDEs by virtue of variation of constants method. Subsequently, based on impulsive delayed Grammian matrix, we study the relative controllability for the addressed linear equations. Moreover, with the help of Krasnoselskii’s fixed point theorem, relative controllability for the considered semilinear equations is proposed. Finally, two examples with numerical simulations are given to illustrate the main results.

The Solution of Caputo Fractional Partial Differential Equations By Sumudu Transform

In this paper, we obtained an explicit solutions of the fractional diffusion-wave equations involving partial Caputo fractional derivative by using Sumudu transform. The solutions of the fractional diffusion-wave equations obtained in terms of Mittag-Leffler function and generalized Wright function.Some illustrative examples are also given.

Harmonic Flow of Quaternion-Kähler Structures

We formulate the gradient Dirichlet flow of Sp(2)Sp(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Sp}(2)\ extrm{Sp}(1)$$\\end{document}-structures on 8-manifolds, as the first systematic study of a geometric quaternion-Kähler (QK) flow. Its critical condition of harmonicity is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document} case. We also draw several comparisons with Spin(7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Spin}(7)$$\\end{document}-structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}-regularity.We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold SU(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SU}(3)$$\\end{document}. We also study compactness and the formation of singularities.

Analytical solution of the Poiseuille flow of a De Kee viscoplastic fluid

We provide an explicit analytical solution of the planar Poiseuille flow of a viscoplastic fluid governed by the constitutive equation proposed by De Kee and Turcotte (1980). Formulae for the velocity and the flow rate are derived, making use of the Lambert W function. It is shown that a solution does not always exist because the flow curve is bounded from above and hence the rheological model can accommodate stresses only up to a certain limit. In fact, the flow curve reaches a peak at a critical shear rate, beyond which it exhibits a negative slope, giving rise to unstable solutions.

Iterated Robin problem, the case of various parameters

ABSTRACT Prescribing Robin boundary conditions for the iterated Poisson equation ( ∂ z ∂ z ¯ ) n w = f leads to Robin-n problems. Extending previous results by allowing an independent choice for the parameters α k , β k for every iteration k , 1 ≤ k ≤ n , leads to explicit integral representations depending on the data of the Robin-n problem. Parting from these integral representation explicit solutions with their respective solvability conditions are derived. For the unit disc of the complex plane, the Robin functions for n = 2 and 3 are explicitly constructed.