Approximate Closed-form Solution Research Articles

Approximate Closed-Form Solutions for Pricing Zero-Coupon Bonds in the Zero Lower Bound Framework

Approximate Closed-Form Solutions for Pricing Zero-Coupon Bonds in the Zero Lower Bound Framework

Ergodic Rate Analysis of Simultaneous Transmitting and Reflecting Reconfigurable Intelligent Surface-Assisted Rate-Splitting Multiple Access Systems Based on Discrete Phase Shifts.

In this paper, we combine simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) with rate-splitting multiple access (RSMA) technology and investigate the ergodic rate performance of an STAR-assisted RSMA system. Considering the discrete phase shifts of the STAR-RIS in practice, the downlink performance of STAR-RIS-assisted RSMA with discrete phase shifts is compared to that with continuous phase shifts. Firstly, the cumulative distribution function of signal-to-interference-plus-noise ratio (SINR) of users is analyzed. Then, the total ergodic rate of the system and its approximate closed-form solution are, respectively, derived based on the cumulative distribution function of users. The simulation results validate the effectiveness of the theoretical analysis, showing good agreement between the derived theoretical ergodic rate and the corresponding simulations. Although the system performance with discrete phase shifts is inferior to that with continuous phase shifts due to quantization errors, the performance of the continuous phase shift system is well approximated when the quantization bit of the phase shift system reaches 3 in the simulations. Additionally, the impact of the number of STAR-RIS elements on the system's performance is analyzed.

Pricing Asian options under the mixed fractional Brownian motion with jumps

The mixed fractional Brownian motion (mfBm) has gained popularity in finance because it can effectively model long-range dependence, self-similarity, and is arbitrage-free. This paper focuses on mfBm with jumps modeled by the Poisson process and derives an analytical formula for valuing geometric Asian options. Additionally, approximate closed-form solutions for pricing arithmetic Asian options and arithmetic Asian power options are obtained. Numerical examples are provided to demonstrate the accuracy of these formulas, which rely on a convenient approximation of the option strike price. The proposed approximation demonstrates significantly higher computational efficiency compared to Monte Carlo simulation.

A new method for determining crack propagation

This paper presents a new method for obtaining approximate closed-form solutions for crack propagation problems. In general, it is not possible to obtain explicit solutions of the parameters and elementary functions of these problems, due to their terms complexity, present in their definitions. In this way, we used numerical methods to obtain approximate numerical solutions. The method presented in this article provides a proposal, which, when defined by differential equations, is applicable and extensible to other problems. The approximate solutions are obtained as an exact solution of a modified initial value problem. The performance is evaluated in relation to problems with known exact solution.

Semi-Analytical Closed-Form Solutions for Dynamical Ro¨ssler-Type System

Mathematical models and numerical simulations are necessary to understand the functions of biological rhythms, to comprehend the transition from simple to complex behavior and to delineate the conditions under which they arise. The aim of this work is to investigate the Ro¨ssler-type system. This system could be proposed as a theoretical model for biological rhythms, generalizing this formula for chaotic behavior. It is assumed that the Ro¨ssler-type system has a Hamilton–Poisson realization. To semi-analytically solve this system, a Bratu-type equation was explored. The approximate closed-form solutions are obtained using the Optimal Parametric Iteration Method (OPIM) using only one iteration. The advantages of this analytical procedure are reflected through a comparison between the analytical and corresponding numerical results. The obtained results are in a good agreement with the numerical results, and they highlight that our procedure is effective, accurate and usefully for implementation in applicationssuch as an oscillator with cubic and harmonic restoring forces, the Thomas–Fermi equation and the Lotka–Voltera model with three species.

Approximate closed-form solutions for free vibration of circular arches with discrete lateral braces

Discrete lateral braces are commonly employed to enhance the out-of-plane stability of arches. However, these braces generate coupling effects among vibration modes, which significantly impact the vibration characteristics of the arches. This paper presents an analytical solution for the free vibration analysis of two-hinged arches with equally-spaced lateral translational and/or rotational braces. A simplified method of evaluating the fundamental frequency for the out-of-plane vibration of arches with equally-spaced lateral braces is introduced. The method relies on the conventional approach of representing arch braces through elastic springs, but here the out-of-plane deflection of the braced arch is constructed by adding related coupled sinusoidal functions to that of the unbraced arch, satisfying both the boundary and kinematic conditions. Subsequently, the fundamental frequency and threshold bracing stiffness or stiffness requirement are derived using the Rayleigh–Ritz method. Finite element analysis (FEA) is applied to investigate the flexural-torsional vibration mode and fundamental frequency of arches with discrete braces, and the results from FEA closely match the theoretical predictions. The effects of number of lateral translational and/or rotational brace, out-of-plane slenderness, included angle, bracing stiffness and bracing position on the fundamental frequency are comprehensively analyzed. It is found that when the fundamental frequency of the braced arch increases due to changes in bracing stiffness, the mode shape displays more intricate behavior, rather than following a straightforward low-to-high mode growth pattern. Furthermore, it is observed that in some cases, the addition of rotational bracing stiffness can markedly enhance the maximum fundamental frequency of the arch with both translational and rotational braces.

The humanitarian vehicle routing problem with non-routineness of trips

In this study, we address a pressing yet underexplored problem in humanitarian logistics (HLs), which is enhancing the safety and security of humanitarian personnel in conflict zones, through the introduction of non-routineness of trips in vehicle routing models. This research is critically justified, as evidenced by the alarming statistics from 2020, where over 100 road ambushes were reported, affecting over 200 humanitarian workers through kidnappings, injuries, and fatalities. Moving beyond traditional approaches that primarily focus on increasing convoy sizes or incorporating ambush probabilities, our research poses a pivotal question: How can existing knowledge from security journals and manuals be effectively leveraged to address this problem? In response, we have developed a groundbreaking vehicle routing model that ensures variability in routes and travel times, significantly reducing the predictability of humanitarian convoys and enhancing their security. Our approach also integrates additional security measures, including avoiding night travel and implementing convoy camping strategies, to further mitigate risks. A central innovation of our study is the creation of a first-of-its-kind index, designed to quantify the routineness of trips – a critical yet previously overlooked metric in HLs. The efficacy of our model is demonstrated through a comprehensive case study in South Sudan, a region afflicted by ongoing civil unrest, underscoring the real-world applicability and urgency of our research. Additionally, we provide an approximate closed-form solution for the aid allocation subproblem, optimizing for both fairness and effectiveness. This research marks a significant leap forward in logistics optimization, especially within HLs in conflict zones, addressing a vital gap in the literature and offering innovative perspectives and practical solutions to complex security problems faced by humanitarian logisticians.

Nonlinear consolidation of arbitrary layered soil with continuous drainage boundary: An approximate closed-form solution

Based on the double nonlinear consolidation constitutive associated with the compression and permeability coefficients, presented by Mesri and Rokhsar (1974), this paper derives an approximate closed-form solution for the one-dimensional nonlinear consolidation of the arbitrary layered soils incorporating the continuous drainage boundary condition. The approximate closed-form solution is obtained by the homogenization of the boundary conditions and eigenfunction method. A model test is conducted to justify the rationality of the approximation and the continuous drainage condition utilized in this study. The calculated results are also compared with those acquired from the simplified analytical solution and the finite difference method. A parametric study is conducted to investigate the influence of various parameters on the consolidation process. The most significant finding is that the influence of Nq appears to be completely different for the cases when Cc/Ck>1 and Cc/Ck<1. When Cc/Ck>1, the increase of Nq shows an adverse influence on the consolidation, whereas the influence becomes positive when Cc/Ck<1. The approximate solution derived herein offers a rigorous analytical approach for the double nonlinear consolidation problems of arbitrary layered soils, providing an effective benchmark for comparison and verification of future sophisticated numerical approaches.

Minimum-Energy Transfer Optimization between Near-Circular Orbits Using an Approximate Closed-Form Solution

This paper explores the use of the averaging method in the optimal control problem related to the multirevolution orbital transfer of a spacecraft with low-thrust capabilities. The regularized equations of motion are expressed using modified equinoctial elements with the eccentric longitude as a fast variable. The control function is represented as a Fourier series relative to the eccentric longitude. The classical averaging technique’s usage results in the averaged trajectory depending only on a limited number of optimization parameters. Moreover, when transferring between near-circular orbits, the averaged motion can be estimated using analytical formulas. As such, the optimal multiorbit flight problem is simplified to nonlinear programming with fewer parameters, thereby accelerating the optimal solution’s derivation. Two practical examples illustrate the technique’s application: orbital transfer near the geostationary orbit and circular orbit raising maneuver. The solutions derived are compared with Pontryagin extremals.

Approximate Closed-Form Solutions for a Class of 3D Dynamical Systems Involving a Hamilton–Poisson Part

The goal of this paper is to build some approximate closed-form solutions for a class of dynamical systems involving a Hamilton–Poisson part. The chaotic behaviors are neglected. These solutions are obtained by means of a new version of the optimal parametric iteration method (OPIM), namely, the modified optimal parametric iteration method (mOPIM). The effect of the physical parameters is investigated. The Hamilton–Poisson part of the dynamical systems is reduced to a second-order nonlinear differential equation, which is analytically solved by the mOPIM procedure. A comparison between the approximate analytical solution obtained with mOPIM, the analytical solution obtained with the iterative method, and the corresponding numerical solution is presented. The mOPIM technique has more advantages, such as the convergence control (in the sense that the residual functions are smaller than 1), the efficiency, the writing of the solutions in an effective form, and the nonexistence of small parameters. The accuracy of the analytical and corresponding numerical results is illustrated by graphical and tabular representations. The same procedure could be successfully applied to more dynamical systems.

A non-singular fractional-order logistic growth model with multi-scaling effects to analyze and forecast population growth in Bangladesh

This paper is primarily concerned with data analysis employing the nonlinear least squares curve fitting method and the mathematical prediction of future population growth in Bangladesh. Available actual and adjusted census data (1974–2022) of the Bangladesh population were applied in the well-known autonomous logistic population growth model and found that all data sets of the logistic (exact), Atangana-Baleanu-Caputo (ABC) fractional-order derivative approach, and logistic multi-scaling approximation fit with good agreement. Again, the existence and uniqueness of the solution for fractional-order and Hyers-Ulam stability have been studied. Generally, the growth rate and maximum environmental support of the population of any country slowly fluctuate with time. Including an approximate closed-form solution in this analysis confers several advantages in assessing population models for single species. Prior studies predominantly employed constant growth rates and carrying capacity, neglecting the investigation of fractional-order methods. Thus, the current study fills a crucial gap in the literature by introducing a more formal approach to analyzing population dynamics. Therefore, we bank on the findings of this article to contribute to accurate population forecasting and planning, national development, and national progress.

Learning solution of a bond-based linear peridynamic model using LS-SVR method

In this paper, we develop an efficient least squares support vector regression (LS-SVR) method for a steady-state bond-based linear Peridynamic (PD) model in two space dimensions. To minimize a residual function associated with PD model, we introduce some dual variables to rewrite the optimization problem to a linear system and obtain a closed form approximate solution of the considered problem. The method is suitable to solve PD problem involving singular kernel, irregular geometrical domains. Numerical experiments are provided to show the accuracy and efficiency of the proposed method.

An Approximate Closed-Form Solution for Mass Transfer Flow Due to a Rotating Rough and Porous Disk

An Approximate Closed-Form Solution for Mass Transfer Flow Due to a Rotating Rough and Porous Disk

Uniform space-charge-limited current for a two-dimensional planar emitter with nonzero monoenergetic initial velocity

While characterizing space-charge-limited current (SCLC) is important for numerous applications, no analytical solutions for SCLC with monoenergetic initial velocity exist for two-dimensional (2D) geometries. Here, we derive approximate closed-form solutions for uniform SCLC with monoenergetic emission of electrons in a 2D planar diode, where emission is restricted to a long patch of width W for electrodes separated by a distance D. We also derive a semiempirical approach for estimating the SCLC for these cases by treating the geometric and velocity correction factors as multiplicative corrections to the SCLC for a one-dimensional vacuum diode given by the Child–Langmuir (CL) law. We show that the SCLC for a finite patch with nonzero velocity can exceed the CL law by three orders of magnitude. The theoretically calculated SCLCs for various emission widths and initial velocities in the 2D diode agree well with particle-in-cell simulations using the over-injection method in XOOPIC; they agree with the semiempirical relationship for lower initial velocities. In the limit of high initial velocity, the geometry and velocity corrections to the CL law cannot be decoupled, invalidating the assumption of the semiempirical approach and causing it to diverge from the theoretical solution and XOOPIC simulations. These results provide valuable estimates for determining the onset of virtual cathode formation for photocathodes and thermionic cathodes, which operate in the over-injection regime to avoid beam quality degradation.

A computationally efficient sequential convex programming using Chebyshev collocation method

This paper aims to develop a computationally efficient sequential convex programming algorithm for a class of nonlinear systems. A Chebyshev collocation discretization (CCD) technique is proposed that transforms the continuous convex optimization problem into a finite-dimensional discrete optimization problem. The CCD technique uses Chebyshev polynomials to construct a closed-form approximate solution to the convex dynamic equation. Moreover, it establishes a linear mapping between the control inputs and the system states. This enables the constraints of the dynamics equations to be externalized from the optimization problem and computed using efficient linear equation solving algorithms. On the one hand, this strategy significantly reduces the number of constraint equations and optimization variables, improving the speed of the optimizer. On the other hand, it preserves the path constraints on the system states in the optimization problem. A numerical simulation example of the perching maneuver for a fixed-wing unmanned aerial vehicle is presented to validate the efficiency of the algorithm. The result shows significant improvements in both speed and accuracy compared to the Euler discretization, and greatly improves the solution speed while maintaining almost the same solution accuracy compared to the Gauss pseudospectral optimization method.

Resource allocation and BER performance analysis of NOMA based cooperative networks

Resource sharing and management can significantly improve the performance and spectral efficiency of wireless systems. In power domain non-orthogonal multiple access (NOMA) systems, users share a common bandwidth for simultaneous data transmission and therfore, the spectral efficiency of the system will be improved at the expense of increased complexity. Provided proper power allocation, it is possible to detect symbols at receivers. In this paper, the downlink of a cellular system is considered where a base station (BS) communicates with two users using NOMA with the assistance of a decode-and-forward (DF) relay. In the proposed scheme, receivers combine received signals from direct and cooperative links, and decode symbols by employing successive interference cancellation (SIC). The bit error rate (BER) performance of the proposed scheme is analyzed and the optimal power allocation is also derived through a Min–Max optimization, and a closed form approximate solution is proposed for binary phase shift keying (BPSK) modulation. Furthermore, it is proved that the proposed receiver achieves full diversity order for all users for proper choice of power allocation. Simulation results also corroborate BER and diversity analysis.

Analytical investigation of the coupled fractional models for immersed spheres and oscillatory pendulums

In this research, coupled fractional models for immersed spheres and oscillatory pendulums have been proposed. We deploy the Laplace transform method together with the negative binomial formula to analytically investigate the dynamics of the systems. Approximate closed-form solutions are successfully revealed with the help of the convolution theorem. Additionally, we graphically illustrate the variational effects of the fractional-orders and the coupling parameters on the paired fields.

Service staffing with delay probabilities

The square-root staffing model generates the number of servers s by summing the “offered load” k (minimum staffing level) and the “safety staff” (square root of k multiplied by the parameter β reflecting the desired service level). We treat the service level as one minus the delay probability (Erlang-C). Due to its iterative nature, we present a model for the Erlang-C which allows us to construct an approximate closed-form solution for the number of servers at the desired service level.

Fast geolocation solution for bistatic interferometric synthetic aperture radar configuration of inclined geosynchronous transmitter with low earth orbit receivers

AbstractSpaceborne interferometric synthetic aperture radar (InSAR) technology is an effective method to obtain digital elevation model (DEM) data. The bistatic InSAR configuration of the inclined‐geosynchronous (InGEO) transmitter with low earth orbit (LEO) receivers (InGEO‐LEO) is a novel InSAR system to acquire terrain information. This novel system is characterised with high resolution, wide swath and timeliness, but the Newton iterative method is time‐consuming to solve the bistatic InSAR equations for fast DEM generation. For the conventional LEO bistatic InSAR system, the closed‐form solution is an effective method to improve the efficiency of solving InSAR equations, which is invalid in the InGEO‐LEO InSAR system because of the significant geometry difference caused by the orbits of InGEO transmitter and LEO receivers. To address this issue, we analyse the bistatic InGEO‐LEO geometry in detail and exploit the bistatic InSAR equations to propose an approximate closed‐form solution (ACS) for the novel system. Compared with the general Newton iterative method, the ACS significantly improves the efficiency of geolocation for the bistatic InGEO‐LEO InSAR system with high precision. Simulation experiments are carried out to verify the effectiveness and superiority of ACS.

Optimization-Based Block Coordinate Gradient Coding for Mitigating Partial Stragglers in Distributed Learning

Gradient coding schemes effectively mitigate full stragglers in distributed learning by introducing identical redundancy in coded local partial derivatives corresponding to all model parameters. However, they are no longer effective for partial stragglers as they cannot utilize incomplete computation results from partial stragglers. This paper aims to design a new gradient coding scheme for mitigating partial stragglers in distributed learning. Specifically, we consider a distributed system consisting of one master and N workers, characterized by a general partial straggler model and focuses on solving a general large-scale machine learning problem with L model parameters using gradient coding. First, we propose a coordinate gradient coding scheme with L coding parameters representing L possibly different diversities for the L coordinates, which generates most gradient coding schemes. Then, we consider the minimization of the expected overall runtime and the maximization of the completion probability with respect to the L coding parameters for coordinates, which are challenging discrete optimization problems. To reduce computational complexity, we first transform each to an equivalent but much simpler discrete problem with N\llL variables representing the partition of the L coordinates into N blocks, each with identical redundancy. This indicates an equivalent but more easily implemented block coordinate gradient coding scheme with N coding parameters for blocks. Then, we adopt continuous relaxation to further reduce computational complexity. For the resulting minimization of expected overall runtime, we develop an iterative algorithm of computational complexity O(N^2) to obtain an optimal solution and derive two closed-form approximate solutions both with computational complexity O(N). For the resultant maximization of the completion probability, we develop an iterative algorithm of...